Static large deviations for a reaction–diffusion model

Abstract

We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.

1 Introduction

Nonequilibrium thermodynamics has aroused a lot of interest in the last decades. Since the beginning of the 2000’s, much attention has been devoted to the investigation of nonequilibrium stationary states which describe a steady flow through a system, [6, 18] and references therein.

Over the last years, a general approach to examine nonequilibrium stationary states, called the Macroscopic Fluctuation Theory, has been developed based on a dynamical large deviations principle for the empirical current [2, 4, 8]. Among the major achievements of the MFT was the deduction of a time-independent variational formula for the quasi-potential, the functional obtained by minimizing the dynamical large deviations rate functional over all trajectories which start from the stationary density profile and produces a fixed fluctuation [3, 20], and the proof that the quasi-potential is Gâteaux differentiable at some density profile if and only if the time-dependent variational formula which defines the quasi-potential has a unique minimizer [5].

At the same time, adapting to the infinite-dimensional setting the strategy proposed by Freidlin and Wentzell [24] for stochastic perturbations of finite-dimensional dynamical systems, Bodineau and Giacomin [10] and Farfán [21] proved a large deviations principle for the empirical measure under the nonequilibrium stationary state for conservative dynamics in contact with reservoirs in which the large deviations rate functional is given by the quasi-potential.

We consider in this article the stochastic evolution obtained by superposing a speeded-up symmetric simple exclusion process with a spin-flip dynamics on a one-dimensional interval with periodic boundary conditions. The hydrodynamic equation induced by the microscopic dynamics, the partial differential equation which describes the macroscopic evolution of the density, is given by a reaction–diffusion equation of type

$$\begin{aligned} \partial _t \rho \,=\, (1/2)\, \varDelta \rho + B(\rho ) - D(\rho ), \end{aligned}$$

(1.1)

where $\varDelta$ represents the Laplacian and where B and D are non-negative polynomials.

We investigate the static large deviations of the empirical measure under the stationary state. In contrast with the previous dynamics [10, 21], in which the hydrodynamic equation has a unique stationary solution which is a global attractor of the dynamical system generated by the PDE, in reaction–diffusion models, for appropriate choices of the functions B, D, the hydrodynamic equation possesses more than one stationary solution.

The existence of multiple stationary solutions to the PDE (1.1) raises new problems and new questions. For instance, the conjecture that the stationary state does not put mass on the unstable solutions of the hydrodynamic equation (1.1). For reaction–diffusion models it has only been proved in [31] that the stationary state is concentrated on the set of classical solutions to the semilinear elliptic equation

$$\begin{aligned} (1/2) \, \varDelta \rho + B(\rho ) - D(\rho ) \;=\; 0. \end{aligned}$$

(1.2)

The main result of this article, Theorem 4, establishes a large deviations principle for the empirical measure under the stationary state. The quasi-potential, the rate functional of this large deviations principle, is represented through a time-dependent variational problem involving the dynamical large deviations rate functional. The value of the quasi-potential at a measure $\varrho$ is given in terms of the infimum of the dynamical large deviations rate functional over all trajectories which start from a stationary solution of the hydrodynamic equation and end at $\varrho$.

A consequence of this result is that the stationary measure is concentrated at the stable, stationary solutions of the hydrodynamic equation. This is the content of Theorem 5, the second main result of the article.

The proof of Theorem 5 is based on two properties of the reaction–diffusion equation (1.1). First, it is known from [14] that all solutions of (1.1) converge to solutions of the semilinear elliptic equation (1.2). In particular, there are no time-periodic solutions. Second, we assume that Eq. (1.2) has only a finite number of solutions, modulo translations. This property holds when the polynomial $F(\rho ) = B(\rho ) - D(\rho )$ satisfies the hypotheses of Lemma 2. Theorem 5 further requires a characterization of the unstable stationary solutions of (1.1). Under the conditions of Lemma 3 on F, this set consists of all non-constant solutions and all constant solutions associated to local maxima of the potential V, where $V'(\rho ) = - F(\rho )$.

We conclude this introduction with some comments. This stochastic dynamics has been introduced by De Masi, Ferrari and Lebowitz in [16]. The authors proved the hydrodynamic limit of the system by duality arguments and the fluctuations of the density field. The dynamical large deviations principle for the empirical density starting from a product measure appeared in [27], following the ideas presented in [29]. Bodineau and Lagouge in [11, 12] proved the dynamical large deviations principle for the empirical current, while two of the authors of this article extended in [31] the dynamical large deviations principle to the case in which the process starts from a deterministic configuration.

The stationary states of the symmetric simple exclusion process are the Bernoulli product measures. The introduction of the spin-flip dynamics creates long range correlations. Although the local distribution of particles remains very close to a Bernoulli product measure due to the speeding-up of the exclusion dynamics, the long range correlations affect substantially the macroscopic behavior of the system. The purpose of this article is to study this effect at the level of the large deviations.

There is a huge literature on large deviations for reaction–diffusion equations perturbed by Gaussian or Lévy noise in finite and infinite dimensions after the seminal paper by Faris and Jona-Lasinio [22]. We refer to the recent books [15, 19] for references on the subject. The noise created by the microscopic spin-flip dynamics considered in this article is of a different nature. This is reflected in the dynamical large deviations rate functions in which singular exponential terms appear. This is one of the sources of technical problems faced in order to prove the regularity conditions of the dynamical rate functional needed to derive the static large deviations principle.

We leave to the end of the next section technical comments and remarks on the proofs and on the assumptions, and we mention here some open problems for future research. It would be interesting to extend to this model the results described at the beginning of this introduction which were obtained from the MFT for one-dimensional conservative interacting particle systems in contact with reservoirs: an alternative time-independent variational formula for the quasi-potential, and a description of the optimal trajectory which solves the time-dependent variational formula defining the quasi-potential. This has been done in [26] in the case where the reaction–diffusion model is reversible, but it remains an open problem in the non-reversible setting. In this general situation the only available information is an expansion of the quasi-potential around a constant stable stationary point obtained by Basile and Jona-Lasinio [1]. A description of the metastable behavior of the reaction–diffusion model when the difference $B(\rho ) - D(\rho )$ forms a double-well potential is also a challenging open problem.

2 Notation and results

Throughout this article, we use the following notation. $\mathbb {N}_{0}$ stands for the set $\{0,1,\ldots \}$. For a function $f: X\rightarrow {\mathbb {R}}$, defined on some set X, let $\Vert f\Vert _\infty = \sup _{x\in X}{|f(x)|}$. We sometimes denote the interval $[0,\infty )$ by $\mathbb {R}_{+}$.

2.1 Reaction–diffusion model

Let $\mathbb {T}_{N}=\mathbb {Z}/N\mathbb {Z}$, $N\ge 1$, be the one-dimensional discrete torus with N points. Denote by $X_{N}$ the set $\{0,1\}^{\mathbb {T}_{N}}$ and by $\eta$ the elements of $X_{N}$, called configurations. For each $x\in \mathbb {T}_{N}$, $\eta (x)$ represents the occupation variable at site x so that $\eta (x)=1$ if the site x is occupied for the configuration $\eta$, and $\eta (x)=0$ if the site is vacant. For each $x \not =y\in \mathbb {T}_{N}$, denote by $\eta ^{x,y}$, resp. by $\eta ^{x}$, the configuration obtained from $\eta$ by exchanging the occupation variables $\eta (x)$ and $\eta (y)$, resp. by flipping the occupation variable $\eta (x)$:

$$\begin{aligned} \eta ^{x,y}(z) \;=\; {\left\{ \begin{array}{ll} \eta (y) &{} \text {if }z=x, \\ \eta (x) &{} \text {if }z=y, \\ \eta (z) &{} \text {otherwise}, \end{array}\right. } \quad \eta ^x(z) \;=\; {\left\{ \begin{array}{ll} \eta (z) &{} \text {if }z\ne x, \\ 1-\eta (z) &{} \text {if }z=x. \end{array}\right. } \end{aligned}$$

Consider the superposition of the speeded-up symmetric simple exclusion process with a spin-flip dynamics. The generator of this $X_{N}$-valued, continuous-time Markov chain acts on functions $f:X_{N}\rightarrow \mathbb {R}$ as

$$\begin{aligned} {\mathcal {L}}_Nf \;=\; N^2 {\mathcal {L}}_Kf + {\mathcal {L}}_Gf, \end{aligned}$$

where ${\mathcal {L}}_K$ is the generator of a symmetric simple exclusion process (Kawasaki dynamics),

$$\begin{aligned} ({\mathcal {L}}_Kf)(\eta ) \;=\; (1/2) \sum _{x\in \mathbb {T}_N} [f(\eta ^{x,x+1}) - f(\eta )], \end{aligned}$$

and ${\mathcal {L}}_G$ is the generator of a spin-flip dynamics (Glauber dynamics),

$$\begin{aligned} ({\mathcal {L}}_Gf)(\eta ) \;=\; \sum _{x\in \mathbb {T}_N} c(\tau _x \eta )[f(\eta ^x) - f(\eta )]. \end{aligned}$$

In the last formula, $c(\eta )$ represents a strictly positive, cylinder function, that is, a function $c: \{0,1\}^\mathbb {Z}\rightarrow {\mathbb {R}}_+$ which depends only on a finite number of coordinates $\eta (y)$. For a sufficiently large N, c can be regarded as a function on $X_N$. $\{\tau _x : x\in {\mathbb {Z}}\}$ represents the group of translations defined by $(\tau _x \eta )(y) = \eta (x+y)$, $y\in {\mathbb {T}}_N$, where the sum is carried modulo N.

Note that the Kawasaki dynamics has been speeded-up by a factor $N^{2}$, which corresponds to the diffusive scaling. Setting the jump rates of the Glauber part to be 0, we retrieve the symmetric simple exclusion dynamics speeded up by $N^{2}$, whose static large deviation principle has been derived with several different boundary conditions in [3, 10,11,12, 21].

Fix a topological space X. Let D(I, X), $I=[0,T]$, $T>0$, or $I = \mathbb {R}_+$, be the space of right-continuous trajectories from I to X with left-limits, endowed with the Skorohod topology. Let $\{\eta _t^N:N\ge 1\}$ be the continuous-time Markov process on $X_{N}$ whose generator is given by ${\mathcal {L}}_{N}$. For a probability measure $\nu$ on $X_{N}$, denote by $\mathbb {P}_{\nu }$ the probability measure on $D(\mathbb {R}_{+},X_{N})$ induced by the process $\eta ^{N}_{t}$ starting from $\nu$. The expectation with respect to $\mathbb {P}_{\nu }$ is represented by ${\mathbb {E}}_{\nu }$. Denote by $\mathbb {P}_{\eta }$ the measure $\mathbb {P}_{\nu }$ when the probability measure $\nu$ is the Dirac measure concentrated on the configuration $\eta$. Analogously, ${\mathbb {E}}_{\eta }$ stands for the expectation with respect to ${\mathbb {P}}_{\eta }$.

2.2 Hydrodynamics

Let $\mathbb {T}$ be the one-dimensional continuous torus $\mathbb {T}=\mathbb {R}/\mathbb {Z}=[0,1)$. Denote by $L^p(\mathbb {T})$, $p\ge 1$, the space of all real p-th integrable functions $G:\mathbb {T}\rightarrow \mathbb {R}$ with respect to the Lebesgue measure $d\theta$: $\int _\mathbb {T}|G(\theta )|^p d\theta <\infty$. The corresponding norm is denoted by $\Vert \cdot \Vert _{p}$:

$$\begin{aligned} \Vert G\Vert _{p}^{p}\;:=\;\int _\mathbb {T}|G(\theta )|^p d\theta . \end{aligned}$$

In particular, $L^2(\mathbb {T})$ is a Hilbert space equipped with the inner product

$$\begin{aligned} \langle G, H \rangle \;=\; \int _\mathbb {T}G(\theta )H(\theta ) d\theta . \end{aligned}$$

For a function G in $L^{2}(\mathbb {T})$, we also denote by $\langle G \rangle$ the integral of G with respect to the Lebesgue measure: $\langle G \rangle :=\int _{\mathbb {T}} G(\theta ) d\theta$.

Let $\mathcal {M}_{+}=\mathcal {M}_{+}(\mathbb {T})$ be the space of all nonnegative measures on $\mathbb {T}$ with total mass bounded by 1, endowed with the weak topology. For a measure $\varrho$ in ${\mathcal {M}}_+$ and a continuous function $G:\mathbb {T}\rightarrow \mathbb {R}$, denote by $\langle \varrho , G \rangle$ the integral of G with respect to $\varrho$:

$$\begin{aligned} \langle \varrho , G \rangle \;=\; \int _\mathbb {T}G(\theta ) \varrho (d\theta ). \end{aligned}$$

The space $\mathcal {M}_+$ is metrizable. Indeed, if $e_{0}(\theta )=1$, $e_{k}(\theta )= \sqrt{2}\cos (2\pi k \theta )$ and $e_{-k}(\theta )= \sqrt{2}\sin (2\pi k\theta )$, $k\in \mathbb {N}$, one can define a distance d on $\mathcal {M}_{+}$ by

$$\begin{aligned} d(\varrho _{1},\varrho _{2}) \;:=\; \sum _{k\in \mathbb {Z}} \dfrac{1}{2^{|k|}} \, |\langle \varrho _{1},e_{k}\rangle - \langle \varrho _{2}, e_{k}\rangle |, \end{aligned}$$

(2.1)

and one can check that the topology induced by this distance corresponds to the weak topology.

Note that $\mathcal {M}_{+}$ is compact under the weak topology, and that, by Schwarz inequality, for all density profiles $\gamma$, $\gamma ':{\mathbb {T}}\rightarrow [0,1]$,

$$\begin{aligned} d(\gamma , \gamma ') \;\le \; 3 \Vert \gamma - \gamma '\Vert _2. \end{aligned}$$

(2.2)

In the previous formula we abuse of notation by writing $d(\gamma , \gamma ')$ for $d(\gamma (\theta ) d\theta , \gamma '(\theta ) d\theta )$.

Denote by $C^m(\mathbb {T})$, m in $\mathbb {N}_{0}\cup \{\infty \}$, the set of all real functions on $\mathbb {T}$ which are m times differentiable and whose m-th derivative is continuous. Given a function G in $C^2(\mathbb {T})$, we shall denote by $\nabla G$ and $\varDelta G$ the first and second derivatives of G, respectively.

Let $\nu _\rho =\nu ^N_{\rho }$, $0\le \rho \le 1$, be the Bernoulli product measure on $X_N$ with the density $\rho$. Define the continuous functions $B,D:[0,1]\rightarrow \mathbb {R}$ by

$$\begin{aligned} B(\rho )\;=\;\int [1-\eta (0)]\, c(\eta )\, d\nu _\rho , \quad D(\rho )\;=\;\int \eta (0)\, c(\eta )\, d\nu _\rho . \end{aligned}$$

Let $\pi ^N: X_N\rightarrow \mathcal {M}_+$ be the function which associates to a configuration $\eta$ the positive measure obtained by assigning mass $N^{-1}$ to each particle of $\eta$,

$$\begin{aligned} \pi ^N(\eta )\;=\;\frac{1}{N}\sum _{x\in \mathbb {T}_N}\eta (x)\delta _{x/N}, \end{aligned}$$

where $\delta _\theta$ stands for the Dirac measure which has a point mass at $\theta \in \mathbb {T}$. Let $\pi ^{N}_{t} = \pi ^N (\eta ^N_t)$, $t\ge 0$. The next result was proved by De Masi, Ferrari and Lebowitz in [16] for the first time. We refer to [16, 27, 28] for its proof.

Theorem 1

Fix a measurable function $\gamma :\mathbb {T}\rightarrow [0,1]$. Let $\nu _N$ be a sequence of probability measures on $X_N$ associated to $\gamma$, in the sense that

$$\begin{aligned} \lim _{N\rightarrow \infty } \nu _N \left( |\langle \pi ^{N},G\rangle -\int _{\mathbb {T}} G(\theta )\gamma (\theta )d\theta |>\delta \right) \;=\;0, \end{aligned}$$

(2.3)

for every $\delta >0$ and every continuous function $G:\mathbb {T}\rightarrow \mathbb {R}$. Then, for every $t\ge 0$, every $\delta >0$ and every continuous function $G:\mathbb {T}\rightarrow \mathbb {R}$,

$$\begin{aligned} \lim _{N\rightarrow \infty } {\mathbb {P}}_{\nu _N} \left( |\langle \pi _{t}^{N},G\rangle -\int _{\mathbb {T}} G(\theta )\rho (t,\theta )d\theta |>\delta \right) \;=\;0, \end{aligned}$$

where $\rho :[0,\infty )\times \mathbb {T}\rightarrow [0,1]$ is the unique weak solution of the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho \;=\; (1/2) \varDelta \rho + F(\rho ) \ \text { on }\ \mathbb {T},\\ \rho (0,\cdot )\;=\;\gamma (\cdot ), \end{array}\right. } \end{aligned}$$

(2.4)

where $F(\rho )=B(\rho )-D(\rho )$.

The definition, the existence and the uniqueness of weak solutions of the Cauchy problem (2.4) are discussed in Sect. 3.

2.3 The reaction–diffusion equation

We present in this subsection the results on the reaction–diffusion equation (2.4) needed in this section. Let S be the set of all classical solutions of the semilinear elliptic equation:

$$\begin{aligned} (1/2)\varDelta \rho + F(\rho ) \;=\; 0 \ \text { on }\ \mathbb {T}. \end{aligned}$$

(2.5)

Classical solution means a [0, 1]-valued function $\rho$ in $C^{2}(\mathbb {T})$ which satisfies the Eq. (2.5) for any $\theta \in \mathbb {T}$. We also denote by $\mathcal {M}_\mathrm{sol}$ the set of all absolutely continuous measures whose density is a classical solution of (2.5):

$$\begin{aligned} \mathcal {M}_\mathrm{sol}\;:=\;\{{{\bar{\varrho }}}\in \mathcal {M}_+:{{\bar{\varrho }}}(d\theta )={\bar{\rho }}(\theta )d\theta ,\ {\bar{\rho }}\in S \}. \end{aligned}$$

Next lemma is Theorem D of [14].

Lemma 1

Let $\rho :[0,\infty )\times \mathbb {T}\rightarrow [0,1]$ be the unique weak solution of the Cauchy problem (2.4). Then, there exists a density profile $\rho _{\infty }$ in S such that $\rho _{t}$ converges to $\rho _{\infty }$ as $t\rightarrow \infty$ in $C^{2}(\mathbb {T})$.

This result excludes the existence of time-periodic solutions of Eq. (2.4), a phenomenon which occurs if the function F is allowed to depend on $\nabla \rho$ as well (cf. [23] and references therein).

We turn to the description of the set S. Denote by ${\mathfrak {R}}$ the set of roots of F in [0, 1]. It is clear that for all $r\in {\mathfrak {R}}$, the constant function $\rho : {\mathbb {T}} \rightarrow {\mathbb {R}}$ given by $\rho (\theta ) = r$, $\theta \in {\mathbb {T}}$, is an element of S. There might be also non-constant periodic solutions.

Let $V:[0,1] \rightarrow {\mathbb {R}}$ be a potential such that $F(\rho ) = - V'(\rho )$. If the polynomial F has degree 1, as $V'(0)<0<V'(1)$, Eq. (2.5) has a unique solution, which is a global attractor for the dynamical system induced by the reaction–diffusion equation (2.4), and given by $\rho (\theta ) = r$, where r is the unique root of F.

Assume that the degree of F is larger than or equal to 2. Denote by $m_1, \dots , m_{{\mathfrak {n}}}$ the local minima of V in [0, 1] and by $M_1, \dots , M_{{\mathfrak {m}}}$ the local maxima in this interval. Since $V'(0)<0<V'(1)$, ${\mathfrak {n}} ={\mathfrak {m}}+1 \ge 1$ and $m_1<M_1< \cdots< M_{{\mathfrak {n}}-1}<m_{\mathfrak {n}}$.

Denote by $\sim$ the equivalence relation in $C^2({\mathbb {T}})$ defined by $\rho \sim \rho '$ if there exists $\theta '\in {\mathbb {T}}$ such that $\rho ' (\theta ) = \rho (\theta +\theta ')$ for all $\theta \in {\mathbb {T}}$. Of course, if $\rho$ is a periodic solution and $\rho '\sim \rho$, then $\rho '$ is also a solution.

Lemma 2

Suppose that all zeros of F are real and that all critical points of V are local minima or local maxima. Then, the elliptic equation (2.5) with periodic boundary conditions has at most a finite number of solutions, modulo the equivalence relation introduced above.

Since we could not find the previous result explicitly stated in the literature, we sketch the proof of this result. The terminology employed can be found in [32]. By Proposition 1.5.2 in [32], F is an $A-B$ function on all intervals $(m_1,M_1)$, $\dots , (M_{{\mathfrak {n}}-1},m_{{\mathfrak {n}}})$. The diagram of the Hamiltonian system $\dot{p} = q$, $\dot{q} = V'(p)$ shows that the periodic solutions of (2.5) are bounded below and above by two consecutive minima of the potential V.

Solutions of (2.5) with periodic boundary conditions can be mapped to solutions of (2.5) with Dirichlet boundary conditions. Indeed, fix two consecutive minima $m_j$, $m_{j+1}$ of V, and a solution $\rho$ of (2.5) taking values in $[m_j, m_{j+1}]$. Let $F_j(r) = F(M_j+r)$, so that $F_j(0)=0$ because $M_j$ is a local maximum of V. Note that $F_j$ is an $A-B$ function on $(m_j -M_j, 0) \cup (0, m_{j+1}-M_j)$. Let $\theta _0 = \min \{\theta \ge 0 : \rho (\theta ) = M_j\}$. Define $\phi :[0,1]\rightarrow [m_j-M_j, m_{j+1}-M_j]$ by $\phi (\theta ) = \rho (\theta +\theta _0)-M_j$. It is clear that $\phi$ is a solution of $(1/2) \varDelta v + F_j(v)=0$ with Dirichlet boundary conditions.

Since $F_j$ is an $A-B$ function on the intervals $(m_j- M_{j},0)$ and $(0, m_{j+1}-M_j)$, by Propositions 3.1.3, 3.1.4 and Theorem 3.1.9 in [32], the time-map of the Eq. (2.5) with Dirichlet boundary conditions is strictly convex and converges to $+\infty$ at the boundary. In particular, for each branch there exist at most two distinct solutions if $V''(M_j)=0$ and at most one solution if $V''(M_j)<0$. Since there is a finite number of branches whose time-map takes value less than or equal to 1, there is a finite number of distinct solutions of (2.5) with Dirichlet boundary conditions. As all solutions with periodic boundary conditions can be mapped to solutions with Dirichlet boundary conditions, the lemma is proved.

We turn to the heteroclinic orbits of (2.4). A complete description has been obtained in [23]. We state here a partial result which fulfills our needs. It asserts that all non-constant stationary solutions are unstable, as well as all constant solutions associated to local maxima of V.

Fix two stationary solutions $\phi \ne \psi$ of (2.4). A trajectory $\rho (t,\cdot )$, $t\in {\mathbb {R}}$, is called a heteroclinic orbit from $\phi$ to $\psi$ if $\lim _{t\rightarrow -\infty } = \phi$, $\lim _{t\rightarrow +\infty } = \psi$ and if $\rho$ solves (2.4) for every $t\in {\mathbb {R}}$. Convergences are meant in $C^1({\mathbb {T}})$. A solution $\phi$ of (2.5) is said to be unstable if there exist $\psi \not \sim \phi$ and a heteroclinic orbit from $\phi$ to $\psi$.

For a solution $\phi$ of (2.5), denote by ${\mathcal {L}}_\phi$ the linear operator on $C^2({\mathbb {T}})$ given by

$$\begin{aligned} {\mathcal {L}}_\phi h \; =\; (1/2) \varDelta h \;-\; V''(\phi ) h. \end{aligned}$$

(2.6)

If $\phi$ is not constant, $\nabla \phi$ is an eigenfunction associated to the eigenvalue 0. A non-constant solution $\phi$ of (2.5) is said to be hyperbolic if all eigenvalues of ${\mathcal {L}}_\phi$ have non-zero real parts, except the eigenvalue $\lambda =0$, whose associated eigenspace has dimension 1.

The eigenvalue 0 of the operator ${\mathcal {L}}_\phi$ is associated to the orbit $\rho (t,\theta ) = \phi (\theta + t)$. Actually, we prove in Lemma 16 that the cost of this orbit along a stationary set vanishes. Moreover, the existence of a positive eigenvalue of ${\mathcal {L}}_\phi$ is related to the existence of a heteroclinic orbit starting from $\phi$ and, therefore, to the instability of $\phi$.

Lemma 3

Assume the conditions of Lemma 2 and that all local maxima of V are non-degenerate: $V''(M_j)\not = 0$, $1\le j<{\mathfrak {n}}$. Then, for each non-constant solution $\phi$ of (2.5), there exist heteroclinic orbits from $\phi$ to $\phi _j$ and from $\phi$ to $\phi _{j+1}$, where $\phi _k(\theta ) = m_k$, $\theta \in {\mathbb {T}}$, and $j=\max \{k< {\mathfrak {n}}: m_k < \phi (\theta ) \; \forall \, \theta \in {\mathbb {T}}\}$. There exist also heteroclinic orbits from $\psi _j$ to $\phi _j$ and from $\psi _j$ to $\phi _{j+1}$, $1\le j < {\mathfrak {n}}$, where $\psi _j(\theta ) = M_j$, $\theta \in {\mathbb {T}}$.

This result follows from Theorems 1.3 and 1.4 in [23]. We just have to show that the hypotheses of these theorems are in force. As $V'(0)<0<V'(1)$, the solutions are bounded below by 0 and above by 1, so that F is dissipative.

We claim that all non-constant solution $\phi$ of (2.5) are hyperbolic. Indeed, fix such a function. As we have seen in the sketch of the proof of Lemma 2, there exists $1\le j < {\mathfrak {n}}$ such that $m_j \le \phi (\theta ) \le m_{j+1}$ for all $\theta \in {\mathbb {T}}$. Denote by $\varPi$ the orbit map associated to the polynomial F (cf. [32, page 51]). Since $V''(M_j)<0$, by [32, Proposition 1.5.2], F is an $A-B$ function in $(m_j,m_{j+1})$. Therefore, by [32, Theorem 2.1.3], $\varPi '(r)\not = 0$ for $r\not = M_j$. Hence, by the proof of [23, Lemma 4.4], $\phi$ is hyperbolic.

We turn to the proof of the lemma. Fix $1\le j<{\mathfrak {n}}$ and a non-constant solution $\phi$ of (2.5) taking values in the interval $[m_j, m_{j+1}]$. We show that there exist heteroclinic orbits from $\phi$ to $\phi _{j+1}$ and from $\psi _j$ to $\phi _{j+1}$. Similar arguments permit to replace $\phi _{j+1}$ by $\phi _{j}$.

The diagram of the Hamiltonian system $\dot{p} = q$, $\dot{q} = V' (p)$ shows that the periodic solutions of (2.5) which takes value in the interval $[m_j, m_{j+1}]$ are either (i) $\phi _j$, $\phi _{j+1}$, $\psi _j$ or (ii) a non-constant periodic solution whose maximal value belongs to $(M_j, m_{j+1})$ and minimal value to $(m_{j}, M_j)$. Moreover, if $\phi$, $\psi$ are such non-constant periodic solutions, either $\min _x \psi (x)< \min _x \phi (x)<M_j<\max _x \phi (x) < \max _x \psi (x)$ or the opposite.

We start with a heteroclinic orbit from $\psi _j$ to $\phi _{j+1}$. We may use the heteroclinic orbit from $M_j$ to $m_{j+1}$ for the ODE ${\dot{x}}(t) = V'(x(t))$ to obtain a heteroclinic orbit from $\psi _j$ to $\phi _{j+1}$ which remains constant in space.

Consider now a non-constant solution $\phi$ of (2.5) such that $m_j \le \phi (\theta ) \le m_{j+1}$. We repeat here the arguments of the proof of Theorem 1.3 in [23] presented at the end of page 111. Let z(h) be the number of strict sign changes of a function $h: {\mathbb {T}}\rightarrow {\mathbb {R}}$. Since $z(\nabla \phi ) \ge 2$, by [23, Proposition 3.1(b)], the unstable dimension of $\phi$, denoted by $i(\phi )$ in [23], is larger than or equal to 1. By the positivity of the first eigenfunction of the operator ${\mathcal {L}}_\phi$, one obtains a trajectory $\rho (t,\theta )$, $t\in {\mathbb {R}}$, which solves (2.4) and such that $\rho (t,\theta ) > \phi (\theta )$, $\lim _{t\rightarrow - \infty } \rho (t) = \phi$. Let $\psi = \lim _{t\rightarrow +\infty } \rho (t)$, which exists in view of Lemma 1. As $m_j\le \phi (\theta )\le m_{j+1}$, we have that $m_j \le \psi (\theta ) \le m_{j+1}$. By the Sturm property, $z(\rho (t) - \phi )$ decreases in time. Since it is equal to 0 for t close to $-\infty$, $z(\psi - \phi )= 0$. Hence, $\psi$ can not be $\psi _j$ or one of the non-constant solutions taking values in the interval $[m_j, m_{j+1}]$. Thus, $\psi$ must be $\phi _j$ or $\phi _{j+1}$. Since $\psi \ge \phi$, $\psi = \phi _{j+1}$, which proves the lemma.

We conclude this subsection with an example which fulfills the assumptions of Lemma 3. Fix $0<a<b$ and consider the reaction–diffusion equation

$$\begin{aligned} \partial _t \rho \;=\; (1/2) \varDelta \rho \;-\; V'(\rho ), \quad \text {where}\quad V(\rho ) \;=\; \frac{b}{4}\,(2\rho -1)^4 \,-\, \frac{a}{2} \, (2\rho -1)^2.\nonumber \\ \end{aligned}$$

(2.7)

This is the so-called Chafee–Infante equation [13]. It is clear that the potential V satisfies the assumptions of Lemma 3. Actually, in this case all stationary solutions and all heteroclinic orbits are known. We examine this example in Sect. 8, where we present microscopic jump rates which fulfill the hypotheses of Theorem 4 below and whose hydrodynamic equation is given by (2.7) with $0<a<b$.

2.4 Hydrostatics

Since the jump rate $c(\eta )$ is strictly positive, the Markov process $\eta ^{N}_{t}$ is irreducible in $X_N$. We denote by $\mu ^{N}$ the unique stationary probability measure under the dynamics. We review in this subsection the asymptotic behavior of the empirical measure under the stationary state $\mu ^N$.

Denote by ${\mathcal {P}}^N$ the probability measure on $\mathcal {M}_{+}$ defined by $\mathcal {P}^{N}:=\mu ^{N}\circ (\pi ^{N})^{-1}$. The following theorem has been established in [31]. It is a consequence of the law of large numbers for the empirical measure, stated in Theorem 1, and of the asymptotic behavior of the solutions of the reaction–diffusion equation, stated in Lemma 1.

Theorem 2

The sequence of measures $\{\mathcal {P}^N:N\ge 1\}$ is asymptotically concentrated on the set $\mathcal {M}_\mathrm{sol}$. Namely, for any $\delta >0$, we have

$$\begin{aligned} \lim _{N\rightarrow \infty }\mathcal {P}^N\left( \varrho \in \mathcal {M}_+ : \inf _{{\bar{\varrho }}\in \mathcal {M}_\mathrm{sol}} d(\varrho , {\bar{\varrho }})\ge \delta \right) \;=\; 0. \end{aligned}$$

Note that this result does not exclude the possibility that the stationary measure gives a positive weight to a neighborhood of an unstable stationary solution of Eq. (2.4).

2.5 Dynamical large deviations

Let $\mathcal {M}_{+,1}$ be the closed subset of $\mathcal {M}_{+}$ consisting of all absolutely continuous measures with density bounded by 1:

$$\begin{aligned} {\mathcal {M}}_{+,1}\;=\;\{\varrho \in {\mathcal {M}}_+: \varrho (d\theta )=\rho (\theta )d\theta ,\ 0\le \rho (\theta )\le 1\ a.e.\ \theta \in \mathbb {T}\}. \end{aligned}$$

Fix $T>0$, and denote by $C^{m,n}([0,T]\times \mathbb {T})$, m, n in $\mathbb {N}_{0}\cup \{\infty \}$, the set of all real functions defined on $[0,T]\times \mathbb {T}$ which are m times differentiable in the first variable and n times in the second one, and whose derivatives are continuous. Let $Q_{T,\eta }=Q^{N}_{T,\eta }$, $\eta \in X_{N}$, be the probability measure on $D([0,T],\mathcal {M}_{+})$ induced by the measure-valued process $\pi ^{N}_{t}$ starting from $\pi ^{N}(\eta )$.

For each path $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,T],{\mathcal {M}}_{+,1})$, define the energy ${\mathcal {Q}}_{T}$ as

$$\begin{aligned} {\mathcal {Q}}_{T}(\pi )\;=\;\sup _{G\in C^{0,1}([0,T]\times \mathbb {T})} \left\{ 2\int _0^Tdt\ \langle \rho _t,\nabla G_t\rangle -\int _0^Tdt\int _{\mathbb {T}}d\theta \ G(t,\theta )^2 \right\} . \end{aligned}$$

(2.8)

It is known (cf. [7, Subsection 4.1]) that the energy ${\mathcal {Q}}_{T}(\pi )$ is finite if and only if $\rho$ has a generalized derivative, denoted by $\nabla \rho$, and this generalized derivative is square integrable on $[0,T]\times \mathbb {T}$:

$$\begin{aligned} \int _{0}^{T} dt \ \int _{\mathbb {T}} d\theta \ |\nabla \rho (t,\theta )|^{2} <\infty . \end{aligned}$$

Moreover, it is easy to see that the energy ${\mathcal {Q}}_{T}$ is convex and lower semicontinuous.

For each function G in $C^{1,2}([0,T]\times \mathbb {T})$, define the functional ${\bar{J}}_{T,G}:D([0,T],{\mathcal {M}}_{+,1})\rightarrow \mathbb {R}$ by

$$\begin{aligned} {\bar{J}}_{T,G}(\pi )&\; =\;\langle \pi _T,G_T\rangle -\langle \pi _{0},G_0\rangle - \int _0^Tdt\ \langle \pi _t, \partial _tG_t+\frac{1}{2}\varDelta G_t\rangle \\&\quad -\frac{1}{2}\int _0^Tdt\ \langle \chi (\rho _t), (\nabla G_t)^2\rangle \\&\quad -\int _0^Tdt\ \big \{\langle B(\rho _t), e^{G_t}-1\rangle + \langle D(\rho _t), e^{-G_t}-1 \rangle \big \}, \end{aligned}$$

where $\chi (r)=r(1-r)$ is the mobility. Let $J_{T,G}: D([0,T],{\mathcal {M}}_+) \rightarrow [0,\infty ]$ be the functional defined by

$$\begin{aligned} J_{T,G} (\pi ) \;=\; {\left\{ \begin{array}{ll} {\bar{J}}_{T,G}(\pi ) &{} \text {if }\pi \in D([0,T],{\mathcal {M}}_{+,1}), \\ \infty &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(2.9)

and let $I_T:D([0,T],{\mathcal {M}}_+)\rightarrow [0,\infty ]$ be the functional given by

$$\begin{aligned} I_T(\pi ) \;=\; {\left\{ \begin{array}{ll} \sup {J_{T,G}(\pi )} &{} \text {if }{\mathcal {Q}}_{T}(\pi )<\infty , \\ \infty &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(2.10)

where the supremum is carried over all functions G in $C^{1,2}([0,T]\times \mathbb {T})$. We sometimes abuse of notation by writing $I_T(\rho )$ for $I_T(\pi )$ and we write $J_{G}$ for $J_{T,G}$ to keep notation simple.

An explicit formula for the functional $I_T$ at smooth trajectories was obtained in Lemma 2.1 of [27]. Let $\rho$ be a function in $C^{2,3}([0,T]\times \mathbb {T})$ with $c\le \rho \le 1-c$, for some $0< c <1/2$. Then, there exists a unique solution $H\in C^{1,2}([0,T]\times \mathbb {T})$ of the partial differential equation

$$\begin{aligned} \partial _t\rho \;=\; (1/2)\varDelta \rho \,-\, \nabla (\chi (\rho )\nabla H) \,+\, B(\rho )e^H \,-\, D(\rho )e^{-H}, \end{aligned}$$

and the rate functional $I_T(\rho )$ can be expressed as

$$\begin{aligned} I_T(\rho ) \;&=\; \frac{1}{2}\int _0^Tdt\ \langle \chi (\rho _t), (\nabla H_t)^2\rangle \\&\quad +\; \int _0^Tdt\ \langle B(\rho _t), 1-e^{H_t}+ H_t e^{H_t} \rangle + \int _0^Tdt\ \langle D(\rho _t), 1 - e^{-H_t} - H_t e^{-H_t} \rangle . \end{aligned}$$

For a measurable function $\gamma :\mathbb {T}\rightarrow [0,1]$, define the dynamical large deviations rate function $I_T(\cdot |\gamma ):D([0,T],{\mathcal {M}}_+)\rightarrow [0,\infty ]$ as

$$\begin{aligned} I_T(\pi |\gamma ) \;=\; {\left\{ \begin{array}{ll} I_{T}(\pi ) &{} \text {if }\pi (0,d\theta )=\gamma (\theta )d\theta , \\ \infty &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The next result, which establishes a dynamical large deviations principle for the measure-valued process $\pi _{\cdot }^{N}$ with rate functional $I_{T}(\cdot |\gamma )$ has been presented in [31] under the assumption that the functions B and D are concave on [0, 1]. We refer to [11, 12, 27] for different versions.

Theorem 3

Assume that the functions B and D are concave in [0, 1]. Fix $T>0$ and a measurable function $\gamma :\mathbb {T}\rightarrow [0,1]$. Consider a sequence $\eta ^{N}$ of initial configurations in $X_{N}$ associated to $\gamma$ in the sense that $\langle \pi ^{N}(\eta ^N),G\rangle$ converges to $\int _{\mathbb {T}} G(\theta )\gamma (\theta )d\theta$, as $N\uparrow \infty$, for all continuous function $G:{\mathbb {T}}\rightarrow {\mathbb {R}}$. Then, the measure $Q_{T,\eta ^{N}}$ on $D([0,T],{\mathcal {M}}_{+})$ satisfies a large deviation principle with the rate function $I_{T}(\cdot |\gamma )$. That is, for each closed subset $\mathcal {C}\subset D([0,T],{\mathcal {M}}_{+})$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {Q_{T, \eta ^{N}}(\mathcal {C})} \;\le \; -\inf _{\pi \in \mathcal {C}}I_{T}(\pi |\gamma ), \end{aligned}$$

and for each open subset $\mathcal {O}\subset D([0,T],{\mathcal {M}}_{+})$,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {Q_{T, \eta ^{N}}(\mathcal {O})} \;\ge \; -\inf _{\pi \in \mathcal {O}}I_{T}(\pi |\gamma ). \end{aligned}$$

Moreover, the rate function $I_{T}(\cdot |\gamma )$ is lower semicontinuous and it has compact level sets.

2.6 Static large deviations

We state in Theorem 4 below the main result of this paper, a large deviations principle for the empirical measure under the stationary measure.

Assume that the semilinear elliptic equation (2.5) admits at most a finite number of solutions, modulo translations. More precisely, assume that there exists $l\ge 1$ and density profiles ${\bar{\rho }}_{1}, \ldots , {\bar{\rho }}_{l}$ in $C^2({\mathbb {T}})$, such that

$$\begin{aligned} \mathcal {M}_\mathrm{sol}\;=\;\{ {\bar{\varrho }}_{i}(d\theta - \omega ) = {\bar{\rho }}_{i}(\theta - \omega )d\theta : 1\le i\le l, \omega \in \mathbb {T}\}. \end{aligned}$$

Lemma 2 provides conditions on the potential V which guarantee that this condition is in force. Let $\mathcal {M}_{i}$, $1\le i\le l$, be the subset of $\mathcal {M}_\mathrm{sol}$ given by $\mathcal {M}_{i}=\{{{\bar{\varrho }}}_{i}(d\theta - \omega ) : \omega \in \mathbb {T}\}$.

Define the functionals $V_{i}:\mathcal {M}_{+}\rightarrow [0,\infty ]$, $1\le i \le l$, by

$$\begin{aligned} V_i(\varrho ) = \inf \Big \{I_T(\pi |\gamma ): T> 0 \,,\, \gamma (\theta )d\theta \in \mathcal {M}_{i} ,\, \pi \in D([0,T],{\mathcal {M}}_{+}) \,\text { and }\, \pi _T = \varrho \Big \},\nonumber \\ \end{aligned}$$

(2.11)

which is the minimal cost to create the measure $\varrho$ from the set $\mathcal {M}_{i}$. We prove in Lemma 16 that in the previous variational formula we may replace the condition $\gamma (\theta )d\theta \in \mathcal {M}_{i}$ by the more restrictive condition $\gamma (\theta ) d\theta = {{\bar{\varrho }}}_{i}$ for some fixed ${{\bar{\varrho }}}_{i} \in \mathcal {M}_{i}$: for all ${{\bar{\varrho }}}_{i} \in {\mathcal {M}}_i$,

$$\begin{aligned} V_i(\varrho ) = \inf \Big \{I_T(\pi |\gamma ): T> 0 \,,\, \gamma (\theta )d\theta = {{\bar{\varrho }}}_{i} \,,\, \pi \in D([0,T],{\mathcal {M}}_{+}) \,\text { and }\, \pi _T = \varrho \Big \}.\nonumber \\ \end{aligned}$$

(2.12)

By an abuse of notation, we sometimes write $V_i(\gamma )$ instead of $V_i(\gamma (\theta ) d\theta )$, where $\gamma :{\mathbb {T}}\rightarrow [0,1]$ is a density profile.

By translation invariance, $V_{i}(\gamma ) = V_{i}(\gamma ')$ if $\gamma ' (\cdot ) = \gamma (\cdot - \omega )$ for some $\omega \in {\mathbb {T}}$. In particular, $V_i$ is constant on the set ${\mathcal {M}}_j$, $j\ne i$, and $v_{ij}=V_{i}({\bar{\varrho }}_{j})$ is well defined, where ${{\bar{\varrho }}}_j$ is any element of ${\mathcal {M}}_j$. Moreover, by choosing $T=1$ and $\pi _t=(1-t){{\bar{\varrho }}}_{i}+t{{\bar{\varrho }}}_j$, $t\in [0,1]$, in the infimum of (2.12) yields that $v_{ij}$ is finite for any $i\ne j$. Finally, by Lemmata 12 and 16, $V_i(\varrho )=0$ for any $\varrho \in \mathcal {M}_i$.

Following [24, Chapter 6], denote by ${\mathcal {T}}(i)$, $i\in {\mathcal {V}} :=\{1, \ldots , l\}$, the set of all oriented, weighted, rooted trees whose vertices are all the elements of ${\mathcal {V}}$ and whose root is i. The edges are oriented from the child to the parent, and the weight $v_{mn}$ is assigned to the oriented edge (m, n). Denote by $\kappa (g)$ the sum of the weights of the tree $g\in {\mathcal {T}}(i)$ and by $w_i$ the minimal weight of all trees in ${\mathcal {T}}(i)$:

$$\begin{aligned} w_{i} \;=\; \min _{g\in {\mathcal {T}} (i)} \kappa (g), \qquad \kappa (g) \;=\; \sum _{(m, n)\in g} v_{mn}. \end{aligned}$$

Since $v_{ij}$ is finite for $i\not = j$, so are $w_i$ and $w=\min _{1\le i \le l} w_{i}$. We will see below in (2.15) that the non-negative parameter $w_i$ corresponds to the exponential weight of a neighborhood of the set ${\mathcal {M}}_i$ under the stationary state.

Note that for all $i\not = j$,

$$\begin{aligned} w_i \;\le \; w_j \;+\; v_{ji}. \end{aligned}$$

(2.13)

Indeed, let g be a graph in ${\mathcal {T}}(j)$ such that $w_j = \kappa (g)$. Denote by (a, b), $a\not = b\in {\mathcal {V}}$, the oriented edge where a is the child and b the parent. Let $i'$ be the parent of i in g. Of course, $i'$ might be j. Denote by $g'$ the tree in ${\mathcal {T}}(i)$ obtained from g by adding the oriented edge (j, i) and removing the the edge $(i,i')$, and note that $\kappa (g) + v_{ji} = \kappa (g') + v_{ii'}$. Since $w_i$ is the minimal value of $\kappa ({\tilde{g}})$, ${\tilde{g}}\in {\mathcal {T}}(i)$, $w_i \le \kappa (g') \le \kappa (g') + v_{ii'} = \kappa (g) + v_{ji} = w_j + v_{ji}$.

For each $1\le i \le l$, define the functions $W_{i}, W:\mathcal {M}_{+}\rightarrow [0,\infty ]$ by

$$\begin{aligned} W_{i}(\varrho ) \;=\; w_{i} - w + V_{i}(\varrho ), \quad W(\varrho ) \;=\; \min _{1\le i \le l} W_{i}(\varrho ). \end{aligned}$$

(2.14)

Note that for all $\varrho \in {\mathcal {M}}_i$,

$$\begin{aligned} W(\varrho ) \;=\; {\overline{w}}_i \;:=\; w_i \,-\, w. \end{aligned}$$

(2.15)

Indeed, fix $\varrho \in {\mathcal {M}}_i$. In view of the definition of W, we have to show that $\min _{1\le j\le l} \{ {\overline{w}}_j + V_j(\varrho )\} = {\overline{w}}_i$. The minimum is less than or equal to ${\overline{w}}_i$ because $V_i(\varrho )=0$. On the other hand, since $V_j(\varrho )=v_{ji}$ and since, by (2.13), $w_i \le w_j + v_{ji}$, ${\overline{w}}_i \le {\overline{w}}_j + V_j(\varrho )$ for $j\not = i$.

The following theorem is the main result of this paper.

Theorem 4

Assume that the jump rates are strictly positive and that the functions B and D are concave on [0, 1]. Assume, furthermore, that the semilinear elliptic equation (2.5) admits at most a finite number of solutions, modulo translations. Then, the sequence of probability measures $\{\mathcal {P}^N; N\ge 1\}$ satisfies a large deviation principle on $\mathcal {M}_{+}$ with speed N and rate function W. Namely, for each closed set ${\mathcal {C}}\subset {\mathcal {M}}_{+}$ and each open set ${\mathcal {O}}\subset {\mathcal {M}}_{+}$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {C}})\le & {} -\inf _{\varrho \in {\mathcal {C}}} W(\varrho ), \\ \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {O}})\ge & {} -\inf _{\varrho \in {\mathcal {O}}} W(\varrho ). \end{aligned}$$

Moreover, the rate functional W is bounded on $\mathcal {M}_{+,1}$, it is lower semicontinuous, and it has compact level sets.

2.7 The support of the stationary measure $\mu ^N$

The next result improves on Theorem 2 and asserts that the stationary measure $\mu ^N$ is concentrated on neighborhoods of stable equilibria of the reaction–diffusion equation (2.4). This result has been conjectured in [11, 12].

As in the previous subsection, assume that the semilinear elliptic equation (2.5) admits at most a finite number of solutions, modulo translations. Denote by ${\mathfrak {M}}$ the set of local minima of V, and by ${\mathcal {S}} \subset {\mathcal {M}}_{\mathrm{sol}}$ the set of associated density profiles:

$$\begin{aligned} {\mathcal {S}} \;=\; \{\varrho (d\theta ) = m\, d\theta : m\in {\mathfrak {M}}\}. \end{aligned}$$

The elements of ${\mathcal {S}}$ are called stable solutions. Lemma 30 justifies this terminology. It states that the quasi-potential associated to each $\varrho \in {\mathcal {S}}$ is strictly positive outside any neighborhood of $\varrho$. More precisely, for every ${\bar{\varrho }}_i (d\theta ) = {\bar{\rho }}_i(\theta ) d\theta \in {\mathcal {S}}$ and $\varepsilon >0$, there exists $c>0$ such that $\inf _{\gamma \not \in {\mathcal {B}}_\varepsilon ({\bar{\varrho }}_i)} V_i(\gamma ) \ge c$, where ${\mathcal {B}}_\varepsilon ({\bar{\varrho }}_i)$ represents a ball in ${\mathcal {M}}_+$ of radius $\varepsilon$ centered at ${\bar{\varrho }}_i$.

Denote by $I_s$, $I_u\subset \{1, \dots , l\}$ the set of indices associated to stable, unstable density profiles, respectively:

$$\begin{aligned} I_s \;=\; \{ j : {\bar{\rho }}_j (\theta ) \, d\theta \in {\mathcal {S}}\}, \quad I_u \;=\; \{1, \dots , l\} \setminus I_s. \end{aligned}$$

Theorem 5

Assume that the hypotheses of Theorem 4 are in force, and that for all $i\in I_u$ there exists $j\in I_s$ such that

$$\begin{aligned} v_{ij}\;=\; 0. \end{aligned}$$

(2.16)

Then, for all $\varepsilon >0$ there exist $c>0$ and $N_0\ge 1$ such that for all $N\ge N_0$,

$$\begin{aligned} {\mathcal {P}}^N \left( \mathcal {M}_+ \Big \backslash \left[ \bigcup _{j\in I_s} {\mathcal {B}}_\varepsilon ({\bar{\varrho }}_j) \right] \right) \;\le \; e^{-c N}. \end{aligned}$$

Of course, one expects the stationary measure $\mu ^N$ to be concentrated on neighborhoods of the density profiles associated to the global minima of the potential V. This problem remains an open question. A finer estimate than the one provided by the large deviations might be needed to answer this open question. We refer to [25] for a similar problem in the context of the pinned Wiener measure, and to [9] and references therein for the study of the concentration of measures in the situation where the rate functional has more than one minimizer.

Lemma 3 provides a set of sufficient conditions, expressed in terms of the potential V, for assumption (2.16) to hold. Indeed, by Lemma 29, if there exists a heteroclinic orbit from ${\mathcal {M}}_i$ to ${\mathcal {M}}_j$, then $v_{ij}=0$. Hence, by Lemma 3, if all zeros of F are real, all critical points of V are local minima or local maxima and all local maxima of V are non-degenerate, hypothesis (2.16) is in force.

Fix a stationary solution $\phi ={\bar{\rho }}_i$ of (2.5). There are at least three different possible definitions of instability: (i) the operator ${\mathcal {L}}_\phi$, defined in (2.6), has an eigenvalue with positive real part. (ii) there exists $\psi \not \sim \phi$ and a heteroclinic orbit from $\phi$ to $\psi$. (iii) $v_{ij}=0$ for some $j\ne i$. We presented in the proof of Lemma 3 a sketch of the proof that (i) $\Rightarrow$ (ii) under some additional hypotheses. Lemma 29 asserts that (ii) $\Rightarrow$ (iii). We believe that the other implications hold, at least with some extra assumptions, but we were not able to prove them.

2.8 Comments and remarks

The characterization of the global attractor of the solutions of reaction–diffusion equations [14, 23] has only been achieved in dimension 1, not to mention the description of the heteroclinic orbits. This is the main obstacle to extend the previous result to higher dimensions. Although it is true that the dynamical large deviations principle has been derived only in one dimension [31], it should not be very difficult to extend it to higher dimensions.

The results presented in this article can be proved for one-dimensional reaction–diffusion models with Dirichlet or Neumann boundary conditions. The description of the heteroclinic orbits in these contexts is simpler than the one with periodic boundary conditions (cf. [13] for the case of Dirichlet boundary conditions).

As mentioned above, it is an open, and very appealing, problem to show that the stationary measure $\mu ^N$ is concentrated on neighborhoods of the density profiles associated to global minima of the potential V. To apply the method presented in the article to solve this question would require a sharp estimate of the cost of the instanton, the trajectory which drives the system from a stable equilibrium to another. In view of Theorem 9 below, it is clear that the instanton in the case of a double well potential with non-constant stationary profiles is the trajectory which crosses the non-constant stationary solution with one period. To estimate the cost of this trajectory seems to be out of reach.

The previous questions lead us to the problem of the metastability of the dynamics. It is challenging to describe the metastable behavior of these reaction–diffusion models.

The hypothesis that the functions B and D are concave is only needed in the proof of the dynamical large deviations principle [31], and we never use it in this paper. If one is able to prove this dynamical result without the concavity assumption, the arguments presented in this article provide a proof of the static large deviations principle without the concavity assumption.

As mentioned in the introduction, the strategy of the proof consists in adapting to our infinite-dimensional setting the Freidlin and Wentzell approach [24] to prove a large deviation principle for the stationary state of a small perturbation of a dynamical system. This has been done before in [10, 21] for conservative evolutions in contact with reservoirs. However, in the context of reaction–diffusion models the existence of several stationary solutions to the hydrodynamic equation introduces additional difficulties.

The proof relies on a representation of the stationary state of the reaction–diffusion model in terms of the invariant measure of a discrete-time Markov chain induced by the successive visits to the neighborhoods of the stationary solutions of the hydrodynamic equation.

The proof of the static large deviations principle can be decomposed in essentially three steps. We first need to derive some regularity properties of the dynamical large deviations rate functional. For instance, that any trajectory which remains in a long time interval far apart (in the $L^2$-topology) from the stationary solutions of the hydrodynamic equation pays a strictly positive cost. Or that the quasi-potential is lower semicontinuous in the weak topology.

The second step consists in obtaining sharp large deviations bounds for the invariant measure of the discrete-time Markov chain. The final step, whose proofs are similar to the ones presented in [10, 21], consists in estimating the minimal cost to create a measure starting from a stationary solution of the hydrodynamic equation.

The topology is one of the main technical difficulties in the argument. The weak topology is imposed by the dynamical large deviations principle which has been derived in this set-up, and one is forced to prove all regularity properties of the rate functionals in this topology. To overcome this obstacle, we systematically use the smoothening properties of the hydrodynamic equation. Lemma 19 is a good illustration of this strategy.

The article is organized as follows. In Sect. 3, we present the main properties of the weak solutions of the Cauchy problem (2.4), and in Sects. 4 and 5, we examine the dynamical and the static large deviations rate functionals. These sections are purely analytical, and no probabilistic argument is used. In Sect. 6, we prove the static large deviations principle, and, in Sect. 7, the concentration of the stationary measure $\mu ^N$. In Sect. 8 we present a reaction–diffusion model which fulfills the hypotheses of Theorem 5.

3 The reaction–diffusion equation

We present in this section several properties of the weak solutions of the Cauchy problem (2.4). When we did not find a reference, we present a proof of the result. Throughout this section and in the next ones, $C_0$ represents a finite, positive constant which depends only on F and which may change from line to line. As mentioned above, this section and the following two ones are purely analytical, and no probabilistic arguments appear.

We first define two concepts of solutions.

Definition 1

A measurable function $\rho :[0,T]\times \mathbb {T}\rightarrow [0,1]$ is said to be a weak solution of the Cauchy problem (2.4) in the layer $[0,T]\times \mathbb {T}$ if for every function G in $C^{1,2}([0,T]\times \mathbb {T})$,

$$\begin{aligned}&\langle \rho _{T}, G_{T}\rangle - \langle \gamma , G_{0} \rangle - \int _{0}^{T} dt \, \langle \rho _{t}, \partial _{t}G_{t} \rangle \nonumber \\&\quad =\;\dfrac{1}{2} \int _{0}^{T} dt \, \langle \rho _{t}, \varDelta G_{t} \rangle + \int _{0}^{T} dt \, \langle F(\rho _{t}), G_{t} \rangle . \end{aligned}$$

(3.1)

Definition 2

A measurable function $\rho :[0,T]\times \mathbb {T}\rightarrow [0,1]$ is said to be a mild solution of the Cauchy problem (2.4) in the layer $[0,T]\times \mathbb {T}$ if for any t in [0, T]

$$\begin{aligned} \rho _{t} \;=\; P_{t}\gamma \;+\; \int _{0}^{t} P_{t-s}F(\rho _{s}) \, ds, \end{aligned}$$

(3.2)

where $\{P_{t} : t\ge 0\}$ stands for the semigroup on $L^{2}(\mathbb {T})$ generated by $(1/2)\varDelta$.

Next proposition asserts that the two notions of solutions are equivalent. We refer to Proposition 6.3 of [31] for the proof.

Proposition 1

Definitions 1 and 2 are equivalent. Moreover, there exists a unique weak solution of the Cauchy problem (2.4).

The next result is contained in Proposition 2.1 of [17].

Proposition 2

Let $\rho$ be the unique weak solution of the Cauchy problem (2.4). Then $\rho$ is infinitely differentiable over $(0,\infty )\times \mathbb {T}$.

Let ${\mathbb {Z}}_* = {\mathbb {Z}}\setminus \{0\}$, and let $c_0: \{0,1\}^{{\mathbb {Z}}_*} \rightarrow {\mathbb {R}}_+$ be the cylinder function defined by $c_0(\xi ) = c(\xi ^{(0)})$, where $\xi ^{(0)}$ is the configuration of $\{0,1\}^{{\mathbb {Z}}}$ defined by $\xi ^{(0)}(x) = \xi (x)$, $x\not = 0$, $\xi ^{(0)}(0) = 0$. The cylinder function $c_1: \{0,1\}^{{\mathbb {Z}}_{*}} \rightarrow {\mathbb {R}}_+$ is defined analogously with $\xi ^{(0)}$ replaced by $\xi ^{(1)}$, where $\xi ^{(1)}(0) = 1$.

Note that $c_0$ and $c_1$ are strictly positive cylinder functions because so is $c(\eta )$. Hence, if $\nu ^*_\rho$ represents the Bernoulli product measure on $\{0,1\}^{{\mathbb {Z}}_*}$ with density $\rho$, the polynomial ${\widehat{B}}(\rho )$ defined by ${\widehat{B}}(\rho ) = E_{\nu ^*_\rho }[c_0 (\eta )]$ is strictly positive. Similarly, the polynomial ${\widehat{D}}(\rho )$ defined by ${\widehat{D}}(\rho ) = E_{\nu ^*_\rho }[c_1 (\eta )]$ is strictly positive.

By definition, $B(\rho ) \;=\; E_{\nu _\rho }[\{1-\eta (0)\} c(\eta )] \;=\; (1-\rho ) \, E_{\nu ^*_\rho }[ c_0 (\eta )] = (1-\rho ) {\widehat{B}}(\rho )$, and $D(\rho ) = \rho {\widehat{D}}(\rho )$. Hence,

$$\begin{aligned} B(\rho ) \;=\; (1-\rho ) \, {\widehat{B}}(\rho ), \quad D(\rho ) \;=\; \rho \, {\widehat{D}}(\rho ), \end{aligned}$$

(3.3)

where ${\widehat{B}}(\rho )$ and ${\widehat{D}}(\rho )$ are strictly positive polynomials. In particular, $F(0) = B(0) - D(0) = {\widehat{B}}(0) >0$ and $F(1) = B(1) - D(1) = - {\widehat{D}}(1) <0$.

Denote by $x_a(t)$, $0\le a\le 1$, the solution of the ODE

$$\begin{aligned} {\dot{x}}(t) \;=\; F(x(t)) \end{aligned}$$

(3.4)

with initial condition $x(0) = a$. Since $F(1)<0<F(0)$, $x_0(t)$ (resp. $x_1(t)$) is strictly increasing (resp. decreasing) and $x_0(t)\rightarrow x_0$ (resp. $x_1(t)\rightarrow x_1$), where $x_0$ (resp. $x_1$) is the smallest (resp. largest) solution of $F(x)=0$. The next result is a simple application of the maximum principle.

Lemma 4

Let $\gamma :{\mathbb {T}}\rightarrow [0,1]$ be a density profile such that $a\le \gamma (\theta ) \le b$ for a.e. $\theta \in {\mathbb {T}}$. Denote by $\rho ^\gamma (t,\theta )$ the unique weak solution of (2.4) with initial condition $\gamma$. Then, $x_a(t) \le \rho (t,\theta ) \le x_b(t)$ for all $t\ge 0$. In particular, for any $t>0$, there exists $\varepsilon = \varepsilon (t)>0$ such that $\varepsilon \le \rho ^\gamma (t,\theta ) \le 1- \varepsilon$ for all $\theta \in {\mathbb {T}}$ and all initial density profiles $\gamma : {\mathbb {T}} \rightarrow [0,1]$. Moreover, there exists $\delta >0$, depending only on F, such that

$$\begin{aligned} \delta \;\le \; {\bar{\rho }}_i \;\le \; 1-\delta \end{aligned}$$

(3.5)

for all $1\le i\le l$, ${\bar{\rho }}_i(\theta )\, d\theta \in {\mathcal {M}}_i$.

Lemma 5

There exists a finite constant $C_0$, depending only on F, such that for any density profile $\gamma :{\mathbb {T}} \rightarrow [0,1]$ and any $t>0$,

$$\begin{aligned} \Vert \rho _t \Vert ^2_2 \;+\; \int _0^t \Vert \nabla \rho _s \Vert ^2_2\, ds \;\le \; C_0 (1+t), \end{aligned}$$

where $\rho (t,\theta )$ stands for the unique weak solution of (2.4) with initial condition $\gamma$.

Proof

Fix a density profile $\gamma :{\mathbb {T}} \rightarrow [0,1]$. By Proposition 2, and since $\rho$ is the weak solution of (2.4), for any $0<s<t$, by an integration by parts,

$$\begin{aligned} \Vert \rho _t \Vert ^2_2 \;=\; \Vert \rho _s \Vert ^2_2 \;-\; \int _s^t \Vert \nabla \rho _r \Vert ^2_2 \, dr \;+\; 2 \int _s^t dr \int _{{\mathbb {T}}} \rho _r (\theta ) F(\rho _r(\theta ))\, d\theta . \end{aligned}$$

Since $\rho _r$ is absolutely bounded by 1, we complete the proof of the lemma by letting $s\downarrow 0$. $\square$

A similar argument provides a bound on the distance between a solution of the hydrodynamic equation and a constant stationary solution. Recall that $\alpha \in (0,1)$ is an attractor of the ODE (3.4) if there exists $\varepsilon >0$ such that the solution x(t) of the ODE with initial condition $x_0$ converges to $\alpha$ as $t\uparrow \infty$ if $|x_0-\alpha |<\varepsilon$. Note in particular that $F(\alpha )=0$ if $\alpha$ is an attractor.

Lemma 6

Let $\varepsilon >0$, let $\alpha$ be an attractor of the ODE (3.4), and let ${\bar{\rho }}_\alpha$ be the density profile given by ${\bar{\rho }}_\alpha (\theta )=\alpha$, $\theta \in {\mathbb {T}}$. There exists $\delta _{10} = \delta _{10}(\varepsilon , \alpha )>0$ such that for any density profile $\gamma : {\mathbb {T}} \rightarrow [0,1]$ such that $\Vert \gamma - {\bar{\rho }}_\alpha \Vert _2 \le \delta _{10}$, $\rho _t$ converges in the sup norm to ${\bar{\rho }}_\alpha$ as $t\uparrow \infty$, where $\rho _t(\theta ) = \rho (t,\theta )$ is the unique weak solution of (2.4) with initial condition $\gamma$. Moreover, for all $t\ge 1$, $\Vert \rho _t - {\bar{\rho }}_\alpha \Vert _\infty \le \varepsilon$.

Proof

Fix $\varepsilon >0$, $\alpha \in (0,1)$ such that $F(\alpha )=0$, a density profile $\gamma :{\mathbb {T}} \rightarrow [0,1]$ and recall the notation introduced in the statement of the lemma. Let $\rho$ be the weak solution of (2.4) with initial condition $\gamma$. Repeating the computation presented in the proof of Lemma 5, we obtain that for every $0<s<t$,

$$\begin{aligned} \Vert \rho _t - {\bar{\rho }}_\alpha \Vert ^2_2 = \Vert \rho _s - {\bar{\rho }}_\alpha \Vert ^2_2 \;- \int _s^t \Vert \nabla \rho _r \Vert ^2_2 \, dr + 2 \int _s^t dr \int _{{\mathbb {T}}} [\rho _r (\theta ) - {\bar{\rho }}_\alpha (\theta )]\, F(\rho _r(\theta ))\, d\theta . \end{aligned}$$

Since $F(\alpha )=0$, we may subtract $F({\bar{\rho }}_\alpha (\theta ))$ from $F(\rho _r(\theta ))$ in the last integral and bound the product by $C_0 [\rho _r (\theta ) - {\bar{\rho }}_\alpha (\theta )]^2$, where $C_0$ is the Lipschitz constant of F. By letting $s\downarrow 0$ and then applying Gronwall inequality, we obtain that

$$\begin{aligned} \Vert \rho _t - {\bar{\rho }}_\alpha \Vert ^2_2 \;+\; \int _0^t \Vert \nabla \rho _r \Vert ^2_2 \, dr \;\le \; \Vert \rho _0 - {\bar{\rho }}_\alpha \Vert ^2_2 \; e^{2C_0t} \end{aligned}$$

(3.6)

for all $t\ge 0$.

Choose $\varepsilon _0>0$ so that $(\alpha -3\varepsilon _0, \alpha + 3\varepsilon _0)$ is contained in the basin of attraction of $\alpha$ for the ODE (3.4) and set $\varepsilon _1 = \min \{\varepsilon /2, \varepsilon _0\}$. Choose $\delta _{10} = \varepsilon _1 e^{-C_0}$ and let $\gamma$ be an initial profile such that $\Vert \gamma - {\bar{\rho }}_\alpha \Vert _2\le \delta _{10}$. By (3.6), for all $0\le t\le 1$,

$$\begin{aligned} \Vert \rho _t - {\bar{\rho }}_\alpha \Vert _2 \;\le \; \varepsilon _1 \quad \text {and}\quad \int _0^1 \Vert \nabla \rho _r \Vert ^2_2 \, dr \;\le \; \varepsilon ^2_1. \end{aligned}$$

(3.7)

In particular, there exists $0\le s\le 1$ such that $\Vert \nabla \rho _s \Vert _2 \le \varepsilon _1$, so that

$$\begin{aligned} \sup _{\theta \not = \omega \in {\mathbb {T}}} \big | \rho (s,\theta ) - \rho (s,\omega ) \big | \;\le \; \Vert \nabla \rho _s \Vert _1 \;\le \; \Vert \nabla \rho _s \Vert _2 \;\le \; \varepsilon _1. \end{aligned}$$

Therefore, by (3.7), for all $\theta \in {\mathbb {T}}$,

$$\begin{aligned} \big | \rho (s,\theta ) - \alpha \big | \;\le \; \sup _{\theta \not = \omega \in {\mathbb {T}}} \big | \rho (s,\theta ) - \rho (s,\omega ) \big | \;+\; \Vert \rho _s - {\bar{\rho }}_\alpha \Vert _1 \;\le \; \varepsilon _1 \;+\; \Vert \rho _s - {\bar{\rho }}_\alpha \Vert _2 \;\le \; \varepsilon \end{aligned}$$

because $2\varepsilon _1 \le \varepsilon$.

Let $x_\pm (t)$, $t\ge 0$, be the solution of the ODE (3.4) with initial condition $x_\pm (0) = \alpha \pm 2 \varepsilon _1$. By the previous estimate, $x_-(0) \le \rho (s,\theta ) \le x_+(0)$ for all $\theta \in {\mathbb {T}}$. Hence, by Lemma 4, $x_-(t) \le \rho (s+t,\theta ) \le x_+(t)$ for all $\theta \in {\mathbb {T}}$, $t\ge 0$. Since the basin of attraction of the ODE is contained in $(\alpha -3\varepsilon _1, \alpha + 3\varepsilon _1)$, $x_\pm (t) \rightarrow \alpha$, as $t\rightarrow \infty$, and $x_-(0) \le x_-(t) \le x_+(t) \le x_+(0)$ for all $t\ge 0$. In particular, $\rho _t$ converges in the sup norm to ${\bar{\rho }}_\alpha$, as $t\rightarrow \infty$, and $\Vert \rho _{s+t} - {\bar{\rho }}_\alpha \Vert _\infty \le \Vert \rho _s - {\bar{\rho }}_\alpha \Vert _\infty \le \varepsilon$ for all $t\ge 0$. This completes the proof of the lemma because $s\le 1$. $\square$

Similar arguments permit to estimate the distance between two solutions of the reaction–diffusion equation (2.4).

Lemma 7

There exists a constant $C_0>0$ such that for any weak solutions $\rho ^{j}$, $j=1,2$, of the Cauchy problem (2.4) with initial profile $\rho _{0}^{j}$ and for any $t>0$,

$$\begin{aligned} \Vert \rho _{t}^{1}-\rho _{t}^{2}\Vert _2 \;\le \; e^{C_0 t}\Vert \rho _{0}^{1}-\rho _{0}^{2}\Vert _2. \end{aligned}$$

Proof

From (3.2), for any $t\ge 0$ and $j=1,2$,

$$\begin{aligned} \rho _{t}^{j}\;=\; P_{t}\rho _{0}^{j} + \int _{0}^{t}P_{t-s}F(\rho _{s}^{j}) ds. \end{aligned}$$

Therefore

$$\begin{aligned} \Vert \rho _{t}^{1}-\rho _{t}^{2}\Vert _{2}&\;\le \; \Vert P_{t}(\rho _{0}^{1}-\rho _{0}^{2})\Vert _{2}+ \int _{0}^{t}\Vert P_{t-s}(F(\rho _{s}^{1})-F(\rho _{s}^{2}))\Vert _{2} \, ds \\&\;\le \; \Vert \rho _{0}^{1}-\rho _{0}^{2}\Vert _{2}+ \Vert F'\Vert _{\infty }\int _{0}^{t}\Vert \rho _{s}^{1}-\rho _{s}^{2}\Vert _{2} \,ds. \end{aligned}$$

In the last inequality, we used the fact that the operator norm of $P_{t}$ is equal to 1. To conclude the proof of the lemma, it remains to apply Gronwall inequality. $\square$

For each function $\rho$ in $L^{2}(\mathbb {T})$, let ${\mathbb {B}}_{\delta }(\rho )$, $\delta >0$, be the $\delta$-open neighborhood of $\rho$ in $L^{2}(\mathbb {T})$. Recall also that we denote by ${\mathcal {B}}_{\delta }(\varrho )$ be the $\delta$-open neighborhood of $\varrho$ in $\mathcal {M}_{+}$. We sometimes represent the neighborhood ${\mathcal {B}}_{\delta }(\varrho )$ by ${\mathcal {B}}_\delta (\gamma )$ when $\varrho (d\theta ) = \gamma (\theta ) d\theta$.

Lemma 8

Let ${\bar{\rho }}:\mathbb {T}\rightarrow [0,1]$ be a classical solution to the Eq. (2.5), and set ${{\bar{\varrho }}}(d\theta )={\bar{\rho }}(\theta )d\theta$. For any $\varepsilon >0$ and $0<T<T'$, there exists $\delta _{11} = \delta _{11}(\varepsilon , T,T') \in (0,\varepsilon )$ such that for any density profile $\rho _{0}:\mathbb {T}\rightarrow [0,1]$, $\rho _{0}(\theta )d\theta$ in ${\mathcal {B}}_{\delta _{11}}({{\bar{\varrho }}})$, it holds that $\rho (t,\theta ) d\theta \in {\mathcal {B}}_{\varepsilon }({\bar{\varrho }})$ for all $0\le t \le T'$ and that $\rho _t\in {\mathbb {B}}_\varepsilon ({\bar{\rho }})$ for all $T\le t \le T'$, where $\rho _{t}(\theta )=\rho (t,\theta )$ is the unique weak solution of the Cauchy problem (2.4) with initial condition $\rho _{0}$.

Proof

Fix $\varepsilon >0$ and $0<T<T'$. Let $\zeta _1 = (1/3) \varepsilon e^{-C_0T'}$, where $C_0$ is the constant appearing in Lemma 7. Fix a density profile $\rho _{0}:{\mathbb {T}}\rightarrow [0,1]$, and let $\pi _t(d\theta ) = \rho (t,\theta ) d\theta$, where $\rho _{t}$ is the unique weak solution of the Cauchy problem (2.4) with initial condition $\rho _{0}$. Recall the definition of the complete orthogonal normal basis $\{ e_k;k\in \mathbb {Z}\}$ introduced just before (2.1). By (2.1) and (3.1), for any $t\ge 0$,

$$\begin{aligned} d(\pi _t, {{\bar{\varrho }}}) \;\le \; d(\pi _0, {{\bar{\varrho }}}) \;+\; \sum _{k\in {\mathbb {Z}}} \frac{1}{2^{|k|}}\, \Big | \dfrac{1}{2} \int _{0}^{t} ds \, \langle \rho _{s}, \varDelta e_k \rangle + \int _{0}^{t} ds \, \langle F(\rho _{s}), e_k \rangle \Big |. \end{aligned}$$

The first term on the right hand side is bounded by $\varepsilon /2$ if $\rho _0 \in {\mathcal {B}}_{\zeta _2}({{\bar{\varrho }}})$, where $\zeta _2=\varepsilon /2$, while the second one is less than or equal to

$$\begin{aligned} t \sum _{k\in {\mathbb {Z}}} \frac{1}{2^{|k|}}\, \big \{ C_0 + (2\pi k)^2 \big \} \;=\; C_0 \,t, \end{aligned}$$

because $\rho _s$ is bounded by 1, F by a constant $C_0$ and $\Vert e_k\Vert _2 =1$. Hence, if $T_1=\varepsilon /2C_0$ and $\pi _{0} \in {\mathcal {B}}_{\zeta _2}({{\bar{\varrho }}})$,

$$\begin{aligned} \pi _t \;\in \; {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}) \text { for all } 0\le t\le T_1. \end{aligned}$$

(3.8)

We turn to the $L^2$-estimate. From the Eq. (3.2), we have

$$\begin{aligned} \Vert \rho _{t}-{\bar{\rho }}\Vert _{2}&\;\le \; \Vert P_{t}(\rho _{0}-{\bar{\rho }})\Vert _{2}+ \int _{0}^{t} \big \Vert P_{t-s} \big [F(\rho _{s})-F({\bar{\rho }})\big ] \, \big \Vert _{2} \, ds \nonumber \\&\;\le \; \Vert P_{t} (\rho _{0}-{\bar{\rho }}) \Vert _{2} + t\Vert F'\Vert _{\infty }, \end{aligned}$$

(3.9)

since the operator norm of $P_{t}$ is equal to 1. Let ${\tilde{\rho }}=\rho _{0}-{\bar{\rho }}$ and, for each $t>0$, ${\tilde{\rho }}_{t}=P_{t}{\tilde{\rho }}$. It is easy to see that, for any $k\in \mathbb {Z}$,

$$\begin{aligned} \langle {\tilde{\rho }}_{t}, e_{k}\rangle \;=\; \langle {\tilde{\rho }}, P_{t}e_{k} \rangle \;=\; e^{-2\pi ^{2}k^{2}t}\langle {\tilde{\rho }}, e_{k} \rangle . \end{aligned}$$

Therefore, from Parseval’s relation,

$$\begin{aligned} \Vert {\tilde{\rho }}_{t}\Vert ^{2}_{2}\;=\; \sum _{k\in \mathbb {Z}}e^{-4\pi ^{2}k^{2}t}\langle {\tilde{\rho }}, e_{k}\rangle ^{2}. \end{aligned}$$

(3.10)

Set $T_2:= \min \{ (\zeta _1/2\Vert F'\Vert _{\infty }), T_1, T\}$ and choose a large enough positive integer $k_{1}$ so that

$$\begin{aligned} \sum _{|k|>k_1} e^{-4\pi ^{2}k^{2}T_2} \;\le \;\zeta ^{2}_1/8. \end{aligned}$$

To estimate the first terms of the series, observe that

$$\begin{aligned} \sum _{|k|\le k_1} e^{-4\pi ^{2}k^{2}t} \langle {\tilde{\rho }}, e_{k}\rangle ^{2} \;\le \; 4^{k_1} \Big ( \sum _{|k|\le k_1} 2^{-|k|} \, \big | \langle {\tilde{\rho }}, e_{k}\rangle \big | \Big )^2 \;\le \; 4^{k_1} d(\rho _0,{\bar{\rho }})^2. \end{aligned}$$

Hence, if we set $\zeta _3 = \zeta _1/2^{k_1+2}$, this last expression is bounded by $\zeta ^2_1/16 \le \zeta ^2_1/8$ provided $\rho _{0}(\theta )d\theta$ belongs to ${\mathcal {B}}_{\zeta _3}({{\bar{\varrho }}})$. Therefore, by (3.9), (3.10) and the choice of $\zeta _3$,

$$\begin{aligned} \Vert \rho _{T_2}-{\bar{\rho }}\Vert _{2} \;\le \; \zeta _1 \end{aligned}$$

(3.11)

if $\pi _{0} \in {\mathcal {B}}_{\zeta _3}({{\bar{\varrho }}})$.

Let $\delta _{11}=\min \{\zeta _2, \zeta _3\}$, and note that $\delta _{11}$ depends only on $\varepsilon$, T, $T'$. By (3.8) and (3.11), and since $T_2\le T_1$, for all $\pi _{0} \in {\mathcal {B}}_{\delta _{11}}({{\bar{\varrho }}})$,

$$\begin{aligned} \pi _t \;\in \; {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}) \text { for all } 0\le t\le T_2 \quad \text {and}\quad \Vert \rho _{T_2}-{\bar{\rho }}\Vert _{2} \;\le \; \zeta _1. \end{aligned}$$

By Lemma 7, by the previous estimate and by definition of $\zeta _1$, for all $T_2\le t\le T'$, $\pi _{0} \in {\mathcal {B}}_{\delta _{11}}({{\bar{\varrho }}})$,

$$\begin{aligned} \Vert \rho _{t}-{\bar{\rho }}\Vert _{2} \;\le \; e^{C_0 (t-T_2)} \Vert \rho _{T_2}-{\bar{\rho }}\Vert _{2} \;\le \; e^{C_0 T'} \zeta _1 \;\le \; \varepsilon /3. \end{aligned}$$

Moreover, by this bound and by (2.2), for all $T_2\le t\le T'$, $\pi _{0} \in {\mathcal {B}}_{\delta _{11}}({{\bar{\varrho }}})$,

$$\begin{aligned} d(\pi _t, {{\bar{\varrho }}}) \;\le \; 3\, \Vert \rho _{t}-{\bar{\rho }}\Vert _{2} \;\le \; \varepsilon . \end{aligned}$$

This completes the proof of the lemma since $T_2\le T$. $\square$

The previous results permit to strengthen Lemma 6.

Lemma 9

Let $\varepsilon >0$, let $\alpha$ be an attractor of the ODE (3.4), and let ${{\bar{\varrho }}}_\alpha (d\theta ) = {\bar{\rho }}_\alpha (\theta ) d\theta$, ${\bar{\rho }}_\alpha (\theta )=\alpha$, $\theta \in {\mathbb {T}}$. There exists $\delta _{12}=\delta _{12}(\varepsilon ,\alpha )>0$ such that for any density profile $\gamma : {\mathbb {T}} \rightarrow [0,1]$ such that $\gamma (\theta ) d\theta \in {\mathcal {B}}_{\delta _{12}}({{\bar{\varrho }}}_\alpha )$, $\rho _t$ converges in the sup norm to ${\bar{\rho }}_\alpha$, as $t\uparrow \infty$, where $\rho _t(\theta ) = \rho (t,\theta )$ is the unique weak solution of (2.4) with initial condition $\gamma$. Moreover, $\pi _t(d\theta ) = \rho (t,\theta ) d\theta$ belongs to ${\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_\alpha )$ for all $t\ge 0$.

Proof

Fix $\varepsilon >0$. Denote by $\zeta _1$ the constant $\delta _{10} = \delta _{10}(\varepsilon , \alpha )$ provided by Lemma 6. Let $\zeta _2=\min \{\zeta _1, \varepsilon \}$, and let $\delta _{12}$ be the constant $\delta _{11} = \delta _{11}(\zeta _2, 1/2, 2)$ provided by Lemma 8 with ${\bar{\rho }} = {\bar{\rho }}_\alpha$.

Fix $\gamma : {\mathbb {T}} \rightarrow [0,1]$ such that $\gamma (\theta ) d\theta \in {\mathcal {B}}_{\delta _{12}}({{\bar{\varrho }}}_\alpha )$. Denote by $\rho _t(\theta ) = \rho (t,\theta )$ the weak solution of the hydrodynamic equation with initial condition $\gamma$. By Lemma 8, $\Vert \rho _1 - \bar{\rho }_\alpha \Vert _2 \le \zeta _1$ and $\pi _t(d\theta ) = \rho (t,\theta ) d\theta$ belongs to ${\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_\alpha )$ for all $t\le 2$.

Since $\Vert \rho _1 - \bar{\rho }_\alpha \Vert _2 \le \zeta _1$, by Lemma 6, $\rho _t$ converges in the sup norm to ${\bar{\rho }}_\alpha$ as $t\uparrow \infty$, and $\Vert \rho _t - \alpha \Vert _\infty \le \varepsilon$ for all $t\ge 2$. In particular, $\pi _t(d\theta ) = \rho (t,\theta ) d\theta$ belongs to ${\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_\alpha )$ for all $t\ge 2$. $\square$

4 The dynamical rate function

We present in this section some features of the dynamical rate function needed to prove the properties of the static rate function stated in the next section. The main result of the section asserts that a trajectory can not remain too long far in the $L^2$-topology from all stationary solutions of the hydrodynamic equation without paying a fixed positive cost.

The first four lemmata have been proved in [31, Section 4] for the rate functional $I_{T}(\cdot |\gamma )$. The same arguments apply the functional $I_{T}$. The first three extract information on the trajectory $\pi (t,d\theta )$ from the finiteness of the large deviations rate functional. Lemma 10 states that a trajectory with finite rate function is a continuous path in $D([0,T],\mathcal {M}_+)$. This lemma is repeatedly used in the rest of this paper, and therefore, is used without any further mention.

Lemma 10

Fix $T>0$. Let $\pi$ be a path in $D([0,T], \mathcal {M}_+)$ such that $I_T(\pi )$ is finite. Then $\pi$ belongs to $C([0,T], \mathcal {M}_{+,1})$.

Lemma 11 states that the density $\rho (t,\theta )$ of a trajectory $\pi (t,d\theta )$ with finite rate function belongs to ${\mathcal {H}}_1$ for almost all t, where ${\mathcal {H}}_1$ represents the space of functions $f: {\mathbb {T}} \rightarrow {\mathbb {R}}$ which have a general derivative in $L^2({\mathbb {T}})$.

Lemma 11

There exists a finite constant $C_0>0$ such that for any $T>0$ and for any path $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,T],{\mathcal {M}}_{+,1})$ with finite energy,

$$\begin{aligned} {\mathcal {E}}_{T}(\rho ):=\int _0^{T} dt\ \int _{\mathbb {T}} d\theta \ \frac{|\nabla \rho (t,\theta )|^{2}}{\chi (\rho (t,\theta ))} \;\le \; C_0\,\{ I_{T}(\pi )+ T + 1\}. \end{aligned}$$

The next result characterizes the weak solutions of the hydrodynamic equation as the trajectories at which the dynamical large deviations rate functional vanishes.

Lemma 12

Fix $T>0$. The density $\rho$ of a path $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,T], \mathcal {M}_{+,1})$ is the weak solution of the Cauchy problem (2.4) with initial profile $\gamma$ if and only if $I_{T}(\pi |\gamma ) = 0$. Moreover, in that case

$$\begin{aligned} {\mathcal {E}}_{T}(\rho ) \;<\; \infty . \end{aligned}$$

The next result is extremely useful. In the expression of the functionals $J_{T,G}$ and $I_T$, terms appearing such as $\int _0^T \langle B(\rho _t), G_t\rangle \, dt$, where G is a smooth function, are not continuous for the weak topology, but only for the $L^1$-topology. The next lemma establishes that if the cost of a sequence $\pi ^n$ of trajectories is uniformly bounded and if this sequence converges weakly to some trajectory $\pi$, then the sequence of density profiles converges in $L^2$-topology. The proof of this result follows from the computations presented in the proof of Theorem 4.7 in [31].

Lemma 13

Fix $T>0$. Let $\{\pi ^{n}(t,d\theta )=\rho ^{n}(t,\theta )d\theta :n\ge 1\}$ be a sequence of trajectories in $D([0,T], \mathcal {M}_{+,1})$. Assume that there exists a finite constant C such that

$$\begin{aligned} \sup _{n\ge 1} \, I_{T}(\pi ^{n}) \;\le \; C. \end{aligned}$$

If $\rho ^{n}$ converges to $\rho$ weakly in $L^{2}(\mathbb {T}\times [0,T])$, then $\rho ^{n}$ converges to $\rho$ strongly in $L^{2}(\mathbb {T}\times [0,T])$.

Recall the definition of the neighborhoods ${\mathbb {B}}_{\delta }(\rho )$ and ${\mathcal {B}}_{\delta }(\varrho )$ introduced just before the statement of Lemma 8. For each $\delta >0$ and $T>0$, denote by ${\mathbb {D}}_{T,\delta }$ the set of trajectories $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,T],\mathcal {M}_{+,1})$ such that $\rho _{t}\notin {\mathbb {B}}_{\delta }({\bar{\rho }})$ for all $0\le t\le T$ and ${\bar{\rho }}\in S$.

Next lemma states that a trajectory can not stay a long time interval far, in the $L^2$-topology, from all stationary solutions of the hydrodynamic equation without paying an appreciable cost. This result plays a fundamental role in the proof of the lower semicontinuity of the functional W. To enhance its interest, note that $L^2$-neighborhoods are much thinner than the neighborhoods of the weak topology.

Lemma 14

For every $\delta >0$ there exists $T=T(\delta )>0$ such that

$$\begin{aligned} \inf _{\pi \in {\mathbb {D}}_{T,\delta }} I_{T}(\pi ) \;>\; 0. \end{aligned}$$

Proof

Assume that the assertion of the lemma is false. Then, there exists some $\delta >0$ such that, for any $n\in \mathbb {N}$,

$$\begin{aligned} \inf _{\pi \in {\mathbb {D}}_{n,\delta }} I_{n}(\pi ) \;=\; 0. \end{aligned}$$

In this case there exists a sequence of trajectories $\{\pi ^{n}(t,d\theta )=\rho ^{n}(t,\theta )d\theta : n\ge 1\}$, $\pi ^n \in {\mathbb {D}}_{n,\delta }$, such that $I_{n}(\pi ^{n})\le 1/n$. Since $I_{T}$ has compact level sets, by using a Cantor’s diagonal argument and passing to a subsequence if necessary, we obtain a path $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D(\mathbb {R}_{+}, {\mathcal {M}}_{+,1})$ such that $\pi ^{n}$ converges to $\pi$ in $D([0,T], {\mathcal {M}}_{+})$ for any $T>0$. Moreover, by Lemma 13, $\rho ^{n}$ converges to $\rho$ strongly in $L^{2}([0,T]\times \mathbb {T})$ for all $T>0$.

Since $I_T$ is lower semicontinuous, $I_{T}(\pi )=0$ for all $T>0$. By Lemma 12, the density of $\pi$, denoted by $\rho$ so that $\pi (t,d\theta ) = \rho (t,\theta ) d\theta$, is the unique weak solution of the Eq. (2.4) with initial condition $\rho (0,\cdot )$. Hence, by Lemma 1, $\rho _{t}$ converges in $C^2({\mathbb {T}})$ to some density profile $\rho _{\infty } \in S$. Therefore, there exists some $T_0>0$ such that

$$\begin{aligned} \Vert \rho _{t}-\rho _{\infty }\Vert _{2}\;\le \; \delta /2, \end{aligned}$$

for any $t\ge T_0$. Hence, since $\pi ^n$ belongs to ${\mathbb {D}}_{n,\delta }$, for $n\ge T_0+1$

$$\begin{aligned} \int _{0}^{T_0+1}\Vert \rho _{t}^{n}-\rho _{t}\Vert _{2} \, dt&\;\ge \; \int _{T_0}^{T_0+1}\Vert \rho _{t}^{n}-\rho _{t}\Vert _{2} \, dt \\&\;\ge \; \int _{T_0}^{T_0+1}\big (\Vert \rho _{t}^{n}-\rho _{\infty }\Vert _{2} - \Vert \rho _{t}-\rho _{\infty }\Vert _{2}\big ) \, dt \\&\;\ge \; \delta - \delta /2 = \delta /2, \end{aligned}$$

which contradicts the strong convergence of $\rho ^{n}$ to $\rho$ in $L^{2}([0,T_0+1]\times \mathbb {T})$ and we are done. $\square$

Analogously, for each $\delta >0$ and $T>0$, denote by ${\mathcal {D}}_{T,\delta }$ the set of trajectories $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,T],{\mathcal {M}}_{+,1})$ such that $\pi _{t}\notin {\mathcal {B}}_{\delta }({{\bar{\varrho }}})$ for all $0\le t\le T$ and ${{\bar{\varrho }}}\in \mathcal {M}_\mathrm{sol}$. A similar result also holds for the set ${\mathcal {D}}_{T,\delta }$.

Corollary 1

For every $\delta >0$, there exists $T>0$ such that

$$\begin{aligned} \inf _{\pi \in {\mathcal {D}}_{T,\delta }} I_{T}(\pi ) \;>\; 0. \end{aligned}$$

Proof

The assertion follows from Lemma 14 and the fact that

$$\begin{aligned} \{\varrho (d\theta )=\rho (\theta )d\theta : \rho \in {\mathbb {B}}_{\delta }({\bar{\rho }})\} \subset {\mathcal {B}}_{3 \delta }({\bar{\rho }}), \end{aligned}$$

for every ${\bar{\rho }}\in S$ and every $\delta >0$ in view of (2.2). $\square$

In the proof of the static large deviations principle, it will be useful to estimate $V_i(\varrho )$ for some measure $\varrho (d\theta ) = \gamma (\theta ) d\theta$. If $\gamma$ is a smooth density profile, this can be achieved by joining ${{\bar{\varrho }}}_{i}$ to $\varrho$ through a linear interpolation $\pi _t = (1-t/T) {{\bar{\varrho }}}_{i} + (t/T) \varrho$, $T>0$, and by estimating the cost of the path $\pi$. This is the content of Lemma 17. For a general measure $\varrho (d\theta ) = \gamma (\theta ) d\theta$, we need first to smooth the density profile $\gamma$. We use the hydrodynamic equation to do that. Fix $\varepsilon >0$ small, and denote by $\rho (t,\theta )$ the solution of the hydrodynamic equation starting from $\gamma$, $0\le t\le \varepsilon$. By Proposition 2, $\rho (\varepsilon , \cdot )$ is smooth. We may use the first part of this argument to joint ${{\bar{\varrho }}}_{i}$ to $\rho (\varepsilon , \theta ) d\theta$. To connect $\rho (\varepsilon , \theta ) d\theta$ to $\gamma (\theta ) d\theta$ we use the backward path $\hat{\pi }_t(d\theta ) = \rho (\varepsilon -t, \theta ) d\theta$, $0\le t\le \varepsilon$. The cost of this path is estimated in the next lemma.

Lemma 15

There exists a constant $C_0>0$ such that for any $T>0$, any weak solution $\rho$ of (2.4), and any classical solution ${\bar{\rho }}$ to the Eq. (2.5),

$$\begin{aligned} I_T(\pi ) \;\le \; C_0 \{T + \Vert \rho _T - {\bar{\rho }}\Vert _1 + \Vert \rho _0 - {\bar{\rho }}\Vert _{1}\}, \end{aligned}$$

where $\pi$ is the trajectory defined by $\pi (t,d\theta ) =\rho (T-t,\theta )d\theta$.

Proof

For any test function $G\in C^{1,2} ([0,T]\times \mathbb {T})$, $J_{T,G}(\pi )$ can be rewritten as

$$\begin{aligned} J_{T,G} (\pi )&= \int _0^Tdt\ \langle \nabla \rho _t, \nabla {\widehat{G}}_t\rangle - \frac{1}{2}\int _0^Tdt\ \langle \chi (\rho _t), (\nabla {\widehat{G}}_t)^2\rangle \\&\quad - \int _0^Tdt\ \langle B(\rho _t), e^{{\widehat{G}}_t}+ {\widehat{G}}_t -1\rangle - \int _0^Tdt\ \langle D(\rho _t), e^{-{\widehat{G}}_t}- {\widehat{G}}_t -1\rangle , \end{aligned}$$

where ${\widehat{G}}(t,\theta ) = G(T-t,\theta )$. The first line on the right hand side is bounded above by

$$\begin{aligned} \frac{1}{2} \int _{0}^{T}dt \int _{\mathbb {T}} d\theta \ \dfrac{|\nabla \rho (t,\theta )|^{2}}{\chi (\rho (t,\theta ))} \;=\; \frac{1}{2}\, {\mathcal {E}}_T(\rho ). \end{aligned}$$

(4.1)

Since for any $0<\rho <1$ and any $a\in {\mathbb {R}}$

$$\begin{aligned} -B(\rho )e^a +D(\rho ) a +D(\rho ) \;\le \; D(\rho ) \log (D(\rho )/B(\rho )), \\ -D(\rho )e^{-a} -B(\rho ) a + B(\rho ) \;\le \; B(\rho ) \log (B(\rho )/D(\rho )), \end{aligned}$$

and since $B(\rho ) = (1-\rho ) {\widehat{B}}(\rho )$, $D(\rho ) = \rho {\widehat{D}}(\rho )$, where ${\widehat{B}}(\rho )$, ${\widehat{D}}(\rho )$ are the strictly positive functions introduced in (3.3), the second line on the right hand side is bounded above by

$$\begin{aligned} {\begin{matrix} &{}\int _0^Tdt\ \langle D(\rho _t)\log (D(\rho _t)/B(\rho _t)) + B(\rho _t)\log (B(\rho _t)/D(\rho _t))\rangle \\ &{}\quad \;\le \; C_0 T - \int _0^Tdt\ \langle D(\rho _t)\log (1-\rho _t)+ B(\rho _t)\log (\rho _t)\rangle \end{matrix}} \end{aligned}$$

(4.2)

To estimate this last term, let $h(x) = x\log x + (1-x) \log (1-x)$ and note that

$$\begin{aligned} \partial _t h(\rho _t) \;&=\; [\log (\rho _t)-\log (1-\rho _t)]\, \partial _t\rho _t\\&=\; [\log (\rho _t)-\log (1-\rho _t)]\, [\frac{1}{2}\varDelta \rho _t + F(\rho _t)]. \end{aligned}$$

This equation is justified since, by Proposition 2, any weak solution of the Eq. (2.4) is smooth for $t>0$. Therefore,

$$\begin{aligned}&- \int _0^Tdt \ \langle D(\rho _t)\log (1-\rho _t) + B(\rho _t)\log (\rho _t)\rangle \;=\; - \; \langle h(\rho _T)\rangle \;+\; \langle h(\rho _0)\rangle \\&\qquad -\frac{1}{2}\, {\mathcal {E}}_T(\rho )- \int _0^Tdt \ \langle D(\rho _t)\log (\rho _t)+ B(\rho _t)\log (1-\rho _t) \rangle . \end{aligned}$$

Adding and subtracting $\langle h({\bar{\rho }})\rangle$, since ${\bar{\rho }}$ takes value in a compact interval of (0, 1), the first two terms on the right hand side can be bounded above by

$$\begin{aligned} C_0 \{\Vert \rho _T - {\bar{\rho }}\Vert _1 + \Vert \rho _0 - {\bar{\rho }}\Vert _{1}\}, \end{aligned}$$

for some $C_0>0$. Since the last term in the penultimate displayed formula is bounded by $C_0T$, we have shown that (4.2) is less than or equal to

$$\begin{aligned} -\frac{1}{2}\, {\mathcal {E}}_T(\rho ) \;+\; C_0 \big \{ T + \Vert \rho _T - {\bar{\rho }}\Vert _1 + \Vert \rho _0 - {\bar{\rho }}\Vert _{1}\big \}. \end{aligned}$$

This estimate together with (4.1) completes the proof of the lemma. $\square$

The last result of this section states that the cost to move inside a set of static solutions is zero.

Lemma 16

Fix $1\le i\le l$. For all ${{\bar{\varrho }}}_1, {{\bar{\varrho }}}_2\in {\mathcal {M}}_i$,

$$\begin{aligned} \inf \big \{ I_T (\pi |{{\bar{\varrho }}}_1) : T>0 \,,\, \pi \in D([0,T], {\mathcal {M}}_+) \,,\, \pi _T = {{\bar{\varrho }}}_2 \} \;=\; 0. \end{aligned}$$

Proof

If ${\mathcal {M}}_i$ is a singleton, then the conclusion is clear. Assume that ${\mathcal {M}}_i$ is not a singleton and fix ${{\bar{\varrho }}}_1$, ${{\bar{\varrho }}}_2\in {\mathcal {M}}_i$ so that ${{\bar{\varrho }}}_k(d\theta ) = {\bar{\rho }}_k(\theta ) d\theta$, $k=1$, 2, and ${\bar{\rho }}_2 (\theta ) = {\bar{\rho }}_1(\theta +\theta _0)$ for some $0<\theta _0<1$.

Fix $a>0$ small, and let $\rho (t,\theta ) = {\bar{\rho }}_1(\theta + a t)$ so that $\rho (0, \cdot ) = {\bar{\rho }}_1 (\cdot )$, $\rho (\theta _0/a, \cdot ) = {\bar{\rho }}_2 (\cdot )$. Let $T=\theta _0/a$, $\pi _t(d\theta ) = \rho (t,\theta ) d\theta$. Since $\partial _t \rho = a \nabla \rho$ and $(1/2) \varDelta \rho + F(\rho ) = 0$, an integration by parts gives that for any smooth function $G:[0,T] \times {\mathbb {T}} \rightarrow {\mathbb {R}}$,

$$\begin{aligned} J_{T,G}(\pi ) \; =\;&- \, \int _0^T dt\, \langle a \rho _t, \nabla G_t \rangle - \frac{1}{2} \int _0^T dt\, \langle \chi (\rho _t), (\nabla G_t)^2 \rangle \\&-\; \int _0^T dt\, \langle B(\rho _t), e^{G_t} -1 - G_t \rangle -\, \int _0^T dt\, \langle D(\rho _t), e^{-G_t} -1 + G_t \rangle . \end{aligned}$$

The second line is negative, while the first one, by Young’s inequality and by Lemma 4, is less than or equal to

$$\begin{aligned} \dfrac{a^2T}{2} \int _{{\mathbb {T}}} \frac{{\bar{\rho }}_1 (\theta )}{1- {\bar{\rho }}_1 (\theta )}\, d\theta \;\le \; C_0 \, a\, \theta _0. \end{aligned}$$

To complete the proof it remains to let $a\rightarrow 0$. $\square$

5 The static rate functional W

In this section, we present some properties of the quasi-potential W. The main result asserts that the functional W, introduced in (2.14), is lower semicontinuous for the weak topology.

The first main result states that W is continuous at each measure ${{\bar{\varrho }}}_{i} \in {\mathcal {M}}_i$ in the $L^2$-topology. The second one states that W is lower semicontinuous in the weak topology.

We start with an estimate of $V_i(\varrho )$ for measures $\varrho (d\theta ) = \gamma (\theta ) d\theta$ whose density is close to ${\mathcal {M}}_i$ in the $L^2$-topology. This estimate together with Lemma 15 will allow us to prove that $V_i$ is continuous for the $L^2$-topology.

Let ${\mathbb {D}}$ be the space of measurable functions on $\mathbb {T}$ bounded below by 0 and bounded above by 1, endowed with the $L^2$-topology:

$$\begin{aligned} {\mathbb {D}} \;=\; \{\rho :\mathbb {T}\rightarrow [0,1] \;:\; 0\le \rho (\theta ) \le 1\;a.e.\ \theta \in \mathbb {T}\}. \end{aligned}$$

For each $1\le i \le l$, let ${\mathbb {V}}_i:{\mathbb {D}}\rightarrow {\mathbb {[}}0,+\infty ]$ be the functional given by ${\mathbb {V}}_i(\rho ) = V_i(\rho (\theta )d\theta )$. Note that the topology of $\mathcal {M}_{+,1}$ is the weak toplogy, while the one of ${\mathbb {D}}$ is the $L^2$-toplogy. Recall that we denote by ${\mathcal {H}}_1$ the Sobolev space of functions G with generalized derivatives $\nabla G$ in $L^{2}(\mathbb {T})$. For each $h>0$ and each $\delta >0$, let ${\mathbb {D}}^h_{\delta }$ be the subset of ${\mathbb {D}}$ consisting of those profiles $\rho$ satisfying the following conditions:

• (A) $\rho \in {\mathcal {H}}_1$ and $\displaystyle \int _{\mathbb {T}}(\nabla \rho (\theta ))^2d\theta \le h$.

• (B) $\delta \le \rho (\theta )\le 1-\delta$ a.e. in $\mathbb {T}$.

Lemma 17

For each $1\le i\le l$, $h>0$, $\delta >0$ and an increasing $C^1$-diffeomorphism $\alpha :[0,1]\rightarrow [0,1]$, there exist constants $C_{1}=C_{1}(\delta , h)>0$ and $C_{2}=C_{2}(\delta , \alpha )>0$ such that for any $\rho$ in ${\mathbb {D}}_{\delta }^{h}$ and ${\bar{\rho }}_i(\theta )d\theta$ in ${\mathcal {M}}_i$

$$\begin{aligned} {\mathbb {V}}_i(\rho ) \;\le \; C_{1} \int _{0}^{1} \alpha (t)^{2} \, dt \;+\; C_{2} \, \Vert \rho -{\bar{\rho }}_{i}\Vert _{1}. \end{aligned}$$

Proof

Fix $1\le i\le l$, $h>0$, $\delta >0$ and let $\alpha :[0,1]\rightarrow [0,1]$ be an increasing $C^1$-diffeomorphism. Let $\rho \in {\mathbb {D}}_{\delta }^{h}$ and ${\bar{\rho }}_i(\theta )d\theta \in {\mathcal {M}}_i$. Consider the path $\pi _{t}^{\alpha }(d\theta )=\rho ^{\alpha }(t,\theta )d\theta$ in $C([0,1], \mathcal {M}_{+})$ with density given by $\rho _{t}^{\alpha } = (1-\alpha (t)){\bar{\rho }}_i+\alpha (t)\rho$. It is clear that $\pi ^{\alpha }$ belongs to $D([0,1],\mathcal {M}_{+,1})$, and it follows from condition (A) that ${\mathcal {Q}}_{1}(\pi ^{\alpha })$ is finite. From the definition of $\rho ^{\alpha }$ it follows that $\nabla \rho _{t}^{\alpha }=\alpha (t)(\nabla \rho -\nabla {\bar{\rho }}_{i})+\nabla {\bar{\rho }}_{i}$ and that $\partial _{t}\rho _{t}^{\alpha }=\alpha '(t)(\rho -{\bar{\rho }}_{i})$. Since $(1/2) \varDelta {\bar{\rho }}_{i} = -F({\bar{\rho }}_{i})$, $J_{1,G}(\pi ^{\alpha })$ can be rewritten as

$$\begin{aligned}&J_{1,G}(\pi ^\alpha )\;=\; \frac{1}{2} \int _{0}^{1}dt\ \Big \{ \alpha (t) \langle (\nabla \rho -\nabla {\bar{\rho }}_{i}), \nabla G_{t}\rangle - \langle \chi (\rho _{t}^{\alpha }), (\nabla G_{t})^{2}\rangle \Big \} \nonumber \\&\quad +\; \int _{0}^{1}dt \ \Big \langle \big \{\alpha '(t)(\rho -{\bar{\rho }}_{i})+F({\bar{\rho }}_{i}) \big \} G_{t}-B(\rho _{t}^{\alpha })(e^{G_{t}}-1) - D(\rho _{t}^{\alpha }) (e^{-G_{t}}-1) \Big \rangle . \end{aligned}$$

(5.1)

By Young’s inequality, the first term on the right hand side of (5.1) is bounded by

$$\begin{aligned} \frac{1}{8} \int _{0}^{1} \alpha (t)^2 \Big \langle \frac{(\nabla \rho -\nabla {\bar{\rho }}_{i})^2}{\chi (\rho _{t}^{\alpha })} \Big \rangle \, dt \;\le \; C_{1}\int _{0}^{1}\alpha (t)^{2}\ dt \end{aligned}$$

for some finite constant $C_{1}=C_{1}(\delta , h)$. To derive the last inequality we used the fact that ${\bar{\rho }}_i$ is bounded away from 0 and 1 and conditions (A) and (B) on $\rho$.

To conclude the proof it is enough to show that the second term on the right hand side of (5.1) is bounded by

$$\begin{aligned} C_{2}\Vert \rho -{\bar{\rho }}_{i}\Vert _{1} \, \end{aligned}$$

for some constant $C_{2}=C_{2}(\delta , \alpha )$.

Consider the function $\varPhi :\mathbb {R}\times (0,1)\times \mathbb {R}\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \varPhi (H, \rho , G)\;=\; HG-B(\rho )(e^{G}-1)-D(\rho )(e^{-G}-1). \end{aligned}$$

If we set $H_{t}=\alpha '(t)(\rho -{\bar{\rho }}_{i})+F({\bar{\rho }}_{i})$, it is clear that the second term on the right hand side of (5.1) can be expressed as

$$\begin{aligned} \int _{0}^{1}\langle \varPhi (H_{t}, \rho _{t}^{\alpha }, G_{t})\rangle \ dt. \end{aligned}$$

It follows from a straightforward computation that for any fixed $H \in \mathbb {R}$ and $\rho \in (0,1)$, the function $\varPhi (H, \rho , \cdot )$ reaches a maximum at

$$\begin{aligned} G(H,\rho )\;=\; \log \left( \dfrac{H+\sqrt{H^{2}+4B(\rho )D(\rho )}}{2B(\rho )}\right) . \end{aligned}$$

From condition (B) and Lemma 4, there exists a constant $c_{\delta }>0$ such that $c_{\delta }\le \rho ^{\alpha } \le 1-c_{\delta }$. On the other hand, since $\varPhi (H, \rho , 0)=0$ and since $G(F(\rho ), \rho )=0$, $\varPhi (F(\rho ), \rho , G(F(\rho ), \rho ))=0$ for any $\rho \in \mathbb {R}$. Therefore, as $(H,\rho ) \mapsto \varPhi (H,\rho ,G(H,\rho ))$ is a Lipschitz-continuous function on the interval $[-\Vert \alpha '\Vert _\infty -\Vert F\Vert _{\infty }, \Vert \alpha '\Vert _\infty +\Vert F\Vert _{\infty }]\times [c_{\delta }, 1-c_{\delta }]$,

$$\begin{aligned} \varPhi (H_{t}, \rho _{t}^{\alpha }, G_{t})&\;\le \; \varPhi (H_{t}, \rho _{t}^{\alpha }, G(H_{t},\rho _{t}^{\alpha })) \\&\;=\; \varPhi (H_{t}, \rho _{t}^{\alpha }, G(H_{t},\rho _{t}^{\alpha })) -\varPhi (F(\rho _{t}^{\alpha }), \rho _{t}^{\alpha }, G(F(\rho _{t}^{\alpha }),\rho _{t}^{\alpha })) \\&\;\le \; C_{2}|H_{t}-F(\rho _{t}^{\alpha })| \\&\;\le \; C_{2} \big \{ \alpha '(t) + \Vert F'\Vert _{\infty }\alpha (t)\big \} |\rho -{\bar{\rho }}_{i}| \end{aligned}$$

for some finite constant $C_{2}=C_{2}(\delta , \alpha )$. These bounds give the desired conclusion. $\square$

We are now in a position to prove that the functional $V_i$ is continuous in the $L^2$-topology.

Theorem 6

For each $1\le i\le l$, the function ${\mathbb {V}}_i$ is continuous at ${\bar{\rho }}_i$ in ${\mathbb {D}}$.

Proof

Fix $1\le i \le l$, and let $\{\rho ^{n} : n\ge 1\}$ be a sequence in ${\mathbb {D}}$ converging to ${\bar{\rho }}_{i}$. Denote by $\lambda ^{n}$, $n\ge 1$, the weak solution to the Eq. (2.4) with initial condition $\rho ^{n}$.

By Lemma 5, there exists a constant $C_{0}>0$, independent of n, such that

$$\begin{aligned} \int _0^1 dt \int _{\mathbb {T}} |(\nabla \lambda _{t}^{n})(\theta )|^{2} d\theta \;\le \; C_{0} \end{aligned}$$

for all $n\ge 1$. Fix $0<\zeta <1$. For each $n\ge 1$, there exists $\zeta \le T_{n}\le 2\zeta$ such that

$$\begin{aligned} \int _{\mathbb {T}} |(\nabla \lambda _{T_{n}}^{n})(\theta )|^{2} d\theta \;\le \; C_{0}/\zeta . \end{aligned}$$

Moreover, by Lemma 4, there exists a constant $0<c_{\zeta }<1/2$, independent of n, such that $c_{\zeta }\le \lambda _{T_{n}}^{n}(\theta ) \le 1-c_{\zeta }$ for all $n\ge 1$ and $\theta$ in $\mathbb {T}$. Therefore, the density profiles $\lambda _{T_{n}}^{n}$, $n\ge 1$, belong to the set ${\mathbb {D}}_{c_{\zeta }}^{C_{0}/\zeta }$ introduced just above Lemma 17.

By definition (2.11) of the functional $V_{i}$,

$$\begin{aligned} {\mathbb {V}}_{i}(\rho ^{n}) \;\le \; {\mathbb {V}}_{i}(\lambda ^{n}_{T_{n}}) \;+\; I_{T_{n}}(\pi ^{n}), \end{aligned}$$

where $\pi ^{n}(t,d\theta )=\lambda ^{n}(T_{n}-t,\theta )d\theta$, $0\le t\le T_n$. Therefore, it is enough to prove that

$$\begin{aligned} \limsup _{n\rightarrow \infty } {\mathbb {V}}_{i}(\lambda ^{n}_{T_{n}}) \;=\; \limsup _{\zeta \downarrow 0}\limsup _{n\rightarrow \infty } I_{T_{n}}(\pi ^{n}) \;=\; 0. \end{aligned}$$

Since $\lambda _{T_{n}}^{n}$ belongs to the set ${\mathbb {D}}_{c_{\zeta }}^{C_{0}/\zeta }$, by Lemma 17, there exist constants $C_{1}=C_{1}(\zeta )>0$ and $C_{2}=C_{2}(\zeta ,\alpha )>0$ such that

$$\begin{aligned} {\mathbb {V}}_{i}(\lambda ^{n}_{T_{n}}) \;\le \; C_{1}\int _{0}^{1}\alpha (t)^{2}\, dt \;+\; C_{2}\, \Vert \lambda _{T_{n}}^{n}-{\bar{\rho }}_{i}\Vert _{2}, \end{aligned}$$

for any increasing $C^1$-diffeomorphism $\alpha :[0,1]\rightarrow [0,1]$. By Lemma 7 and since $T_n\le 2\zeta \le 2$,

$$\begin{aligned} \Vert \lambda _{T_{n}}^{n}-{\bar{\rho }}_{i}\Vert _{2} \;\le \; C_0\, \Vert \rho ^{n}-{\bar{\rho }}_{i}\Vert _{2} \end{aligned}$$

(5.2)

for some finite constant $C_0>0$, independent of n, and whose value may change from line to line. Since $\rho ^{n}$ converges to ${\bar{\rho }}_{i}$ in $L^2$-topology,

$$\begin{aligned} \limsup _{n\rightarrow \infty }{\mathbb {V}}_{i}(\lambda ^{n}_{T_{n}}) \;\le \; C_{1}\inf _{\alpha } \Big \{\int _{0}^{1}\alpha (t)^{2}\, dt \Big \} \;=\; 0. \end{aligned}$$

It remains to prove that

$$\begin{aligned} \limsup _{\zeta \downarrow 0} \limsup _{n\rightarrow \infty } I_{T_{n}}(\pi ^{n}) \;=\; 0. \end{aligned}$$

(5.3)

By Lemma 15, by (5.2), and since $T_n\le 2\zeta$,

$$\begin{aligned} I_{T_{n}}(\pi ^{n}) \;\le \; C_0\, \Big \{T_{n} \,+\, \Vert \lambda _{T_{n}}^{n}-{\bar{\rho }}_{i}\Vert _{2} \,+\, \Vert \rho ^{n}-{\bar{\rho }}_{i}\Vert _{2} \Big \} \;\le \; C_0\, \big \{\zeta + \Vert \rho ^{n}-{\bar{\rho }}_{i}\Vert _{2} \big \}. \end{aligned}$$

To complete the proof of (5.3) and the one of the lemma, it remains to recall that $\rho ^n \rightarrow {\bar{\rho }}_{i}$ in $L^2({\mathbb {T}})$. $\square$

The proof of the next result is similar to the one of Proposition 4.9 in [21]. The lemma asserts that the functional $V_i$ is uniformly bounded in ${\mathcal {M}}_{+,1}$.

Lemma 18

The function W is finite if and only if $\varrho$ belongs to $\mathcal {M}_{+,1}$. Moreover, for all $1\le i\le l$,

$$\begin{aligned} \sup _{\varrho \in \mathcal {M}_{+,1}} V_i(\varrho ) \;<\; \infty , \end{aligned}$$

so that $\sup _{\varrho \in \mathcal {M}_{+,1}} W (\varrho ) < \infty$.

Proof

Fix $\varrho \in \mathcal {M}_{+}$, and suppose that $W(\varrho )<\infty$. By the definition (2.14) of the functional W, there exists $1\le i\le l$ such that $V_i(\varrho )<\infty$. Hence, by (2.11), there exist ${\bar{\rho }}\in {\mathcal {M}}_i$, $T<\infty$ and a trajectory $\pi _t(d\theta )$, $0\le t\le T$, such that $\pi _T = \varrho$, $I_T(\pi | {\bar{\rho }})<\infty$. By (2.10) and (2.9), $\pi \in D([0,T], {\mathcal {M}}_{+,1})$, proving that $\varrho = \pi _T$ belongs to ${\mathcal {M}}_{+,1}$, as claimed.

To prove the reciprocal assertion and the uniform bound, fix $1\le i \le l$, $\varrho \in \mathcal {M}_{+,1}$, $\varrho (d\theta ) = \rho (\theta ) d\theta$, and denote by $\lambda$ the weak solution of the Eq. (2.4) with initial condition $\rho$. By Lemmata 4 and 5, there exist constants $0<a<1/2$ and $C_{0}>0$ such that $a \le \lambda _{t}(\theta ) \le 1-a$ for all $t\ge 1$, $\theta \in {\mathbb {T}}$, and

$$\begin{aligned} \int _0^2 dt \int _{\mathbb {T}} |(\nabla \lambda _{t})(\theta )|^{2} d\theta \;\le \; C_{0}. \end{aligned}$$

In particular, there exists $1\le T\le 2$ such that $\lambda _{T}$ belongs to the set ${\mathbb {D}}_{a}^{C_{0}}$.

By the definition (2.11) of the functional $V_{i}$,

$$\begin{aligned} V_{i}(\rho ) \;\le \; V_{i}(\lambda _{T}) \;+\; I_{T}(\pi ), \end{aligned}$$

where $\pi (t,d\theta )=\lambda (T-t,\theta )d\theta$, $0\le t\le T$.

As $\lambda _{T}$ belongs to the set ${\mathbb {D}}_{a}^{C_{0}}$, by Lemma 17, $V_{i}(\lambda _{T})\le C_1$ for some finite constant $C_1$ which depends only on F. On the other hand, by Lemma 15 and since $\Vert \gamma \Vert _1 \le 1$ for all density profile $\gamma$, $I_{T}(\pi ) \le C_2$, which completes the proof of the lemma in view of the definition of the functional W. $\square$

We now turn to the proof that the functionals $V_i$ are lower semicontinuous for the weak topology. The idea of the proof is very simple. Let $\varrho ^n$ be a sequence converging to $\varrho$. Since, by Lemma 18, $V_i$ is finite, there exists a trajectory $\pi ^n_t$, $0\le t\le T_n$, such that $\pi ^n_0 \in {\mathcal {M}}_i$, $\pi ^n_{T_n} = \varrho ^n$, $V_i(\varrho ^n) \le I_{T_n} (\pi ^n) + 1/n \le C$. We will now use the lower semicontinuity of $I_T$ and the fact that the level sets are compact to conclude. If the sequence $T_n$ is uniformly bounded, say by T, we may add a piece of length $T-T_n$ to the trajectory $\pi ^n$ letting it to stay at $\pi ^n_0 \in {\mathcal {M}}_i$ in the time interval $[0,T-T_n]$. In this way, we obtain a new sequence, denoted by $\overline{\pi }^n_t$, of trajectories such that $\overline{\pi }^n_0 \in {\mathcal {M}}_i$, $\overline{\pi }^n_{T} = \varrho ^n$, $V_i(\varrho ^n) \le I_{T_n} (\pi ^n) + 1/n = I_{T} (\overline{\pi }^n) + 1/n \le C$. Since the level sets are compact, we may extract a converging subsequence. Denote by $\pi$ the limit and observe that $\pi _0 \in {\mathcal {M}}_i$, $\pi _T=\varrho$. By the lower semicontinuity and by definition of $V_i$, $V_i(\varrho ) \le I_{T} (\pi ) \le \liminf _n V_i(\varrho ^n)$, and we are done.

Of course, it might happen that the sequence $T_n$ is not bounded, and this is the main difficulty. In this case, we will use Lemma 14 to claim that the trajectory $\pi ^n$ may not spend too much time outside an $L^2$-neighborhood of a stationary solution. Hence, all the proof consists in replacing the long intervals of time at which the trajectory stays close to a stationary profile ${\bar{\rho }}_i$ by one which remains only a time interval of length 2. This is done by showing in Lemma 19 below that it is possible to go from a neighborhood of ${\bar{\rho }}_i$ to ${\bar{\rho }}_i$ in time 1 by paying a small cost and by using Theorem 6 to obtain a trajectory which goes from ${\bar{\rho }}_i$ to a neighborhood of ${\bar{\rho }}_i$ in time 1 by paying a small cost.

Lemma 19

Fix $1\le i\le l$ and ${{\bar{\varrho }}}_{i}(d\theta )={\bar{\rho }}_i(\theta )d\theta \in {\mathcal {M}}_i$. For any $\varepsilon >0$ there exists $\delta _{13}=\delta _{13}(\varepsilon )>0$ such that for any $\varrho (d\theta )=\gamma (\theta )d\theta$ in ${\mathcal {B}}_{\delta _{13}}({{\bar{\varrho }}}_{i})$ there exists a path $\pi (t,d\theta )=\rho (t,\theta )d\theta$ in $D([0,1], \mathcal {M}_{+})$ such that $\pi _{0}=\varrho$, $\pi _{1}={{\bar{\varrho }}}_{i}$, $I_{1}(\pi )\le \varepsilon$, and $\pi _t \in {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})$ for all $0\le t\le 1$.

Proof

Fix $1\le i\le l$, $\varepsilon >0$, and a density profile $\gamma : {\mathbb {T}}\rightarrow {\mathbb {R}}$. Let $\rho (t,\theta )$ be the unique weak solution of (2.4) with initial condition $\gamma$, and let $\pi _t(d\theta ) =\rho (t,\theta ) d\theta$.

Fix $0<\zeta _1<\varepsilon$, to be chosen later, and set $T= 1/4$, $T'=1/2$. Let $\zeta _2>0$ be the constant $\delta _{11}(\zeta _1, 1/4, 1/2)$ given by Lemma 8 for ${\bar{\rho }}= {\bar{\rho }}_i$. Assume that $\gamma \in {\mathcal {B}}_{\zeta _2} ({{\bar{\varrho }}}_{i})$. According to Lemma 8, $\rho _s(\theta )\, d\theta$ belongs to ${\mathcal {B}}_{\zeta _1} ({{\bar{\varrho }}}_{i})$ for all $0\le s\le 1/2$ and $\Vert \rho _s - {\bar{\rho }}_i\Vert _2 \le \zeta _1$ for all $1/4\le s\le 1/2$.

By Proposition 2, $\rho _t$ belongs to $C^\infty ({\mathbb {T}})$ for all $t>0$. By Lemma 4, there exists $a>0$, depending only on F, such that $a\le \rho (s,\theta ) \le 1-a$ for all $\theta \in {\mathbb {T}}$ and $1/4\le s\le 1/2$. On the other hand, by Lemma 5, there exists a finite constant $C_0$, depending only on F, such that

$$\begin{aligned} \int _0^{1/2} dt \int _{{\mathbb {T}}} |(\nabla \rho ) (t,\theta )|^2 \, d\theta \;\le \; C_0. \end{aligned}$$

In particular, there exists $T_1\in [1/4,1/2]$ such that

$$\begin{aligned} \int _{{\mathbb {T}}} |(\nabla \rho ) (T_1,\theta )|^2 \, d\theta \;\le \; 4 C_0, \end{aligned}$$

(5.4)

so that $\rho _{T_1}$ belongs to ${\mathbb {D}}^{4C_0}_a$

Recall that $T_1\le 1/2$. Let $\alpha : [0,1/2] \rightarrow [0,1]$ be an increasing $C^1$-diffeomorphism, and define the trajectory $\rho ^\alpha _t$, $T_1\le t\le T_1+1/2$, by $\rho ^\alpha (T_1+s, \theta ) = \alpha (s) {\bar{\rho }}_i + [1-\alpha (s)] \rho _{T_1}$, $0\le s \le$ 1/2. By a similar computation to the one presented in the proof of Lemma 17, and since $\Vert \rho _{T_1}-{\bar{\rho }}_{i}\Vert _{1} \le \Vert \rho _{T_1}-{\bar{\rho }}_{i}\Vert _{2} \le \zeta _1$

$$\begin{aligned} I_{[T_1,T_1+1/2]} (\rho ^\alpha ) \;\le \; C_1 \int _{0}^{1/2} [1-\alpha (t)]^{2} \, dt \; \Vert \nabla \rho _{T_1} - \nabla {\bar{\rho }}_{i}\Vert ^2_{2} \;+\; C_{2}(\alpha ) \, \zeta _1, \end{aligned}$$

where $C_1$ is a finite constant depending only on F, and $C_2(\alpha )$ one which also depends on $\alpha$. In view of (5.4),

$$\begin{aligned} I_{[T_1,T_1+1/2]} (\rho ^\alpha ) \;\le \; C_3 \int _{0}^{1/2} [1-\alpha (t)]^{2} \, dt \;+\; C_{2}(\alpha ) \, \zeta _1, \end{aligned}$$

for some finite constant $C_3$ independent of $\gamma$.

Choose an increasing $C^1$-diffeomorphism $\alpha : [0,1/2] \rightarrow [0,1]$ which turns the first term on the right hand side bounded by $\varepsilon /2$. Note that this diffeomorphism does not depend on $\gamma$. For this fixed $\alpha$, choose $\zeta _1$ small enough for the second term to be less than or equal to $\varepsilon /2$. To complete the proof of the first assertion of the lemma, juxtapose the trajectories $\rho _t$, $0\le t\le T_1$, $\rho ^\alpha _t$, $T_1\le t\le T_1+1/2$, and the constant one ${\bar{\rho }}_i$, $T_1 +1/2\le t\le 1$.

We turn to the assertion that $\pi _t \in {\mathcal {B}}_{\varepsilon } ({{\bar{\varrho }}}_{i})$ for all $0\le t\le 1$. By the second paragraph of the proof, and by definition of $T_1$, $\pi _t \in {\mathcal {B}}_{\zeta _1} ({{\bar{\varrho }}}_{i}) \subset {\mathcal {B}}_{\varepsilon } ({{\bar{\varrho }}}_{i})$ for all $0\le t\le T_1$. By definition of the trajectory $\rho ^\alpha$, $d(\pi _t, \bar{\varrho }_{i}) \le d(\pi _{T_1}, \bar{\varrho }_{i}) < \varepsilon$ for $T_1\le t \le T_1+1/2$. This completes the proof of the lemma since $\pi _t = \bar{\varrho }_{i}$ for $T_1+1/2 \le t\le 1$. $\square$

We have now all the elements to prove the main result of the section.

Theorem 7

For each $1\le i\le l$, $V_{i}$ is lower semicontinuous. In particular, the rate function W is also lower semicontinuous.

Proof

To keep notation simple, we prove the theorem in the case where all solutions of (2.5) are constant in space, or equivalently, assume that

$$\begin{aligned} {\mathcal {M}}_\mathrm{sol} \;=\; \{ {{\bar{\varrho }}}_{i}(d\theta )={\bar{\rho }}_id\theta : i=1,\ldots , l \}. \end{aligned}$$

It is not difficult to extend the argument to the general case by invoking Lemma 16.

Fix $1\le i \le l$, $q\in \mathbb {R}_{+}$, and let

$$\begin{aligned} {\mathcal {V}}_i^{(q)} \;=\; \{ \varrho \in \mathcal {M}_{+} : V_{i}(\varrho )\le q \}. \end{aligned}$$

By the proof of Lemma 18, ${\mathcal {V}}_i^{(q)}\subset \mathcal {M}_{+,1}$. We claim that ${\mathcal {V}}_i^{(q)}$ is a closed subset of $\mathcal {M}_{+}$. To see this, let $\{\varrho ^{n}(d\theta )=\rho ^{n}(\theta )d\theta : n\ge 1\}$ be a sequence in ${\mathcal {V}}_i^{(q)}$ converging to some $\varrho (d\theta )=\rho (\theta )d\theta$ in $\mathcal {M}_{+}$.

From (2.11), for each $n\ge 1$, there exist $T_n>0$ and a path $\pi ^{n}$ in $C([0,T_{n}], \mathcal {M}_{+,1})$ such that $\pi _{0}^{n}={{\bar{\varrho }}}_{i}$, $\pi _{T_{n}}^{n}=\varrho ^{n}$ and

$$\begin{aligned} I_{T_{n}}(\pi ^{n} | {{\bar{\varrho }}}_{i} ) \;\le \; V_{i}(\varrho ^{n}) + 1/n \;\le \; q+1. \end{aligned}$$

(5.5)

Assume first that the sequence $\{ T_{n} : n\ge 1\}$ is bounded above by some $T<\infty$. In this case, let ${\widehat{\pi }}^{n}$ be the trajectory which remains at ${{\bar{\varrho }}}_{i}$ in the time interval $[0, T-T_n]$ and then follows the trajectory $\pi _n$:

$$\begin{aligned} {\widehat{\pi }}^{n}_t \;=\; {\left\{ \begin{array}{ll} {{\bar{\varrho }}}_{i} &{} 0\le t\le T-T_n, \\ \pi _n(t-T+T_n) &{} T-T_n \le t\le T. \end{array}\right. } \end{aligned}$$

Note that ${\widehat{\pi }}^{n}_T = \varrho ^{n}$ for all $n\ge 1$, and that $I_{T}({\widehat{\pi }}^{n} | {{\bar{\varrho }}}_{i}) = I_{T_{n}}(\pi ^{n} | {{\bar{\varrho }}}_{i}) \le q+1/n$ for all $n\ge 1$.

By Theorem 3, the functional $I_{T}(\cdot | {{\bar{\varrho }}}_{i})$ has compact level sets and is lower semicontinuous. There exists, in particular, a subsequence $n_j$ and a trajectory $\pi \in D([0,T], {\mathcal {M}}_+)$ such that ${\widehat{\pi }}^{n_j} \rightarrow \pi$, $I_{T}(\pi | {{\bar{\varrho }}}_{i})\le q$. Since $\pi _T = \lim _j {\widehat{\pi }}^{n_j}_{T} = \lim _j \pi ^{n_j}_{T_{n_j}} = \lim _j \varrho ^{n_j} = \varrho$, by definition of $V_i$, $V_i(\varrho )\le q$, which completes the proof of the theorem in the case where the sequence $T_n$ is bounded.

Suppose now that the sequence $\{ T_{n} : n\ge 1\}$ is unbounded. In view of (5.5), by Lemma 14, the path $\pi ^n$ may not remain too long outside an $L^2$-neighborhood of one of the stationary profiles. We use this observation, Lemma 19 and Theorem 6, to construct from $\pi ^n$ a new path on a bounded time interval by replacing the long intervals of time in which $\pi ^n$ remained close to a stationary profile by a path defined in a time interval of length 2 which connects the entrance time in a neighborhood of a stationary profile ${\bar{\rho }}_i$ to ${\bar{\rho }}_i$ and from this profile to the exit time of the neighborhood. The details are given below.

Fix $\varepsilon >0$. By Theorem 6, there exists $\zeta _1$, such that $V_j(\pi ) \le \varepsilon$ if $\pi (d\theta ) = \rho (\theta ) d\theta$ and $\Vert \rho -{\bar{\rho }}_j\Vert _2\le \zeta _1$ for any $1\le j\le l$. Let $\zeta _2$ be the constant $\delta _{13}(\varepsilon )$ given by Lemma 19 and set $\zeta = \min \{\zeta _1, \zeta _2\}$.

Let ${\mathcal {L}}_{j}$, $1 \le j \le l$, be the closed $L^2$-neighborhood of ${\bar{\rho }}_{j}$: ${\mathcal {L}}_{j} = \{\varrho (d\theta )=\rho (\theta )d\theta : \rho \in {\mathbb {B}}_{\zeta } [{\bar{\rho }}_{j}]\}$, where ${\mathbb {B}}_{\zeta } [{\bar{\rho }}_{j}]$ represents the closure of ${\mathbb {B}}_{\zeta } ({\bar{\rho }}_{j})$, and let ${\mathcal {L}} = \cup _{1\le j\le l} {\mathcal {L}}_j$. Assume that $\zeta$ is sufficiently small so that $\{{\mathcal {L}}_j\}_{j=1}^l$ are mutually disjoint. Note that ${\mathcal {L}}_{j}$ is a closed subset of $\mathcal {M}_{+,1}$. We define a sequence of entrances and exit times associated to the sets ${\mathcal {L}}_j$. Recall that $\pi ^n_0= {{\bar{\varrho }}}_{i}$, and set $\tau ^n_1 =0$. Let $\sigma ^n_1$ be the last exit time from ${\mathcal {L}}_i$:

$$\begin{aligned} \sigma ^n_1 \;=\; \sup \big \{t\le T_n : \pi ^n_t \in {\mathcal {L}}_i \big \}. \end{aligned}$$

Note that the path $\pi ^n_t$ may visit several neighborhoods ${\mathcal {L}}_j$, $j\not = i$, in the time interval $[0, \sigma ^n_1]$, and that it does not return to ${\mathcal {L}}_i$ after $\sigma ^n_1$. Suppose that $\tau ^n_k$, $\sigma ^n_k$, $1\le k < p$, have already been introduced. Define

$$\begin{aligned} \tau ^n_p \;:=\; \inf \big \{\sigma ^n_{p-1}\le t\le T_{n}: \pi ^n_{t}\in {\mathcal {L}} \big \}, \quad \sigma ^n_p \;:=\; \sup \big \{t\le T_{n}: \pi ^n_{t}\in {\mathcal {L}}_{{\mathfrak {j}}(p)} \big \}, \end{aligned}$$

where ${\mathfrak {j}}(p)$ is the index of the neighborhood visited at time $\tau ^n_p$: ${\mathfrak {j}}(p)= a$ if $\pi ^n_{\tau ^n_p} \in {\mathcal {L}}_a$. By convention, if $\pi ^n_{t}\not \in {\mathcal {L}}$ for all $\sigma ^n_{p-1}\le t\le T_{n}$, we set $\tau ^n_{p'}$ and $\sigma ^n_{p'}$ to be $\infty$ for all $p'\ge p$ and we do not define ${\mathfrak {j}}(p)$. Note that ${\mathfrak {j}}(1) = i$ and that ${\mathfrak {j}}(p) \not = {\mathfrak {j}}(q)$ if $q\not = p$.

Denote by $S_n$ the set of neighborhoods visited by $\pi ^n$, $S_n = \{ {\mathfrak {j}}(p) : \tau _p(\pi ^{n})<\infty \} = : \{{\mathfrak {j}}(1), \dots , {\mathfrak {j}}(b)\}$. By the choice of $\zeta$ and by Lemma 19, there exist paths $\pi ^{1,+}$, $\pi ^{m,-}$, $\pi ^{m,+}$, $2\le m< b$, $\pi ^{b,-}$ such that

$$\begin{aligned} \pi _{0}^{k,-} \;=\; \pi ^n(\tau _{{\mathfrak {j}}(k)}),\quad \pi _{1}^{k,-} \;=\; {{\bar{\varrho }}}_{{\mathfrak {j}}(k)}, \quad \pi _{0}^{k,+} \;=\; {{\bar{\varrho }}}_{{\mathfrak {j}} (k)}, \quad \pi _{1}^{k,+}=\pi ^n(\sigma _{{\mathfrak {j}}(k)}), \end{aligned}$$

and such that $I_{1}(\pi _{1}^{1,+}) \le 2\varepsilon$, $I_{1}(\pi _{1}^{m,-}) + I_{1}(\pi _{1}^{m,+})\le 3\varepsilon$, $2\le m< b$, $I_{1}(\pi _{1}^{b,-}) \le \varepsilon$. Here and below, to keep notation simple, we denote sometimes $\pi ^n_t$ by $\pi ^n(t)$. If $\sigma ^n_b \le T_n$ there exists also a path $\pi ^{b,+}$, such that

$$\begin{aligned} \pi _{0}^{b,+} \;=\; {{\bar{\varrho }}}_{{\mathfrak {j}} (b)}, \quad \pi _{1}^{b,+}=\pi ^n(\sigma _{{\mathfrak {j}}(b)}), \end{aligned}$$

and such that $I_{1}(\pi _{1}^{b,+}) \le 2\varepsilon$.

We construct below a path ${\widetilde{\pi }}^{n}$ from the previous paths and from $\pi ^n$ under the assumption that $\sigma ^n_b \le T_n$. The construction can be easily adapted to the cases $\sigma ^n_b=+\infty$. For $1\le c<b$, let

$$\begin{aligned} R_{c} \;=\; 2c \;+\; \sum _{a=1}^{c} (\tau _{{\mathfrak {j}}(a+1)} - \sigma _{{\mathfrak {j}}(a)}). \end{aligned}$$

This sequence represents the times at which the path ${\widetilde{\pi }}^{n}$ visits the measures ${{\bar{\varrho }}}_{{\mathfrak {j}}(c+1)}$. Set

$$\begin{aligned} {\widetilde{T}}_n \;=\; R_{b-1} \;+\; 1 \;+\; T_n \;-\; \sigma _{{\mathfrak {j}}(b)}, \end{aligned}$$

and define the path ${\widetilde{\pi }}^{n}$ in $C([0,\widetilde{T}_{n}], \mathcal {M}_{+,1})$ as follows:

$$\begin{aligned} {\widetilde{\pi }}^{n} (t) \;=\; {\left\{ \begin{array}{ll} \pi ^{1,+} (t) &{} 0\le t\le 1,\\ \pi ^n(\sigma _{{\mathfrak {j}}(1)} + t -1) &{} 1\le t\le 1+ \tau _{{\mathfrak {j}}(2)} - \sigma _{{\mathfrak {j}}(1)}, \\ \pi ^{2,-} (t - 1 - \tau _{{\mathfrak {j}}(2)} + \sigma _{{\mathfrak {j}}(1)}) &{} 1+ \tau _{{\mathfrak {j}}(2)} - \sigma _{{\mathfrak {j}}(1)}\le t\le R_1,\\ \pi ^{2,+} (t - R_1) &{} R_1 \le t\le R_1 +1, \\ \cdots &{} \cdots \\ \pi ^{b,+} (t-R_{b-1}) &{} R_{b-1}\le t\le R_{b-1} + 1, \\ \pi ^n(\sigma _{{\mathfrak {j}}(b)} + t - R_{b-1} - 1) &{} R_{b-1} + 1\le t \le R_{b-1} + 1 + T_n - \sigma _{{\mathfrak {j}}(b)}. \end{array}\right. } \end{aligned}$$

From the definition of ${\widetilde{\pi }}^{n}$ it is clear that ${\widetilde{\pi }}_{0}^{n}={{\bar{\varrho }}}_{i}$, ${\widetilde{\pi }}^{n}(\widetilde{T}_{n}) =\pi ^{n}(T_{n}) = \varrho$ and that

$$\begin{aligned} I_{\widetilde{T}_{n}}({\widetilde{\pi }}^{n}) \;\le \; I_{T_{n}}(\pi ^{n}) \;+\; 3b\varepsilon \;\le \; I_{T_{n}}(\pi ^{n}) \;+\; 3l\varepsilon . \end{aligned}$$

(5.6)

In particular, by (5.5) and (5.6), $I_{\widetilde{T}_{n}}({\widetilde{\pi }}^{n})$ is uniformly bounded. The time spent by ${\widetilde{\pi }}^{n}$ in ${\mathcal {L}}^{c}$ is at least $\widetilde{T}_{n}-2l$. Therefore, by Lemma 14 and by the previous uniform bound on $I_{\widetilde{T}_{n}}({\widetilde{\pi }}^{n})$, the sequence $\widetilde{T}_{n}$ is uniformly bounded. At this point, we may repeat the arguments presented in the first part of the proof, which are solely based on a uniform bound for the sequence $I_{\widetilde{T}_{n}}({\widetilde{\pi }}^{n})$, provided by (5.5) and (5.6), and on a uniform bound of the sequence ${\widetilde{T}}_n$, to conclude. $\square$

We conclude this section with a result needed in the next one. In Lemma 19 we constructed paths from ${\mathcal {B}}_\delta ({\mathcal {M}}_i)$ to ${\mathcal {M}}_i$ whose costs are small. In the next result, we prove a partial converse statement by showing that there are measures $\varrho$ at distance $\delta$ from ${\mathcal {M}}_i$ for which there exist paths from ${\mathcal {M}}_i$ to $\varrho$ whose cost is small.

Lemma 20

For every $1\le i\le l$ and $\varepsilon >0$, there exists $\delta _{14}=\delta _{14}(\varepsilon )>0$ such that for all $0\le \delta <\delta _{14}$, there exists a measure $\varrho \in {\mathcal {M}}_{+,1}$ such that $d(\varrho , {\mathcal {M}}_i) = \delta$ and $V_i(\varrho ) \le \varepsilon$.

Proof

Fix $1\le i\le l$, ${{\bar{\varrho }}}_{i}\in {\mathcal {M}}_i$, and $\varepsilon >0$. Recall the definition of the constant $c_1$ introduced in (3.5), and let $0<a\le c_1/2$. Let $\pi (t,d\theta )$, $0\le t\le 1$, be the trajectory $\pi (t,d\theta ) = \rho (t,\theta ) d\theta$, $\rho (t,\theta ) = {\bar{\rho }}_i(\theta ) + at$. On the one hand, by definition (2.1) of the distance, $d({\mathcal {M}}_i, \pi _1) = d({{\bar{\varrho }}}_{i}, \pi _1) = a$. On the other hand, for every smooth function $G:[0,1]\times {\mathbb {T}} \rightarrow {\mathbb {R}}$, since $(1/2)\varDelta \rho _t = (1/2)\varDelta {\bar{\rho }}_i = -F({\bar{\rho }}_i)$,

$$\begin{aligned} J_{1,G}(\pi ) \; \le \; \int _0^1 dt\, \Big \{ \big \langle \big \{ a + [F({\bar{\rho }}_i) - F(\rho _t)] \big \} \, G_t \big \rangle \;&-\; \langle B(\rho _t), e^{G_t}-1 - G_t\rangle \Big \}. \end{aligned}$$

Note that we omitted in the previous expression the terms $\chi (\rho _t) [\nabla G_t]^2$ and $D(\rho _t) [e^{-G_t}-1 + G_t]$ which are positive. By definition of $\rho _t$, $F({\bar{\rho }}_i) - F(\rho _t)$ is absolutely bounded by $C_F a$, where $C_F$ stands for the Lipschitz constant of the function F. By definition, $a\le c_1/2$ and by (3.5), $c_1 \le {\bar{\rho }}_i \le 1-c_1$. Thus, $c_1/2 \le \rho _t \le 1-c_1/2$. Let $b = \inf \{ B(x) : c_1/2 \le x \le 1-c_1/2\}>0$, so that

$$\begin{aligned} J_{1,G}(\pi ) \; \le \; \int _0^1 dt\, \Big \langle (1+h_t)\, a \, G_t - b\, [ e^{G_t}-1 - G_t] \Big \rangle , \end{aligned}$$

where $h_t$ is absolutely bounded by $C_F$. Assume that $0\le a<a_1$ where $a_1$ is chosen so that $(C_F-1)a_1 < b$. The right hand side of the previous expression is bounded by $\psi _b((1\pm C_F) a)$, uniformly in G, where $\psi _b(x) := (x+b) \log [1 + (x/b)] - x$. Therefore, $V_i(\varrho ) \le I_1(\pi ) \le \psi _b((1\pm C_F) a)$. Since $\psi _b (0)=0$, there exists $a_0$ such that $\psi _b((1\pm C_F) a) \le \varepsilon$, for any $0 \le a < a_0$. The assertion of the lemma holds provided we choose $\delta _{14} = a_0 \wedge a_1$ and $\varrho = {{\bar{\varrho }}}_{i} + a d\theta$ for $0\le a < \delta _{14}$. $\square$

Remark 1

Actually, we proved the existence of trajectory $\pi _t$, $0\le t\le 1$, such that $I_1(\pi )\le \varepsilon$, $\pi _0 = {{\bar{\varrho }}}_{i} \in {\mathcal {M}}_{i}$, $d(\pi _1, {\mathcal {M}}_i) = \delta$.

6 The static large deviations principle

We prove in this section Theorem 4. As we said before, the proof is based on a representation of the stationary state of the reaction–diffusion model in terms of the invariant probability measure of a discrete-time Markov chain. In the first part of this section, we introduce the discrete-time Markov chain and we prove in Proposition 3 sharp upper and lower bounds for its invariant probability measure.

For any $0<\beta _0<\beta _1$, let $B_i$ be the open neighborhoods, and $\varGamma _i$ be the closed neighborhoods given by

$$\begin{aligned} B&= \bigcup _{i=1}^l B_i, \quad \text {where}\quad B_i \;:=\; {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i) \;:=\; \{\varrho \in \mathcal {M}_{+}:\, \inf _{{{\bar{\varrho }}}\in \mathcal {M}_{i}} d(\varrho , {{\bar{\varrho }}}) < \beta _0\}. \\ \varGamma&= \bigcup _{i=1}^l \varGamma _i, \quad \text {where}\quad \varGamma _i \;=\; \{\varrho \in \mathcal {M}_{+}:\, \beta _1\le \inf _{{{\bar{\varrho }}}\in \mathcal {M}_{i}} d(\varrho , {{\bar{\varrho }}})\le 2\beta _1\}. \end{aligned}$$

To stress the dependence of B and $\varGamma$ on $\beta _0$, $\beta _1$ we sometimes denote B, $\varGamma$ by $B(\beta _0)$, $\varGamma (\beta _1)$, respectively.

For $N\ge 1$ and a subset A of ${\mathcal {M}}_{+}$, let $A^N = (\pi ^N)^{-1}(A)$ and let $H_A^N:D({\mathbb {R}}_+, X_N)\rightarrow [0,+\infty ]$ be the hitting time of $A^N$:

$$\begin{aligned} H^N_A \;=\; \inf \big \{t\ge 0: \eta _t\in A^N\big \}. \end{aligned}$$

The first result of the section states that the reaction–diffusion model reaches the set B in finite time with high probability.

Lemma 21

For every $\delta >0$, there exist $T_0$, $C_0$, $N_0 >0$, depending on $\delta$, such that

$$\begin{aligned} \sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H_{B(\delta )}^N \ge kT_0\right] \;\le \; \exp \left\{ -k C_0 N \right\} \end{aligned}$$

for all $N\ge N_0$ and all $k\ge 1$.

Proof

Fix $\delta >0$. By Corollary 1, there exist $T_0>0$ and $C_0>0$, which depend on $\delta$, such that

$$\begin{aligned} \inf _{\pi \in {\mathcal {D}}_{T_0,\delta }} I_{T_0}(\pi ) \;>\; C_0, \end{aligned}$$

where ${\mathcal {D}}_{T_0,\delta } = D([0,T_0],{\mathcal {M}}_{+}\backslash B(\delta ))$. For each integer $N\ge 1$, denote by $\eta ^N$ a configuration in $X_N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}\left[ H_B^N \ge T_0\right] \;=\;\sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H_B^N \ge T_0\right] . \end{aligned}$$

By the compactness of $\mathcal {M}_{+}$, every subsequence of $\pi ^N(\eta ^N)$ contains a sub-subsequence converging to some $\varrho$ in $\mathcal {M}_{+}$. Moreover, since each configuration in $X_N$ has at most one particle per site, any limit point $\varrho$ belongs to $\mathcal {M}_{+,1}$. From this observation and since ${\mathcal {D}}_{T_0,\delta }$ is a closed subset of $D([0,T_0],\mathcal {M}_{+})$, by the dynamical large deviations upper bound, there exists a measure $\varrho (d\theta ) = \gamma (\theta )d\theta$ in $\mathcal {M}_{+,1}$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}\left[ H_B^N\ge T_0\right] \;&\le \; \limsup _{N\rightarrow \infty }\frac{1}{N}\log {Q}_{T_0, \eta ^N}({\mathcal {D}}_{T_0,\delta }) \\ \;&\le \; -\inf _{\pi \in {\mathcal {D}}_{T_0,\delta }} I_{T_0}(\pi |\gamma ) \;<\; -C_0. \end{aligned}$$

In particular, there exists $N_0\ge 1$ such that for every integer $N\ge N_0$,

$$\begin{aligned} \sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H^N_B\ge T_0\right] \;\le \; \exp \{-C_0N\}. \end{aligned}$$

To complete the proof, we proceed by induction, applying the strong Markov property. Suppose that the statement of the lemma is true for all integers $j<k$. Let $N\ge N_0$ and let $\hat{\eta }$ be a configuration in $X_N$. By the strong Markov property,

$$\begin{aligned} {\mathbb {P}}_{\hat{\eta }}\left[ H^N_B\ge kT_0\right] \;=\;&{\mathbb {E}}_{\hat{\eta }}\left[ \mathbf 1 \left\{ H^{N}_{B}\ge T_{0}\right\} \, {\mathbb {P}}_{\eta _{ _{T_0}}}\left[ H^N_B\ge (k-1)T_0\right] \right] \\ \;\le \;&{\mathbb {P}}_{\hat{\eta }}\left[ H_B^N\ge T_0\right] \; \sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H^N_B\ge (k-1)T_0\right] \\ \;\le \;&\exp \left\{ -k\, C_0N\right\} , \end{aligned}$$

which completes the proof. $\square$

Corollary 2

For every $\delta >0$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \sup _{\eta \in X_N} {\mathbb {E}}_{\eta }\left( H^N_{B(\delta )}\right) \;\le \; 0. \end{aligned}$$

Proof

Fix $\delta >0$ and denote by $T_0$, $C_0$, $N_0\ge 1$ the constants provided by Lemma 21. For every integer $N\ge N_0$ and for every configuration $\eta$ in $X_N$,

$$\begin{aligned} {\mathbb {E}}_{\eta }\left( H_B^N\right) \;\le \; T_0\sum _{k=0}^{\infty }{\mathbb {P}}_{\eta }\left( H_B^N\ge kT_0\right) \;\le \; T_0\sum _{k=0}^{\infty }\exp \left\{ -kC_0N\right\} \;\le \; \frac{T_0}{1-e^{-C_0N}}, \end{aligned}$$

which proves the corollary. $\square$

We have now all elements to introduce the discrete-time Markov chain. Let $\partial B^N$ (which depends on $\beta _0$)¹ be the set of configurations $\eta$ in $X_N$ for which there exists a finite sequence of configurations $\{\eta ^i: 0\le i\le k\}$ in $X_N$ with $\eta ^0$ in $\varGamma ^N$, $\eta ^k = \eta$, and such that

• (a) For every $1\le i\le k$, the configuration $\eta ^{i}$ can be obtained from $\eta ^{i-1}$ by a jump of the dynamics (either from the stirring mechanism or from the non-conservative spin flip dynamics).

• (b) The unique configuration of the sequence which can enter into $B^N$ after a jump is $\eta ^k$.

We similarly define the set $\partial B_{i}^{N}$, $1\le i\le l$. It is clear that for N large enough and $\beta _1$ small enough,

$$\begin{aligned} \partial B^N \;=\; \bigcup _{i=1}^l\partial B^N_i. \end{aligned}$$

Let $\tau = \tau ^N: D({\mathbb {R}}_+, X_N)\rightarrow [0,\infty ]$ be the stopping time given by

$$\begin{aligned} \tau \;=\;\inf \left\{ t> 0: \hbox { there exist } s < t \hbox { such that } \eta _s\in \varGamma ^N \hbox { and } \eta _t\in \partial B^N\right\} . \end{aligned}$$

(6.1)

Set $\tau _1:=\tau$. We recursively define the sequence of stopping times $\{\tau _k : k\ge 1\}$ by

$$\begin{aligned} \tau _k\;=\;\inf \left\{ \tau _k> 0: \hbox { there exist }s< t \hbox { such that }\tau _{k-1} < s, \eta _s\in \varGamma ^N \hbox { and } \eta _t\in \partial B^N\right\} . \end{aligned}$$

This sequence generates a discrete-time Markov chain $\xi _k$ on $\partial B^N$ by setting $\xi _k = \eta _{\tau _k}$. The arguments presented before Lemma 5.1 in [21] show that this chain is irreducible. Denote by $\nu ^{N}$ its unique invariant probability measure.

Define ${\tilde{v}}_{ij}$ by

$$\begin{aligned} {\tilde{v}}_{ij}&= \inf \{I_T(\pi |{\bar{\rho }}): T> 0 \,,\, {\bar{\rho }}(\theta )d\theta \in \mathcal {M}_{i} \,,\, \pi \in {\mathcal {A}}_T \,,\, \pi _T \in {\mathcal {M}}_{j} \}, \\ \text {where}\quad {\mathcal {A}}_T&= \big \{ \pi \in C([0,T],{\mathcal {M}}_{+}) : \pi _{t}\notin \mathcal {M}_\mathrm{sol} \text { for all }0<t<T \big \}. \end{aligned}$$

The jumps of the Markov chain $\{\xi _k : k\ge 1\}$ correspond to paths from $\partial B^N_i$ to $\partial B^N_j$ which do not visit other boundaries. They are thus related to the dynamical large deviations principle. In Lemmata 22 and 24 we estimate the probability of a jump from $\partial B^N_i$ to $\partial B^N_j$ for $j\not = i$. These estimates provide sharp bounds for the invariant probability measure of the chain $\xi _k$, alluded to at the beginning of this section.

Lemma 22

For every $1\le i \not = j\le l$, $\varepsilon >0$ and $0<\beta _1< (1/4) \min _{a\not = b} d({\mathcal {M}}_a, {\mathcal {M}}_b)$, there exists $0<\delta _{15}< \beta _1$ such that for all $\beta _0<\delta _{15}$

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log \inf _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta } (\eta _{\tau }\in \partial B_j^N) \;\ge \; -\, {\tilde{v}}_{ij} \,-\, \varepsilon . \end{aligned}$$

Proof

Fix $\varepsilon >0$, $1\le i \not = j\le l$, and $0<\beta _1< (1/4) \min _{a\not = b} d({\mathcal {M}}_a, {\mathcal {M}}_b)$. By definition of ${\tilde{v}}_{ij}$, there exist $T>0$, ${{\bar{\varrho }}}_{i} (d\theta ) = {\bar{\rho }}_i (\theta ) d\theta \in {\mathcal {M}}_i$, ${{\bar{\varrho }}}_j \in {\mathcal {M}}_j$, and $\pi \in {\mathcal {A}}_T$ such that $\pi _0 = {{\bar{\varrho }}}_{i}$, $\pi _T = {{\bar{\varrho }}}_j$, $I_T(\pi | {\bar{\rho }}_i) \le {\tilde{v}}_{ij} + \varepsilon$. Note that $\pi _t\not \in {\mathcal {M}}_i$ for all $0<t\le T$ because $\pi \in {\mathcal {A}}_T$.

Let $t_0$ be the first time the path $\pi _t$ is at distance $\beta _1$ from ${\mathcal {M}}_i$: $t_0 = \min \{ t\ge 0 : d(\pi _t, {\mathcal {M}}_i) \ge \beta _1\}$. Since $\pi$ belongs to ${\mathcal {A}}_T$ and $\pi _T\in {\mathcal {M}}_j$, $\zeta _2 = \inf _{t_0\le t\le T} d(\pi _t, \cup _{k\not =j} {\mathcal {M}}_k) >0$. Let $\zeta _3$ be the constant $\delta _{13}(\min \{\beta _1, \varepsilon \})$ given by Lemma 19. Set $\delta _{15}= (1/2) \min \{\beta _1,\zeta _2, \zeta _3\}$ and fix $\beta _0<\delta _{15}$.

For each integer $N>0$, let $\eta ^N$ be a configuration in $\partial B_i^N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}(\eta _{\tau }\in \partial B_{j}^{N}) \;=\; \inf _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta }(\eta _{\tau } \in \partial B_{j}^{N}). \end{aligned}$$

(6.2)

Recall that every subsequence of $\pi ^N(\eta ^N)$ contains a sub-subsequence converging in $\mathcal {M}_{+}$ to some measure $\varrho$ which belongs to $\mathcal {M}_{+,1}$. We may therefore assume that $\pi ^N(\eta ^N)$ converges to $\varrho (d\theta )=\gamma (\theta )d\theta$ which belongs to the closure of $B_{i}$: $\varrho \in \overline{{\mathcal {B}}_{\beta _0}({{\bar{\varrho }}}'_{i})}$, for some ${{\bar{\varrho }}}'_{i} \in {\mathcal {M}}_i$.

Since $\beta _0<\zeta _3$, we may apply Lemma 19 to connect $\varrho$ to ${{\bar{\varrho }}}_{i}'\in {\mathcal {M}}_i$. Denote by $\pi '_t$, $0\le t\le 1$, the path given by Lemma 19 and such that $\pi '_0 = \varrho$, $\pi '_1 = {{\bar{\varrho }}}_{i}'\in {\mathcal {M}}_i$, $I_1(\pi '| \gamma ) \le \varepsilon$ and $\pi '_t \in {\mathcal {B}}_{\beta _1}({{\bar{\varrho }}}_{i}')$, $0\le t\le 1$. We may apply Lemma 16 to connect ${{\bar{\varrho }}}_{i}'$ to ${{\bar{\varrho }}}_{i}$, the initial point of the path introduced in the first paragraph of the proof. By Lemma 16, there exists $T'$ and a path $\pi '' \in C([0,T'], {\mathcal {M}}_+)$ such that $\pi ''_0 = {{\bar{\varrho }}}_{i}'$, $\pi ''_{T'} = {{\bar{\varrho }}}_{i}$ and $I_{T'}(\pi ''| {{\bar{\varrho }}}_{i}') \le \varepsilon$.

Concatenate the paths $\pi '$, $\pi ''$ and $\pi$ to obtain a path ${\widetilde{\pi }}$ in $C([0, T+T'+1], {\mathcal {M}}_+)$ such that ${\widetilde{\pi }}_0 = \varrho$, ${\widetilde{\pi }}_{T+T'+1} \in {\mathcal {M}}_j$, $I_{T+T'+1}(\pi ''| \varrho ) \le {\tilde{v}}_{ij} + 3\varepsilon$. Moreover, $d(\pi _t, {\mathcal {M}}_i) < \beta _1$ for $0\le t\le 1+T'$. In particular, ${\widetilde{\pi }}_t$ reaches a distance $\beta _1$ from ${\mathcal {M}}_i$ for the first time at $t= 1+T'+t_0$, and $\inf _{t_0\le t\le T} d({\widetilde{\pi }}_{1+T'+t}, \cup _{k\not =j} {\mathcal {M}}_k) =\zeta _2> \beta _0$.

Denote by ${\mathcal {N}}$ a $\beta _0/2$-neighborhood in $D([0,T+T'+1], {\mathcal {M}}_+)$ of the path ${\widetilde{\pi }}$. It follows from the last observation of the previous paragraph and from the fact that ${\widetilde{\pi }}_{T+T'+1}\in {\mathcal {M}}_j$ that ${\mathcal {N}} \subset \{\eta _{\tau }\in \partial B_j^N\}$. Since ${\mathcal {N}}$ is an open set, by the lower bound of the dynamical large deviations principle, and by (6.2)

$$\begin{aligned}&\liminf _{N\rightarrow \infty }\frac{1}{N} \inf _{\eta \in \partial B^N_i} \log {\mathbb {P}}_{\eta }(\eta _{\tau }\in \partial B_{j}^{N}) \;=\; \liminf _{N\rightarrow \infty }\frac{1}{N} \log {\mathbb {P}}_{\eta ^N}(\eta _{\tau }\in \partial B_{j}^{N}) \\&\quad \;\ge \; \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}({\mathcal {N}}) \;\ge \; - \inf _{\bar{\pi }\in {\mathcal {N}}} I_{T+T'+1}(\bar{\pi }|\gamma ) \;\ge \; - \, I_{T+T'+1}({\widetilde{\pi }}|\gamma ). \end{aligned}$$

This completes the proof of the lemma because $I_{T+T'+1}({\widetilde{\pi }}| \varrho ) \le {\tilde{v}}_{ij} + 3\varepsilon$. $\square$

The proof of the upper bound for ${\mathbb {P}}_{\eta } (\eta _{\tau }\in \partial B_j^N)$, $\eta \in \partial B_i^N$, requires a lower bound for the dynamical large deviations rate functional. For $T>0$ and $\zeta >0$, let ${\mathcal {C}}_j = {\mathcal {C}}_j(T,\zeta )$ be the closed subset of $D([0,T], \mathcal {M}_{+})$ consisting of all paths $\pi$ for which there exists some time $t \in [0, T]$ such that $\pi (t)$ belongs to $\varGamma _j(\zeta )$ or $\pi (t-)$ belongs to $\varGamma _j(\zeta )$.

Lemma 23

For every $1\le i\not = j\le l$, $\varepsilon >0$, there exist $\delta _{16} = \delta _{16}(\varepsilon )>0$ and $T=T(\varepsilon )>0$ such that for all $\delta '<\delta _{16}$, $T'\ge T$, $\gamma (\theta )d\theta \in \varGamma _i(\delta ')$,

$$\begin{aligned} \inf _{\pi \in {\mathcal {C}}_j(T',\delta ')} I_{T'}(\pi |\gamma ) \;\ge \; v_{ij} \;-\; \varepsilon . \end{aligned}$$

Proof

Fix $1\le i\not = j\le l$ and assume that the assertion of the lemma is false. In that case, there exists $\varepsilon >0$ such that for for every $\zeta >0$ and $T>0$, there exist $\zeta '<\zeta$, $T'\ge T$, $\gamma (\theta )d\theta \in \varGamma _i(\zeta ')$ and $\pi \in {\mathcal {C}}_j(T',\zeta ')$ with

$$\begin{aligned} I_{T'}(\pi |\gamma ) \; < \; v_{ij} \;-\; \varepsilon /2. \end{aligned}$$

In particular, taking the sequences $\zeta _n = 1/n$, $T_n=1$, $n\ge 1$, there exist $\zeta '_n <1/n$, $T'_n\ge 1$, $\gamma _n\in \varGamma _i(\zeta '_n)$ and $\pi _n\in {\mathcal {C}}_j(T'_n, \zeta '_n) \cap C([0,T'_n],\mathcal {M}_{+,1})$ with

$$\begin{aligned} I_{T'_n}(\pi ^n|\gamma _n)\;<\; v_{ij} \,-\, \varepsilon /2. \end{aligned}$$

(6.3)

Since $\pi ^n$ belongs to ${\mathcal {C}}_{j}(T'_n, \zeta '_n) \cap C([0,T'_n],{\mathcal {M}}_{+,1})$, there exists $0<{\widetilde{T}}_n\le T'_n$ such that $\pi ^n({\widetilde{T}}_n) \in \{\varrho \in \mathcal {M}_{+,1}:\, \zeta '_n \le \inf _{{{\bar{\varrho }}}\in \mathcal {M}_{j}}d(\varrho , {{\bar{\varrho }}})\le 2\zeta '_n\}$.

Assume first that the sequence of times $\{{\widetilde{T}}_n: \, n\ge 1\}$ is bounded above by some $T>0$. For each integer $n>0$, let ${\widehat{\pi }}^n$ be the path in $C([0,T-{\widetilde{T}}_n],\mathcal {M}_{+,1})$ given by ${\widehat{\pi }}^n_t(d\theta ) = {\widehat{\rho }}^n(t,\theta ) d\theta$, where ${\widehat{\rho }}^n$ is the solution of the hydrodynamic equation (2.4) with initial condition $\rho ^n({\widetilde{T}}_n)$, where $\pi ^n({\widetilde{T}}_n, d\theta ) = \rho ^n({\widetilde{T}}_n, \theta ) d\theta$. Since $d(\pi ^n({\widetilde{T}}_n), {\mathcal {M}}_j) \le 2 \zeta '_n$, by Lemma 8, ${\widehat{\pi }}^n (T-{\widetilde{T}}_n)$ converges to some element of ${\mathcal {M}}_j$.

Let ${\widetilde{\pi }}^n$ be the path in $C([0,T],\mathcal {M}_{+,1})$ given by

$$\begin{aligned} {\widetilde{\pi }}^n_t \;=\; {\left\{ \begin{array}{ll} \pi ^n_t &{} \hbox { if }\; 0\le t\le {\widetilde{T}}_n,\\ {\widehat{\pi }}^n(t-{\widetilde{T}}_n) &{} \hbox { if }\; {\widetilde{T}}_n\le t\le T. \end{array}\right. } \end{aligned}$$

By definition of ${\widetilde{\pi }}^n$, $I_{T}({\widehat{\pi }}^n) = I_{T}({\widehat{\pi }}^n|\gamma _n) = I_{T'_n}(\pi ^n|\gamma _n) < v_{ij} \,-\, \varepsilon /2$. Since $I_T$ has compact level sets and since $\pi ^n_0(d\theta ) = \gamma _{n}(\theta )d\theta$ belongs to $\varGamma _{i}(\zeta '_n)\cap \mathcal {M}_{+,1}$, there exists a subsequence of ${\widetilde{\pi }}^n$ converging to some $\pi$ in $C([0,T],\mathcal {M}_{+,1})$ such that $\pi _0 \in {\mathcal {M}}_i$, $\pi _T \in {\mathcal {M}}_j$, and $I_T(\pi )\le v_{ij}-\varepsilon /2$, which contradicts the definition of $v_{ij}$.

If the sequence $\{{\widetilde{T}}_n: \, n\ge 1\}$ is not bounded, we may repeat the reasoning presented in the proof of Theorem 7 to replace the path $\pi ^{n}$ by a path ${\bar{\pi }}^{n}$ which satisfies an inequality analogous to (6.3) (with extra factors of $\varepsilon$) and whose entry time to the set $\{\varrho \in \mathcal {M}_{+}:\, \zeta '_n \le \inf _{{{\bar{\varrho }}}\in \mathcal {M}_{j}} d(\varrho , {{\bar{\varrho }}})\le 2 \zeta '_n \}$ is uniformly bounded in n. This completes the proof of the lemma, since the bounded case has been treated above. $\square$

We now prove the upper bound.

Lemma 24

For every $1\le i \not = j\le l$, $\varepsilon >0$, there exists $\delta _{17}=\delta _{17}(\varepsilon )$ such that for all $0<\beta _0< \beta _1< \delta _{17}$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log \sup _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta } (\eta _{\tau }\in \partial B_j^N) \;\le \; -\, v_{ij} + \varepsilon . \end{aligned}$$

Proof

Fix $1\le i \not = j\le l$, $\varepsilon >0$. Let $\zeta _{1}>0$, $T>0$ be chosen according to Lemma 23, and fix $0<\beta _0<\beta _1<\zeta _1$. By the strong Markov property,

$$\begin{aligned} \sup _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta }(\eta _{\tau }\in \partial B_j^N) \;\le \; \sup _{\eta \in \varGamma _i^N} {\mathbb {P}}_{\eta }(\eta (H_{\partial B^N}) \in \partial B_j^N), \end{aligned}$$

where $H_D$, $D\subset X_N$, represents the hitting time of the set D and $\eta (t) = \eta _t$. For each integer $N>0$, fix a configuration $\eta ^N$ in $\varGamma _i^N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}(\eta (H_{\partial B^N}) \in \partial B_j^N) \;=\; \sup _{\eta \in \varGamma _i^N} {\mathbb {P}}_{\eta }(\eta (H_{\partial B^N}) \in \partial B_j^N). \end{aligned}$$

By Lemma 18, $v_{ij}<\infty$. Thus, by Lemma 21, there exists $T_{\beta _0}>0$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty } \frac{1}{N}\log \sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H_{B}^N \ge T_{\beta _0} \right] \;\le \; -\, v_{ij}. \end{aligned}$$

We may assume that $T_{\beta _0}>T$, where T is the time introduced at the beginning of the proof. On the other hand, since $\eta ^N \in \varGamma _i^N$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N} \Big (\eta (H_{\partial B^N}) \in \partial B_j^N ,\, H^N_B \le T_{\beta _0} \Big ) \le \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}(H^N_{\varGamma _j}\le T_{\beta _0}). \end{aligned}$$

By intersecting the set $\{\eta (H_{\partial B^N}) \in \partial B_j^N \}$ with the set $\{H^N_B \le T_{\beta _0}\}$ and its complement, since

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log \{a_N+b_N\} \;\le \; \max \left\{ \limsup _{N\rightarrow \infty }\frac{1}{N}\log a_N, \limsup _{N\rightarrow \infty }\frac{1}{N}\log b_N\right\} , \end{aligned}$$

(6.4)

it follows from the two previous estimates that

$$\begin{aligned} \begin{aligned}&\limsup _{N\rightarrow \infty } \frac{1}{N} \log \sup _{\eta \in \varGamma _i^N} {\mathbb {P}}_{\eta } \big (\eta (H_{\partial B^N}) \in \partial B_j^N \big ) \\&\qquad \; \le \; \max \Big \{\limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}(H^N_{\varGamma _j}\le T_{\beta _0}) \,,\, -\, v_{ij} \Big \}. \end{aligned} \end{aligned}$$

(6.5)

Let ${\mathcal {C}}_j$ be the set introduced in Lemma 23 associated to the pair $(\beta _1, T_{\beta _0})$. Since ${\mathcal {C}}_j$ is a closed set, and since $\{H^N_{\varGamma _j}\le T_{\beta _0}\} \subset {\mathcal {C}}_j$, by the dynamical large deviations upper bound and by the compactness of $\mathcal {M}_{+}$, there exists $\gamma (\theta )d\theta \in \varGamma _i$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}(H^N_{\varGamma _j}\le T_{\beta _0}) \le \limsup _{N\rightarrow \infty }\frac{1}{N}\log Q_{T_{\beta _0}, \eta ^N}({\mathcal {C}}_j) \le -\inf _{\pi \in {\mathcal {C}}_j} I_{T_{\beta _0}}(\pi |\gamma ). \end{aligned}$$

By Lemma 23, the last term is bounded above by $-v_{ij} + \varepsilon$. This completes the proof of the lemma in view of (6.5). $\square$

The proof of the next result is similar to the ones of Lemmata 3.1 and 3.2 of chapter 6 in [24]. Recall the notation introduced above Eq. (2.13). Consider a set $\varOmega$, which is not assumed to be countable. Denote by $\varOmega _i$, $1\le i\le l$, a partition of $\varOmega$: $\varOmega = \cup _{1\le i\le l} \varOmega _i$, $\varOmega _i \cap \varOmega _j = \varnothing$ for $i\not = j$. Let $(Z_n : n\ge 0)$ be a discrete-time Markov chain on $\varOmega$ and denote by p(x, dy), $x\in \varOmega$, the transition probability of the chain $Z_n$. Assume that any set $\varOmega _j$ can be reached from any point $x\in \varOmega$: $\sum _{n\ge 0} P_x[Z_n \in \varOmega _j]>0$.

Lemma 25

Suppose that there exist nonnegative numbers $p_{ij}, {\tilde{p}}_{ij}$, $1\le i \not =j\le l$, and a number $a>1$ such that

$$\begin{aligned} \frac{1}{a}\, p_{ij} \;\le \; P(x,\varOmega _j) \;\le \; a \, {\tilde{p}}_{ij} \quad \text {for all } \; x\in \varOmega _i, \, i\,\ne \, j. \end{aligned}$$

Then,

$$\begin{aligned} \frac{1}{a^{2(l-1)}} \, \frac{Q_i}{\sum _{1\le j\le l}{\tilde{Q}}_j} \; \le \; \nu (\varOmega _i) \;\le \; a^{2(l-1)} \frac{{\tilde{Q}}_i}{\sum _{1\le j\le l} Q_j} \end{aligned}$$

for any invariant probability measure $\nu$, where $Q_i$, ${\tilde{Q}}_{i}$ are given by

$$\begin{aligned} Q_i \;=\; \sum _{g\in {\mathcal {T}}(i)}\prod _{(m, n)\in g} p_{mn}\quad \hbox { and } \quad {\tilde{Q}}_i \;=\; \sum _{g\in {\mathcal {T}} (i)}\prod _{(m, n)\in g} {\tilde{p}}_{mn}. \end{aligned}$$

Let

$$\begin{aligned} {\tilde{w}}_{i} \;=\; \min _{g\in {\mathcal {T}}(i)} \sum _{(m, n)\in g} {\tilde{v}}_{mn}. \end{aligned}$$

By the argument presented in the proof of [24, Lemma 4.1], we have $w_{i}={\tilde{w}}_{i}$ for all $1\le i\le l$. We are now in a position to state the main result of this subsection.

Proposition 3

For every $\varepsilon >0$, there exists $\delta _{18} = \delta _{18}(\varepsilon )$ such that for all $1\le i \le l$, $0< \beta _0< \beta _1 < \delta _{18}$,

$$\begin{aligned} \limsup _{N\rightarrow \infty } \frac{1}{N}\log \nu ^N(\partial B_i^N)\le & {} -\, {\overline{w}}_i \,+\, \varepsilon , \\ \liminf _{N\rightarrow \infty } \frac{1}{N}\log \nu ^N(\partial B_i^N)\ge & {} -\, {\overline{w}}_i \,-\, \varepsilon . \end{aligned}$$

Proof

Since $w_i = {\tilde{w}}_i$, the assertion of this proposition is a straightforward consequence of Lemmata 22, 24 and 25. $\square$

6.1 Lower bound

We prove in this subsection the large deviations lower bound, that is, for any open subset ${\mathcal {O}}$ of $\mathcal {M}_{+}$,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log \mathcal P^{N}({\mathcal {O}}) \;\ge \; -\inf _{\varrho \in {\mathcal {O}}} W(\varrho ). \end{aligned}$$

Recall the definition of the stopping time $\tau$ introduced in (6.1). Following [10, 21, 24], we represent the stationary measure $\mu ^{N}$ of a subset A of $X_{N}$ as

$$\begin{aligned} \mu ^{N}(A) \;=\; \frac{1}{C_N} \int _{\partial B^N} {\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in A\}\, ds \right) \, d\nu ^N(\eta ), \end{aligned}$$

(6.6)

where

$$\begin{aligned} C_{N}\;=\;\int _{\partial B^{N}} {\mathbb {E}}_{\eta }(\tau )\, d\nu ^{N}(\eta ). \end{aligned}$$

The first lemma provides an estimate on the normalizing constant $C_{N}$.

Lemma 26

For any $\varepsilon >0$, there exists $\delta _{19}=\delta _{19}(\varepsilon )$ such that for all $0<\beta _0<\beta _1<\delta _{19}$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log C_{N} \;\le \; \varepsilon . \end{aligned}$$

Proof

Fix $\varepsilon >0$ and let $\zeta _1$ be a positive number such that $2\zeta _1$ is smaller than the constants $\delta _{13}(\varepsilon )$ introduced in Lemma 19 and smaller than the constant $\delta _{14}(\varepsilon )$ introduced in Lemma 20. Fix $0<\beta _0<\beta _1<\zeta _1$. Since $H_{\varGamma _{i}}^{N}<\tau$ when the process starts from $\partial B_{i}^{N}$, by the Strong Markov property,

$$\begin{aligned} C_{N}&=\sum _{i=1}^{l}\int _{\partial B_{i}^{N}} {\mathbb {E}}_{\eta }(\tau )\, d\nu ^{N}(\eta ) \;=\;\sum _{i=1}^{l}\int _{\partial B_{i}^{N}} {\mathbb {E}}_{\eta } \big (\tau \, \mathbf 1 \{ H_{\varGamma _{i}}^{N}<\tau \} \big ) \, d\nu ^{N}(\eta ) \\&\le \sum _{i=1}^{l}\int _{\partial B_{i}^{N}} {\mathbb {E}}_{\eta }(H_{\varGamma _{i}}^{N})\, d\nu ^{N}(\eta ) \;+\; \sup _{\eta \in X_{N}} {\mathbb {E}}_{\eta } (H_{B}^{N}). \end{aligned}$$

By Corollary 2 and by (6.4), it remains to show that for every $1\le i\le l$,

$$\begin{aligned} \limsup _{N\rightarrow \infty } \frac{1}{N}\log \sup _{\eta \in \partial B_{i}^{N}} {\mathbb {E}}_{\eta } (H_{\varGamma _{i}}^{N}) \;\le \; 3\varepsilon . \end{aligned}$$

(6.7)

Fix $1\le i\le l$. We claim that there exists $N_0$ such that for all $N\ge N_0$,

$$\begin{aligned} \sup _{\eta \in \varGamma ^N_{i} \cup \partial B_{i}^{N}} {\mathbb {P}}_{\eta }(H_{\varGamma _{i}}^{N} \ge 3) \;\le \; 1-\exp {\{-3N\varepsilon \}}. \end{aligned}$$

(6.8)

To prove this assertion, for each integer $N>0$, consider a configuration $\eta ^{N}$ such that $\pi ^N(\eta ^N)$ belongs to ${\mathcal {B}}_{2\beta _1}[\mathcal {M}_{i}] := \overline{{\mathcal {B}}_{2\beta _1}(\mathcal {M}_{i})}$ (which contains the set $\varGamma ^N_{i} \cup \partial B_{i}^{N}$) and such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^{N}}(H_{\varGamma _{i}}^{N}<3) \;=\; \inf _{\eta \in {\mathcal {B}}_{2\beta _1}^{N}[\mathcal {M}_{i}]} {\mathbb {P}}_{\eta }(H_{\varGamma _{i}^{N}}<3). \end{aligned}$$

Recall that each subsequence of $\pi ^N(\eta ^N)$ contains a sub-subsequence converging in $\mathcal {M}_{+}$ to some $\varrho$ which belongs to $\mathcal {M}_{+,1}$. Therefore, we may assume that $\pi ^N(\eta ^N)$ converges to $\varrho (d\theta )=\gamma (\theta )d\theta$ and that $\varrho$ belongs to the closure of ${\mathcal {B}}_{2\beta _1}({{\bar{\varrho }}}_{i})$: $\varrho \in {\mathcal {B}}_{2\beta _1}[{{\bar{\varrho }}}_{i}] := \overline{{\mathcal {B}}_{2\beta _1}({{\bar{\varrho }}}_{i})}$ for some ${{\bar{\varrho }}}_{i} \in {\mathcal {M}}_i$.

Let $\overline{\pi }_0 = \varrho \in {\mathcal {B}}_{2\beta _1}[{{\bar{\varrho }}}_{i}] \subset {\mathcal {B}}_{2\zeta _{1}}({{\bar{\varrho }}}_{i})$, and let $\overline{\pi }$ be a path in $D([0,1], \mathcal {M}_{+,1})$ provided by Lemma 19. Let ${\widetilde{\pi }}$ be a path in $D([0,1], \mathcal {M}_{+,1})$ provided by Remark 1: ${\widetilde{\pi }}_{0}={{\bar{\varrho }}}_{i}$, ${\widetilde{\pi }}_{1}\in \mathcal {M}_{+}\setminus {\mathcal {B}}_{2\beta _1}[\mathcal {M}_{i}]$ and $I_{1}({\widetilde{\pi }})\le \varepsilon$. Define the path $\pi$ in $D([0,2], \mathcal {M}_{+,1})$ by concatenating the paths $\overline{\pi }$ and ${\widetilde{\pi }}$:

$$\begin{aligned} \pi _{t} \;=\; \overline{\pi }_{t} \quad \text {for}\quad 0\le t \le 1, \qquad \pi _{t} \;=\; {\widetilde{\pi }}_{t-1} \quad \text {for}\quad 1\le t \le 2. \end{aligned}$$

The path $\pi _t$, $0\le t\le 2$, starts from $\varrho$, hits ${\mathcal {M}}_i$ and then ${\mathcal {B}}_{2\beta _1}[\mathcal {M}_{i}]^c$. Its cost $I_{2}(\pi )$ is bounded by $2\varepsilon$.

Denote by $\varLambda _{\beta _1/2}(\pi )$ the $\beta _1/2$-open neighborhood of the trajectory $\pi$ in $D([0,2], \mathcal {M}_{+,1})$. Since $\varLambda _{\beta _1/2}(\pi ) \subset \{H_{\varGamma _{i}}^{N}<3\}$, by the dynamical large deviations lower bound, by definition of the sequence $\eta ^N$ and since $I_{2}(\pi ) \le 2\varepsilon$, for N large enough,

$$\begin{aligned} {\mathbb {P}}_{\eta }(H_{\varGamma _{i}}^{N}<3) \;\ge \; \exp -N \Big \{\inf _{\pi '\in \varLambda _{\beta _1/2}(\pi )}I_{2}(\pi '|\gamma ) \, +\, \varepsilon \Big \} \;\ge \; \exp {\{-3N\varepsilon \}} \end{aligned}$$

for all $\eta \in {\mathcal {B}}_{2\beta _1}^{N}[\mathcal {M}_{i}]$, which proves (6.8).

The estimate (6.8) together with the arguments presented in Lemma 21 and Corollary 2 gives the bound (6.7), which completes the proof. $\square$

Proof of the lower bound of Theorem4. We first claim that for any open set ${\mathcal {O}}$ of $\mathcal {M}_{+}$ containing some ${{\bar{\varrho }}}_{i} \in {\mathcal {M}}_i$, $1\le i \le l$,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {O}}) \;\ge \; - \,{\overline{w}}_i. \end{aligned}$$

(6.9)

Indeed, fix $1\le i \le l$, $\varepsilon >0$ and choose first $\beta _1>0$ and then $0<\beta _0<\beta _1$ satisfying two conditions: (a) $\beta _0 <\zeta _1$, where $\zeta _1$ is the positive constant $\delta _{11}(\beta _1/2, 1/2,1)$ provided by Lemma 8 for ${\bar{\rho }}={\bar{\rho }}_i$, and (b) the pair $(\beta _0, \beta _1)$ fulfills the lower bound of Proposition 3 and Lemma 26. Assume, moreover, that ${\mathcal {B}}_{2\beta _1}(\overline{\varrho }_i)\subset {\mathcal {O}}$. Note that condition (a) entails that $\beta _0<\beta _1/2$.

By (6.6), and since $\tau \ge H^N_{\varGamma _i}$ if the initial configuration belongs to $\partial B_i^N$,

$$\begin{aligned} {\mathcal {P}}^N({\mathcal {O}}) \;&=\; \frac{1}{C_N}\int _{\partial B^N} {\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in \mathcal {O}^{N} \}\, ds\right) \, d\nu ^N(\eta ) \\ \;&\ge \; \frac{1}{C_{N}}\int _{\partial B_i^N}{\mathbb {E}}_{\eta }(H^N_{\varGamma _i}) \, d\nu ^N(\eta ) \;\ge \; \frac{1}{C_{N}}\, \nu ^N(\partial B_i^N)\, \inf _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta }(H^N_{\varGamma _i}\ge 1). \end{aligned}$$

By Lemma 26 and Proposition 3, to conclude the proof of claim (6.9), it remains to show that

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log \inf _{\eta \in \partial B_i^N} {\mathbb {P}}_{\eta }(H^N_{\varGamma _i}\ge 1) \;\ge \; 0. \end{aligned}$$

For each integer $N>0$, let $\eta ^N$ be a configuration in $\partial B_i^N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}(H^N_{\varGamma _i}\ge 1) \;=\; \inf _{\eta \in \partial B_i^N}{\mathbb {P}}_{\eta }(H^N_{\varGamma _i}\ge 1). \end{aligned}$$

Denote by $\eta ^{N_k}$ a subsequence of $\eta ^N$ which transforms the $\liminf$ in a limit and let $\pi _0 (d\theta ) = \gamma (\theta ) d\theta$ be a limit point of $\pi ^{N_k}(\eta ^{N_k})$. Observe that $\pi _0 \in \overline{{\mathcal {B}}_{\beta _0}(\mathcal {M}_{i})}$ and denote by $\rho :[0,1]\times \mathbb {T}\rightarrow [0,1]$ the unique weak solution of the Cauchy problem (2.4) starting from $\gamma$. By Lemma 8 and by the definition of $\beta _0$, $\beta _1$, $\pi _t(d\theta ) := \rho (t,\theta ) d\theta$ belongs to ${\mathcal {B}}_{\beta _1/2}({\mathcal {M}}_i)$ for all $0\le t\le 1$.

Let ${\mathcal {N}}$ be the subset of $D([0,1], \mathcal {M}_{+})$ given by all trajectories $\pi '_t$, $0\le t\le 1$, such that $\sup _{0\le t\le 1} d(\pi '_t,\pi _t) < \beta _1/2$. Note that the set ${\mathcal {N}}$ is open because $\pi _t$ is continuous. In particular, since $\pi _t$ belongs to ${\mathcal {B}}_{\beta _1/2}({\mathcal {M}}_i)$ for any $0\le t\le 1$, $\{H^N_{\varGamma _i}\ge 1\} \supset (\pi ^N)^{-1}(\mathcal N) := \{\eta _\cdot \in D([0,1],X_N):\pi ^N(\eta _\cdot )\in {\mathcal {N}}\}$. Therefore,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}(H^N_{\varGamma _i}\ge 1) \;&\ge \; \liminf _{N\rightarrow \infty }\frac{1}{N}\log {Q}_{1,\eta ^N}({\mathcal {N}}) \\ \;&\ge \; -\inf _{\pi '\in {\mathcal {N}}} I_1(\pi '|\gamma ) \;\ge \; - I_1(\pi |\gamma ) \;=\; 0, \end{aligned}$$

which completes the proof of the claim.

It follows from (6.9) that there exists a sequence $\varepsilon _N\rightarrow 0$ such that

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {B}}_{\varepsilon _N}(\mathcal {M}_{i})) \;\ge \; - \, {\overline{w}}_i. \end{aligned}$$

(6.10)

Fix an open subset ${\mathcal {O}}$ of $\mathcal {M}_{+}$. In order to prove the lower bound, it is enough to show that for any measure $\varrho$ in ${\mathcal {O}}$, $1\le i\le l$, $T>0$, and any trajectory $\pi$ in $D([0,T],{\mathcal {M}}_{+})$ with $\pi _0\in {\mathcal {M}}_i$, $\pi _T = \varrho$,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {O}}) \;\ge \; -\, {\overline{w}}_i \;-\; I_T(\pi ). \end{aligned}$$

(6.11)

To prove this claim, fix an open subset ${\mathcal {O}}$ of $\mathcal {M}_{+}$, a measure $\varrho$ in ${\mathcal {O}}$, $1\le i\le l$, $T>0$, and a trajectory $\pi$ in $D([0,T],{\mathcal {M}}_{+})$ with $\pi _0\in {\mathcal {M}}_i$, $\pi _T = \varrho$. Since $\mu ^N$ is the stationary measure,

$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^N({\mathcal {O}})&= \liminf _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {E}}_{\mu ^{N}} \big [ \, {\mathbb {P}}_{\eta }\left( \pi ^N_T\in {\mathcal {O}}\right) \big ] \\&\ge \liminf _{N\rightarrow \infty }\frac{1}{N}\log \Big ( {\mathcal {P}}^N({\mathcal {B}}_{\varepsilon _N}(\mathcal {M}_{i}))\inf _{\eta \in {\mathcal {B}}_N} {\mathbb {P}}_{\eta }[\pi ^N_T\in {\mathcal {O}}] \Big ), \end{aligned}$$

where ${\mathcal {B}}_N = \{\eta : \pi ^N(\eta ) \in {\mathcal {B}}_{\varepsilon _N}(\mathcal {M}_{i}) \}$. Let $\eta ^N$ be a configuration in ${\mathcal {B}}_N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}\left[ \pi ^N_T\in {\mathcal {O}}\right] \;=\; \inf _{\eta \in {\mathcal {B}}_N} {\mathbb {P}}_{\eta }\left[ \pi ^N_T\in {\mathcal {O}}\right] . \end{aligned}$$

Since $\eta ^N$ belongs to ${\mathcal {B}}_N$ and $\varepsilon _N\rightarrow 0$, we may assume, taking a subsequence if necessary, that $\pi ^N(\eta ^N)$ converges to some ${{\bar{\varrho }}}_{i} (d\theta ) = {\bar{\rho }}_i (\theta ) d\theta \in {\mathcal {M}}_i$. By (6.10), the expression appearing in the penultimate displayed formula is bounded below by

$$\begin{aligned} - \, {\overline{w}}_i \,+\, \liminf _{N\rightarrow \infty }\frac{1}{N} \log {\mathbb {P}}_{\eta ^N}\left[ \pi ^N_T\in {\mathcal {O}}\right] \;=\; -\, {\overline{w}}_i + \liminf _{N\rightarrow \infty }\frac{1}{N} \log {Q}_{T, \eta ^N}({\mathcal {O}}_T), \end{aligned}$$

where ${\mathcal {O}}_{T}=\{\pi '\in D([0,T], {\mathcal {M}}_+) : \pi _{T} \in {\mathcal {O}}\}$. Since the set ${\mathcal {O}}_T$ is open, by the lower bound of the dynamical large deviations principle, the previous expression is bounded below by

$$\begin{aligned} -\, {\overline{w}}_i \;-\; \inf _{\pi '\in {\mathcal {O}}_T} I_T(\pi ') \;\ge \; -\, {\overline{w}}_i \;-\; I_T(\pi ). \end{aligned}$$

In view of (2.12), this completes the proof of (6.11) and the one of the lower bound.

6.2 Upper bound

We prove in this subsection the large deviations upper bound. The proof relies on the next two lemmata. The proof of the first one is similar to the proof of Lemma 23 and is left to the reader.

For a closed subset ${\mathcal {C}}$ of ${\mathcal {M}}_+$ and $T>0$, let $C_T$ be the subset of $D([0, T],\mathcal {M}_{+})$ consisting of all paths $\pi$ for which there exists t in [0, T] such that $\pi (t)$ or $\pi (t-)$ belongs to ${\mathcal {C}}$. Note that $C_T$ is a closed subset of $D([0,T],\mathcal {M}_{+})$.

Lemma 27

Fix a closed subset ${\mathcal {C}}$ of ${\mathcal {M}}_+$ such that $\inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho )<\infty$. For every $\varepsilon >0$, there exist $\delta _{20} = \delta _{20}({\mathcal {C}}, \varepsilon )>0$ and $T_{20}>0$ such that for all $1\le i\le l$, $0<\beta _1<\delta _{20}$, $T'\ge T_{20}$, $\gamma (\theta )d\theta \in \varGamma _i$,

$$\begin{aligned} \inf _{\pi \in C_{T'} } I_{T'}(\pi |\gamma ) \;\ge \; \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) -\varepsilon . \end{aligned}$$

Recall the definition of the set B introduced just before Lemma 21 and recall from Corollary 2 that the set B is attained immediately. In particular, if the reaction–diffusion model has to reach a set ${\mathcal {C}}$ before it hits B, it has to follow straightforwardly the optimal trajectory to ${\mathcal {C}}$. The cost of such trajectory has been estimated in the previous lemma, providing the next result.

Lemma 28

Fix $1\le i\le l$ and a closed subset ${\mathcal {C}}$ of ${\mathcal {M}}_+$. For every $\varepsilon >0$, there exist $\delta _{21} = \delta _{21}({\mathcal {C}}, \varepsilon )>0$ such that for all $0<\beta _0< \beta _1<\delta _{21}$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \sup _{\eta \in \varGamma ^N_i} {\mathbb {P}}_{\eta }\left[ H_{{\mathcal {C}}}^N < H_{B}^N\right] \;\le \; - \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) \,+\, \varepsilon . \end{aligned}$$

Proof

Fix $\varepsilon >0$, $1\le i\le l$, and a closed subset ${\mathcal {C}}$ of ${\mathcal {M}}_+$. We may assume that the left hand side of the inequality appearing in the statement of the lemma is finite. This implies that $\pi ^{-1}_N({\mathcal {C}}) \cap X_N \not = \varnothing$ for infinitely many N’s. Let $\{\eta ^{N_k} : k\ge 1\}$ be a sequence of configurations such that $\pi ^{N_k}(\eta ^{N_k}) \in {\mathcal {C}}$. Since ${\mathcal {M}}_+$ is compact, taking a subsequence, if necessary, we may assume that $\pi ^{N_k}(\eta ^{N_k})$ converges to a measure, denoted by $\varrho$, which belongs to ${\mathcal {M}}_{+,1}$. Since ${\mathcal {C}}$ is closed, $\varrho \in {\mathcal {C}}$ so that ${\mathcal {C}} \cap {\mathcal {M}}_{+,1} \not = \varnothing$. In particular, By Lemma 18, $\inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho )<\infty$.

Let $\zeta _1$, $R_1$ be the constants $\delta _{20}$, $T_{20}$ given by Lemma 27. Fix $0<\beta _0<\beta _1<\zeta _1$. Since $\inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho )<\infty$, by Lemma 21, there exists $R_2>0$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \sup _{\eta \in X_N} {\mathbb {P}}_{\eta }\left[ H_{B}^N \ge R_2 \right] \;\le \; - \, \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ). \end{aligned}$$

(6.12)

Let $T=\max \{R_1, R_2\}$.

Recall the definition of the set $C_T$ introduced just above the statement of Lemma 27 and the fact that $C_T$ is a closed subset of $D([0,T],\mathcal {M}_{+})$. Note also that $\{H^N_{{\mathcal {C}}}\le T\} \subset C_T$.

Let $\eta ^N$ be a configuration in $\varGamma _i^N$ such that

$$\begin{aligned} {\mathbb {P}}_{\eta ^N}\left[ H^N_{{\mathcal {C}}}\le T\right] \;=\; \sup _{\eta \in \varGamma _i^N} {\mathbb {P}}_{\eta }\left[ H^N_{{\mathcal {C}}}\le T\right] . \end{aligned}$$

Taking a subsequence if necessary, we may assume that $\pi ^N(\eta ^N)$ converges to some measure $\varrho (d\theta )=\gamma (\theta )d\theta$ in $\varGamma _i\cap \mathcal {M}_{+,1}$. Since $\{H^N_{{\mathcal {C}}}\le T\} \subset C_T$ and since $C_T$ is a closed set, by the dynamical large deviations upper bound and by Lemma 27,

$$\begin{aligned}&\limsup _{N\rightarrow \infty }\frac{1}{N}\log \sup _{\eta \in \varGamma ^N_i} {\mathbb {P}}_{\eta ^N}\big (H^N_{{\mathcal {C}}} < H^N_B \le T \big ) \;\le \; \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathbb {P}}_{\eta ^N}\big (H^N_{{\mathcal {C}}}\le T \big ) \\&\qquad \;\le \; \limsup _{N\rightarrow \infty }\frac{1}{N} \log {Q}_{T,\eta ^N}(C_T) \;\le \; -\inf _{\pi \in C_T} I_{T}(\pi |\gamma ) \;\le \; - \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) \;+\; \varepsilon . \end{aligned}$$

By (6.12), by this estimate and by (6.4),

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \sup _{\eta \in \varGamma ^N_i} {\mathbb {P}}_{\eta }\left[ H_{{\mathcal {C}}}^N < H_{B}^N\right] \;\le \;- \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) \;+\; \varepsilon , \end{aligned}$$

which completes the proof of the lemma. $\square$

Proof of the upper bound of Theorem4. Let ${\mathcal {C}}$ be a closed subset of ${\mathcal {M}}_+$. Assume first that $\mathcal {M}_{i} \cap {\mathcal {C}} = \varnothing$ for any $1\le i \le l$. In this case, let $\beta _1>0$ be such that $\cup _{1\le i\le l} {\mathcal {B}}_{2\beta _1}(\mathcal {M}_{i})\cap {\mathcal {C}} = \varnothing$.

By the representation (6.6) of the stationary measure $\mu ^N$,

$$\begin{aligned} {\mathcal {P}}^{N}({\mathcal {C}})&= \mu ^N({\mathcal {C}}^N) \;=\; \frac{1}{C_N}\int _{\partial B^N} {\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {C}}^N \}\, ds\right) d\nu ^N(\eta ) \\&\le \frac{1}{C_N}\sum _{i=1}^l\nu ^N(\partial B_i^N) \sup _{\eta \in \partial B^N_i}{\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {C}}^N \}\, ds\right) . \end{aligned}$$

A configuration in $X_N$ can jump to at most 2N different configurations and the jump rates are bounded by $N^2$. Since any trajectory in $D({\mathbb {R}}_+,X_N)$ has to jump at least once before the stopping time $\tau$, the constant $C_N$ appearing in the denominator is bounded below by $c_0/N^{3}$ for some positive constant $c_0$. Hence, by (6.4) and by Proposition 3, in order to prove the upper bound it is enough to show that for each $1\le i \le l$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log \sup _{\eta \in \partial B^N_i} {\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {C}}^N\}\, ds\right) \;\le \; -\, \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) \;+\; \varepsilon . \end{aligned}$$

(6.13)

The time integral appearing in the previous formula vanishes if $\tau \le H^N_{{\mathcal {C}}}$. We may therefore introduce the indicator of the set $H^N_{{\mathcal {C}}} \le \tau$. After doing this and applying the strong Markov property, we obtain that the left hand side of the previous inequality is less than or equal to

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log \sup _{\eta \in \partial B^N_i} {\mathbb {P}}_{\eta }\left[ H_{{\mathcal {C}}}^N < \tau \right] \sup _{\eta \in {\mathcal {C}}^N} {\mathbb {E}}_{\eta }\left( \tau \right) . \end{aligned}$$

Since the distance between the empirical measure before and after a jump is bounded by C / N, and since ${\mathcal {B}}_{2\beta _1}(\mathcal {M}_{i})\cap {\mathcal {C}} = \varnothing$, for N large enough, any trajectory in $D({\mathbb {R}}_+, X_N)$ starting at some configuration in $\partial B_i^N$, ${\mathcal {C}}^N$, satisfies $H^N_{\varGamma _i} \le H^N_{{\mathcal {C}}}$, $\tau \le H^N_B$, respectively. Hence, by the strong Markov property, the previous expression is bounded above by

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \sup _{\eta \in \varGamma ^N_i}{\mathbb {P}}_{\eta }\left[ H_{{\mathcal {C}}}^N < H_{B}^N\right] \sup _{\eta \in {\mathcal {C}}^N} {\mathbb {E}}_{\eta }\left( H_B^N\right) . \end{aligned}$$

By Corollary 2 and Lemma 28, the previous expression is bounded by $-\, \inf _{\varrho \in {\mathcal {C}}} V_{i}(\varrho ) + \varepsilon$, which completes the proof of (6.13) and the one of the upper bound in the case $\mathcal {M}_{i} \cap {\mathcal {C}} = \varnothing$ for $1\le i \le l$.

We turn to the general case. We first claim that for each $1\le i\le l$, $\varepsilon >0$, there exists $\zeta _1>0$ such that for all $0<\beta _0<\zeta _1$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^{N}({\mathcal {B}}_{\beta _0}({\mathcal {M}}_{i})) \;\le \; - \, {\overline{w}}_{i} \;+\; 2 \varepsilon . \end{aligned}$$

(6.14)

Indeed, fix $\varepsilon >0$ and set $\zeta _1 = \min \{\delta _{18}, \delta _{19}\}$, where $\delta _{18}>0$ is the constant provided by Proposition 3 and $\delta _{19}>0$ is the one given by Lemma 26. Fix $\beta _0<\zeta _1$. By the representation (6.6) of the stationary measure $\mu ^N$,

$$\begin{aligned} {\mathcal {P}}^{N}({\mathcal {B}}_{\beta _0}({\mathcal {M}}_{i}))&= \mu ^N({\mathcal {B}}^N_{\beta _0}({\mathcal {M}}_{i})) = \frac{1}{C_N}\int _{\partial B^N} {\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {B}}^N_{\beta _0}({\mathcal {M}}_{i}) \}\, ds\right) d\nu ^N(\eta ) \\&\le \frac{1}{C_N}\sum _{j=1}^l\nu ^N(\partial B_j^N) \sup _{\eta \in \partial B^N_j}{\mathbb {E}}_{\eta }\left( \int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {B}}^N_{\beta _0}({\mathcal {M}}_{i}) \}\, ds\right) . \end{aligned}$$

We have seen in the first part of the proof that $\limsup _N N^{-1} \log C^{-1}_N \le 0$. On the other hand, for $\eta \in \partial B_j^N$, $j\not = i$, $\tau \le H^N_{B_i}$, so that $\int _0^{\tau } \mathbf 1 \{\eta _s\in {\mathcal {B}}^N_{\beta _0}({\mathcal {M}}_{i}) \}\, ds =0$. Finally, denote by $\vartheta (t)$, $t>0$, the time translation of a trajectory by t. For $\eta \in \partial B_i^N$, writing $\tau$ as $H^N_{\varGamma _i} + H^N_B \circ \vartheta (H^N_{\varGamma _i})$, by the strong Markov property, since $\beta _0<\zeta _1$, and by Proposition 3, the left hand side of (6.14) is bounded by

$$\begin{aligned} -\, {\overline{w}}_i \;+\; \varepsilon \;+\; \limsup _{N\rightarrow \infty }\frac{1}{N}\log \Big \{ \sup _{\eta \in \partial B^N_i}{\mathbb {E}}_{\eta } (H^N_{\varGamma _i}) \,+\, \sup _{\eta \in \partial \varGamma ^N_i} {\mathbb {E}}_{\eta } (H^N_{B}) \Big \}. \end{aligned}$$

By (6.4), (6.7) and Corollary 2, the limit superior of the previous equation is bounded by $\varepsilon$, which completes the proof of (6.14).

Let ${\mathcal {C}}$ be a closed subset of ${\mathcal {M}}_+$ and fix $\varepsilon >0$. Let A be the set of indices i such that ${\mathcal {C}} \cap {\mathcal {M}}_i \not = \varnothing$. Let $\zeta _1$ be the positive constant introduced in (6.14), and choose $\beta _0 <\zeta _1$ such that $d({\mathcal {C}}, {\mathcal {M}}_j)>\beta _0$ for all $j\in A^c$. Since ${\mathcal {C}} \subset \cup _{i\in A} {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i) \cup [ {\mathcal {C}} \setminus \{\cup _{i\in A} {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i)\}]$, and since ${\mathcal {C}} \setminus \{\cup _{i\in A} {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i)\}$ is a closed set which does not intersect the set ${\mathcal {M}}_\mathrm{sol}$, by (6.4), by (6.14) and by the first part of the proof,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log {\mathcal {P}}^{N}({\mathcal {C}}) \;\le \; - \min \Big \{ \min _{i\in A} {\overline{w}}_i \,,\, \inf _{\pi \in {\mathcal {C}} \setminus \{\cup _{i\in A} {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i)\}} W(\pi ) \Big \} \;+\; 2\varepsilon . \end{aligned}$$

By (2.15), ${\overline{w}}_i = W({{\bar{\varrho }}}_{i})$ for ${{\bar{\varrho }}}_{i}\in {\mathcal {M}}_i$. On the other hand, since ${\mathcal {M}}_i \cap {\mathcal {C}} \not = \varnothing$,

$$\begin{aligned} \inf _{\pi \in {\mathcal {C}}} W(\pi ) \;\le \; \min _{i\in A} W({{\bar{\varrho }}}_{i}), \quad \inf _{\pi \in {\mathcal {C}}} W(\pi ) \;\le \inf _{\pi \in {\mathcal {C}} \setminus \{\cup _{i\in A} {\mathcal {B}}_{\beta _0}({\mathcal {M}}_i)\}} W(\pi ), \end{aligned}$$

which completes the proof of the upper bound.

7 Proof of Theorem 5

We first show that if there exists a heteroclinic orbit from $\phi \in {\mathcal {M}}_i$ to $\psi \in {\mathcal {M}}_j$, then the cost of going from ${\mathcal {M}}_i$ to ${\mathcal {M}}_j$ vanishes.

Lemma 29

Suppose that there exists a heteroclinic orbit from $\phi \in {\mathcal {M}}_i$ to $\psi \in {\mathcal {M}}_j$. Then, $v_{ij}=0$.

Proof

Fix $i\not = j$ in $\{1, \dots , l\}$ and assume that there exists a heteroclinic orbit from $\phi \in {\mathcal {M}}_i$ to $\psi \in {\mathcal {M}}_j$, denoted by $\rho (t,\theta )$, $t\in {\mathbb {R}}$. By Proposition 2, $\phi$ is smooth, and, by Lemma 4, there exists $0<c<1/2$ such that $c\le \phi (\theta )\le 1-c$. Since $\rho (t)$, converges in $C^1({\mathbb {T}})$ to $\phi$, $\psi$ as $t\rightarrow - \infty$, $t\rightarrow + \infty$ respectively, by Lemmata 17, 19 and since the dynamical large deviations rate functional vanishes along the solution of the hydrodynamic equation, $v_{ij} = 0$. $\square$

We now prove that $\rho (\theta ) = r$ is a stable solution of the reaction–diffusion equation (2.5) if r is a local minimum of V.

Lemma 30

Fix $1\le i\le l$. Let ${{\bar{\varrho }}}_{i}(d\theta )={\bar{\rho }}_i(\theta )d\theta , {\bar{\rho }}_i(\theta ) = r$, where r is a local minimum of V. Then, for all $\varepsilon >0$ there exist $c>0$ such that

$$\begin{aligned} \inf \big \{ V_i(\varrho ): \varrho \not \in {\mathcal {B}}_\varepsilon ({\bar{\varrho }}_i) \big \} \;\ge \; c. \end{aligned}$$

Proof

Suppose that $\inf \{ V_i(\varrho ) : \varrho \in {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c\} = 0$ for some $\delta >0$. In this case there exists a sequence of density profiles $\gamma _n$ and of trajectories $\pi ^n (t,d\theta ) = \rho ^n(t,\theta ) d\theta$, $0\le t\le T_n$, such that $\rho ^n(0,\theta ) = {\bar{\rho }}_i$, $\rho ^n(T_n,\theta ) = \gamma _n(\theta )$, $\gamma _n(\theta ) d\theta \in {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c$ and $I_{T_n} (\pi ^n) \le 1/n$.

By Lemma 9, there exists $0<\varepsilon <\delta$ such that $\pi _t \in {\mathcal {B}}_\delta ({{\bar{\varrho }}}_{i})$ for all $t\ge 0$ if $\pi _0 \in {\mathcal {B}}_{2\varepsilon } ({{\bar{\varrho }}}_{i})$. Let $\tau _n$ be the time the trajectory $\pi ^n$ leaves the set ${\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})$ for ever, and let $\sigma _n$ be the hitting time of the set ${\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c$ after $\tau _n$:

$$\begin{aligned} \tau _n \;=\; \sup \{t\le T_n : \pi ^n_t \in {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})\}, \quad \sigma _n \;=\; \inf \{ t\ge \tau _n : \pi ^n_t \in {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c\}. \end{aligned}$$

Since in the interval $[\tau _n, \sigma _n]$ the trajectory $\pi ^n$ remains in the set ${\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i}) \setminus {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})$, if $\delta$ is small enough for ${\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i}) \cap {\mathcal {B}}_{\delta }({\mathcal {M}}_j) = \varnothing$ for all sets ${\mathcal {M}}_j$, $j\not = i$, by Corollary 1, $\sigma _n - \tau _n$ is uniformly bounded by a finite constant, denoted by T.

Extend the definition of $\pi ^n$ from the interval $[0,T_n]$ to ${\mathbb {R}}_+$ by following the hydrodynamic trajectory after $T_n$: $\pi ^n(T_n+t,d\theta ) = {\tilde{\rho }}^n(t,\theta ) d\theta$, where ${\tilde{\rho }}_n$ is the solution of the hydrodynamic equation with initial condition $\rho ^n_{T_n}$. Let ${\bar{\pi }}^n_t$, $0\le t\le T$, be the trajectory defined by ${\bar{\pi }}^n_t = \pi ^n(\tau _n +t)$. Since $\pi ^n$ belongs to $C([0,T_n],\mathcal {M}_{+,1})$, note that ${\bar{\pi }}^n_0\in \partial {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})$, that ${\bar{\pi }}^n$ hits the set ${\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c$ in the time interval [0, T] and that $I_T({\bar{\pi }}^n) \le 1/n$.

By the compactness of the level sets of $I_T$, the lower semi-continuity of this functional and the compactness of the space ${\mathcal {M}}_+$, there exists a subsequence ${\bar{\pi }}^{n_k}$ which converges to some trajectory $\pi$ such that $\pi _0\in \partial {\mathcal {B}}_{\varepsilon }({{\bar{\varrho }}}_{i})$, $\pi$ hits the set ${\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})^c$ in the time interval [0, T] and $I_T(\pi ) =0$. By Lemma 12, the density of $\pi _t$, denoted by $\rho _t$, is a solution of the hydrodynamic equation. This contradicts the property of $\varepsilon$ and concludes the proof of the lemma. $\square$

Proof of Theorem 5

Recall the definition of the set of indices $I_s$, $I_u$. We claim that ${\overline{w}}_a >0$ for all $a\in I_u$. To prove this statement, it is enough to show that for each $a\in I_u$, there exists $b \in I_s$ such that $w_a > w_b$.

Fix $a\in I_u$. By assumption, there exists $b \in I_s$ such that $v_{ab} =0$. We claim that $w_a > w_b$. Indeed, on the one hand, by Lemma 30, $v_{bc}>0$ for all $c\not = b$. On the other hand, let g be a graph in ${\mathcal {T}}(a)$ such that $w_a = \kappa (g)$. Recall that we denote by (d, e), $e\not = d\in {\mathcal {V}}$, the oriented edge where d is the child and e the parent. Let c be the parent of b in g. Of course, c might be a. Denote by $g'$ the tree in ${\mathcal {T}}(b)$ obtained from g by adding the oriented edge (a, b) and removing the the edge (b, c), and note that $\kappa (g) + v_{ab} = \kappa (g') + v_{bc}$. Since $w_b$ is the minimal value of $\kappa ({\tilde{g}})$, ${\tilde{g}}\in {\mathcal {T}}(b)$, $w_b \le \kappa (g')$ so that $w_b + v_{bc} \le \kappa (g) + v_{ab} = w_a + v_{ab} = w_a$. The last identity follows from the fact that $v_{ab}=0$ and the next to last from the fact that $\kappa (g) = w_a$. Since $v_{bc}>0$, we conclude that $w_b< w_a$, as claimed.

We claim that for every $\delta >0$,

$$\begin{aligned} \inf \left\{ W(\pi ) : \pi \not \in \bigcup _{i\in I_s} {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i}) \right\} \;>\; 0. \end{aligned}$$

(7.1)

Fix $\delta >0$. Since ${\overline{w}}_a>0$ for all $a\in I_u$, in view of the definition of W, we only need to check that

$$\begin{aligned} \inf \left\{ V_j (\pi ) : \pi \not \in \bigcup _{i\in I_s} {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i}) \right\} \;>\; 0 \end{aligned}$$

for each $j\in I_s$. This is the content of Lemma 30, proving (7.1)

To complete the proof of the theorem, it remains to observe that the complement of $\cup _{i\in I_s} {\mathcal {B}}_{\delta }({{\bar{\varrho }}}_{i})$ is a closed set and to apply the upper bound of the static large deviations principle stated in Theorem 4. The theorem is proved. $\square$

8 The Chafee–Infante equation

We present in this section an example of a reaction–diffusion model which fulfills the hypotheses of Theorems 4 and 5. Actually, in this model a complete description of the stationary solutions and of the heteroclinic orbits is available.

Fix $0<a<b$ and recall the definition of the potential $V=V_{a,b}$ introduced in (2.7). Denote by ${\mathbb {T}}_{1/2\pi }$ the one-dimensional torus with length $(2\pi )^{-1}$. Let $p = (1/2) \sqrt{a/b}$, $c = 2(2\pi )^2$ and define $\phi :{\mathbb {R}}_+ \times {\mathbb {T}}_{1/2\pi } \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \phi (t,\theta ) \;=\; \frac{1}{p} \, \Big \{ \rho \big ( c\, t \,,\, 2\pi \theta \big ) \,-\, \frac{1}{2} \Big \}. \end{aligned}$$

A simple computation shows that $\rho$ solves the Eq. (2.7) if and only if $\phi$ solves

$$\begin{aligned} \partial _t \phi \;=\; \varDelta \phi \;+\; \lambda (1-\phi ^2)\phi , \end{aligned}$$

(8.1)

where $\lambda = 4ca= 8 (2\pi )^2 a$. Note that $\phi$ takes values in the interval $[- \sqrt{b/a}, \sqrt{b/a}]$. This is the so-called Chafee–Infante equation [13] with periodic boundary condition.

A complete characterization of the stationary solutions of the Chafee–Infante equation with periodic boundary conditions is presented in [30, Proposition 1.1]. In our context it can be stated as follows. Let $\rho _\pm$ be the minima of V: $\rho _\pm = (1/2) \pm p = (1/2) [1 \pm \sqrt{a/b}]$.

Theorem 8

For all $0<a<b$, the Eq. (8.1) admits three constant stationary solutions: $\psi _{\pm } = \rho _\pm$, $\psi _{1/2}=1/2$. For all nonnegative integers m such that $1\le m^2 < \lambda = 32 \pi ^2 a$, up to translations, there exists a non-constant periodic stationary solution $\phi _m=\phi _{m,\lambda }$ with m periods in ${\mathbb {T}}_{1/2\pi }$. Moreover, $\lim _{\lambda \downarrow m^2} \phi _{m,\lambda } = 1/2$ in $C^2({\mathbb {T}}_{1/2\pi })$. The reaction–diffusion equation (2.4) with potential $V_{a,b}$ has no other stationary solutions.

The heteroclinic orbits of the Chafee–Infante equation with periodic boundary conditions have been characterized in [23]. Next result follows from Theorems 1.3 and 1.4 of [23],

Theorem 9

There are heteroclinic orbits from $\psi _{1/2}$ to $\psi _{\pm }$, and from $\psi _{1/2}$ to $\phi _m$ for all integers $1\le m^2 <\lambda$. Fix $1\le n^2 <\lambda$. There are heteroclinic orbits from $\phi _n$ to $\psi _{\pm }$, and from $\phi _n$ to $\phi _m$ for all integers $1\le m^2<n^2$. There are no other heteroclinic orbits.

Next proposition follows from the previous results and from Theorem 5.

Proposition 4

Consider a reaction–diffusion model which satisfies the assumptions of Theorem 4 and which gives rise to the hydrodynamic equation (2.7) with $0<a<b$. Let ${{\bar{\varrho }}}_\pm (d\theta ) = \rho _\pm d\theta$. Then, for every $\delta >0$, there exist $c>0$ and $N_0\ge 1$ such that for all $N\ge N_0$,

$$\begin{aligned} {\mathcal {P}}^N \big ({\mathcal {B}}_\delta ({{\bar{\varrho }}}_{-}) \cup {\mathcal {B}}_\delta ({{\bar{\varrho }}}_{+}) \big ) \;\ge \; 1 \;-\; e^{-cN}. \end{aligned}$$

If the jump rates are invariant under a global flipping of the configuration: $c(\eta ) = c({\varvec{1}}-\eta )$, where ${\varvec{1}}$ is the configuration with all sites occupied, there is a symmetry between occupied and vacant sites so that ${\mathcal {P}}^N ({\mathcal {B}}_\delta ({{\bar{\varrho }}}_+)) = {\mathcal {P}}^N ({\mathcal {B}}_\delta ({{\bar{\varrho }}}_-))$ for all $\delta >0$. Hence, with this additional assumption, we may refine the previous proposition:

Corollary 3

Under the assumptions of Proposition 4, if $c(\eta ) = c({\varvec{1}}-\eta )$, for every $0<\delta < (1/2) d({{\bar{\varrho }}}_{-}, {{\bar{\varrho }}}_{+})$, there exist $c>0$ and $N_0\ge 1$ such that for all $N\ge N_0$,

$$\begin{aligned} \Big | {\mathcal {P}}^N ({\mathcal {B}}_\delta ({{\bar{\varrho }}}_-)) - \frac{1}{2}\, \Big | \;\le \; e^{-cN}, \quad \Big | {\mathcal {P}}^N ({\mathcal {B}}_\delta ({{\bar{\varrho }}}_+)) - \frac{1}{2}\, \Big | \;\le \; e^{-cN}. \end{aligned}$$

An example. We conclude this section with an example of a reaction–diffusion model satisfying the assumptions of Theorem 4 and whose hydrodynamic equation is given by (2.7).

Consider the reaction–diffusion model whose jump rate $c(\eta )$ is given by

$$\begin{aligned} c(\eta ) \; =\; a_2 {\varvec{1}}\{ \eta _{-1} \not = \eta _1\} \;+\; a_1 {\varvec{1}}\{ \eta _{-1} = \eta _1 = \eta _0 \} \; +\; a_0 {\varvec{1}}\{ \eta _{-1} = \eta _1 \not = \eta _0 \}. \end{aligned}$$

Let $\xi$ be the configuration obtained from $\eta$ by flipping all occupation variables: $\xi (x) = 1 - \eta (x)$, $x\in {\mathbb {T}}_N$. Since $[1-\eta (0)] c(\eta ) = \xi (0) c(\xi )$, $B(\rho ) = D(1-\rho )$. Moreover,

$$\begin{aligned} F(\rho ) \;=\; \frac{1}{4} \Big \{ (a_0-3a_1-2a_2) (2\rho -1) - (a_0 + a_1-2a_2) (2\rho -1)^3 \Big \}. \end{aligned}$$

Fix $0<{\mathfrak {a}}<{\mathfrak {b}}$, and set $a_1={\mathfrak {a}}>0$. Choose $a_2\ge {\mathfrak {a}} + 2 {\mathfrak {b}}>0$ and set $a_0 = 2 a_2 + 4{\mathfrak {b}} - {\mathfrak {a}} \ge {\mathfrak {a}} + 8{\mathfrak {b}} >0$. Since the three parameters are positive, the jump rate is strictly positive as required.

As $a_0-3a_1-2a_2 = 4({\mathfrak {b}} - {\mathfrak {a}})$ and $a_0 + a_1-2a_2 = 4{\mathfrak {b}}$, in the variables ${\mathfrak {a}}$, ${\mathfrak {b}}$ the function F becomes

$$\begin{aligned} F(\rho ) \;=\; ({\mathfrak {b}}-{\mathfrak {a}}) (2\rho -1) \;-\; {\mathfrak {b}} (2\rho -1)^3, \end{aligned}$$

in conformity with (2.7) for $a= ({\mathfrak {b}}-{\mathfrak {a}})/2$, $b={\mathfrak {b}}/2$.

Since $B(\rho ) = D(1-\rho )$, D is concave if and only if B is concave. The functions B is concave if $3a_1 +a_0\le 4a_2 \le 4a_0$. We claim that these inequalities are in force. On the one hand, as ${\mathfrak {b}}> {\mathfrak {a}}>0$ and $a_2>0$, we have that $a_2 < 2 a_2 + 4{\mathfrak {b}} - {\mathfrak {a}} = a_0$. On the other hand, $3a_1 +a_0 -4a_2 = 2a_1 + 4{\mathfrak {b}} - 2a_2 = 2{\mathfrak {a}} + 4{\mathfrak {b}} - 2a_2 \le 0$ from the definition of $a_2$.

This shows that all the assumptions of Theorem 4 are fulfilled.