Abstract
We show that the solutions to the damped stochastic wave equation converge pathwise to the solution of a stochastic heat equation. This is called the Smoluchowski–Kramers approximation. Cerrai and Freidlin have previously demonstrated that this result holds in the cases where the system is exposed to additive noise in any spatial dimension or when the system is exposed to multiplicative noise and the spatial dimension is one. The current paper proves that the Smoluchowski–Kramers approximation is valid in any spatial dimension when the system is exposed to multiplicative noise.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The motion of an elastic material in a region \(D \subset {{\mathbb {R}}}^d\) exposed to friction as well as deterministic and random forcing can be described by the damped stochastic wave equation
In the above equation, \(\mu >0\) is the mass-density of the material. The forcing term \(\varDelta u^\mu \) describes the forces neighboring particles exert on each other, \(-\partial u^\mu /\partial t\) models a constant friction term, b is a nonlinear forcing term, and \(g Q \partial w/ \partial t\) is a space and time dependent stochastic forcing. The noise is driven by w(t), a \(L^2(D)\)-cylindrical Wiener processes [15, Chapter 4.2.1]. The Dirichlet boundary conditions guarantee that the boundary of the elastic material is fixed. Initial conditions are also prescribed.
We study the asymptotics of the solutions to this equation as the mass density \(\mu \rightarrow 0\) and demonstrate that the solutions converge to the solutions of a stochastic heat equation
The heat equation can be thought of as (1.1) with \(\mu \) formally replaced by 0
This limit, the Smoluchowski–Kramers approximation, was first investigated by Smoluchowski [28] and Kramers [24] for finite dimensional diffusions of the form
where \(X^\mu \) is \({\mathbb {R}}^d\)-valued, \(b: [0,+\,\infty ) \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d\) is a vector field and \(g: [0,+\,\infty ) \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}^{d\times k}\), and W(t) is a k-dimensional Wiener process. As \(\mu \rightarrow 0\) the solutions converge pathwise on finite time intervals to the solution of the first-order equation
Furthermore, the first-order equation approximates some longer-time behaviors of the second-order system including invariant measures and exit time problems. Many Smoluchowski–Kramers results for finite dimensional systems are summarized in [18] including pathwise convergence, invariant measures, Wong–Zakai approximation, homogenization, and large deviations. Various generalizations including the presence of state-dependent friction have been investigated in the finite dimensional case [1, 7, 8, 14, 19,20,21,22,23, 25, 29].
The Smoluchowski–Kramers approximation for stochastic partial differential equations such as (1.1) were first investigated by Cerrai and Freidlin [5, 6]. In [5], they considered the additive noise case where \(g(t,x,u) \equiv 1\) and in [6], they considered the multiplicative noise case when the spatial dimension \(d=1\). In each case they show that the solutions \(u^\mu (t,x)\) of (1.1) converge to the solutions of (1.2) pathwise in probability, in the sense that for any \(T>0\) and \(\delta >0\)
The Smoluchowski–Kramers approximation in the presence of a magnetic field and Smoluchowski–Kramer’s interplay with large deviations in the small noise regime for infinite dimensional systems and with invariant measures have also been investigated [9,10,11,12,13, 26, 27].
The main results of this paper fill a gap in the literature by demonstrating that the Smoluchowski–Kramers approximation is valid in the case of multiplicative noise in any spatial dimension \(d\ge 1\) if the noise covariance Q satisfies appropriate assumptions. Furthermore, the methods in this paper allow us to improve from convergence in probability as in (1.5) to \(L^p\) convergence. In particular the main result of this paper, Theorem 4.2, proves that for any \(T>0\) and \(p\ge 1\),
If \(D \subset {\mathbb {R}}^d\) is an open region with smooth boundary then there is a complete orthonormal basis of \(L^2(D)\) consisting of eigenfunctions of \(\varDelta \) such that \(\varDelta e_k(x) = -\,\alpha _k e_k(x)\) for an increasing sequence of eigenvalues \(\alpha _k\ge 0\). Weyl’s Theorem [17, page 356] guarantees that the eigenvalues of \(-\varDelta \) with Dirichlet boundary conditions behave like \(\alpha _k \sim k^{2/d}\) as \(k \rightarrow +\,\infty \). In dimension \(d=1\), the eigenvalues have the useful property that \(\sum _{k=1}^\infty \frac{1}{\alpha _k}< +\,\infty \). A consequence is that (1.1) is well-defined when is exposed to white noise (the case where \(Q=I\) is the identity) (see [6]). In dimensions \(d\ge 2\), the noise must be more regular than white noise in order for (1.1) to be well-defined.
In the additive noise case considered in [5], the Smoluchowski–Kramers approximation is proved under the assumption that Q is diagonalized by the same basis of eigenfunctions as the Laplacian with eigenvalues \(Q e_k = \lambda _k e_k\) and that \(\sum _{k=1}^\infty \frac{\lambda _k^2}{\alpha _k^{1-\theta }} < +\,\infty \) for some \(\theta \in (0,1)\). This is also the minimal condition that guarantees that the solutions to (1.1) and (1.2) are well-defined and function valued.
The minimal conditions on the noise covariance Q that guarantee that the heat equation with multiplicative noise (1.2) is well-defined and function valued are characterized in [2,3,4]. We assume that Q is diagonalized by the same sequence of eigenfunctions as the A, \(Q e_k = \lambda _k e_k\). In the dimension \(d=1\) case, (1.2) is well-defined if the eigenvalues of Q are assumed to be uniformly bounded. In dimensions \(d\ge 2\), (1.2) is well-defined if the eigenvalues of Q are assumed to satisfy
for some \(q,\beta >0\) satisfying \(\frac{\beta (q-2)}{q}<1\). In the case where the eigenfunctions of the Laplacian are equibounded and the \(\alpha _k \sim \frac{2}{d}\), this simplifies to the condition that
In this paper, we show that the solutions to (1.1) exist and are function valued under the same conditions on the eigenvalues of Q. This requires a novel proof because the argument of [2,3,4] relies on the fact that the heat equation semigroup is analytic, but the wave equation semigroup is not analytic. Furthermore, we show that the Smoluchowski–Kramers approximation is valid in the sense that (1.6) holds under these same minimal assumptions on Q.
The proofs of the well-posedness of (1.1) and the Smoluchowski–Kramers approximation (1.6) are both based on a careful analysis of the wave equation semigroup.
The paper is organized as follows. In Sect. 2, we describe the assumptions and notations used in the paper. In Sect. 3, we recall some results about the heat equation. In Sect. 4, we state the main results of this paper. In Sect. 5, we carefully analyze the properties of the wave equation semigroup. In Sect. 6, we analyze the properties of the stochastic convolutions with the wave equation semigroup. In Sect. 7, we apply the results from Sects. 5 and 6 to prove that the stochastic wave equation is well-defined. Finally, in Sect. 8 we prove that the mild solutions to the stochastic wave equation converge to the mild solution of the stochastic heat equation.
2 Assumptions and notations
We consider the damped stochastic wave equation (1.1) under the following assumptions.
Assumption 2.1
The functions \(b: [0,+\,\infty )\times D\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(g: [0,+\,\infty )\times D \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) are uniformly Lipschitz continuous and have sublinear growth in the third variable. There exists \(C\ge 0\) such that for any \(u,v \in {\mathbb {R}}\),
and
Assume that \(D\subset {\mathbb {R}}^d\) is a bounded set with smooth boundary. Define \(H=L^2(D)\) and let A be the realization of the Laplace operator in H with Dirichlet boundary conditions. There exists a sequence of eigenfunctions of A that form a complete orthonormal basis of H. We list the eigenvalues in increasing order \(0< \alpha _1\le \alpha _k\le \alpha _{k+1}\) so that
Because the boundary of D is smooth, the eigenfunctions \(e_k\) are infinitely differentiable functions on the closure of D (see, for example, [17, Thoerem 6.5.1]).
The cylindrical Wiener process w(t) is defined as the formal sum
where \(\{\beta _k(t)\}\) is a sequence of independent one-dimensional Brownian motion on a common probability space. Integration against a cylindrical Wiener process is defined in [15, Chapter 4.2.1].
For a positive self-adjoint operator \(Q \in \mathcal {L}_+(H)\) diagonalized by the basis \(\{e_j\}\) with eigenvalues \(Qe_j = \lambda _j e_j\), define
Assumption 2.2
The operator \(Q \in {\mathscr {L}}_+(H)\) is diagonalized by the same orthonormal basis of H as A. Q has eigenvalues \(\lambda _j\ge 0\) satisfying
There exist constants \(q \in [2,+\,\infty ]\) and \(\beta >0\) satisfying
and
In the case where \(q=+\,\infty \), (2.6) means that \(\beta <1\).
Remark 2.3
By Weyl’s Theorem ( [17, page 356]), the eigenvalues of the Laplacian grow like \(\alpha _k \sim k^{\frac{2}{d}}\) where d is the spatial dimension of the domain D. If the \(e_k\) are equibounded in the \(L^\infty (D)\) norm (which is the case when D is a generalized rectangle) then (2.5)–(2.6) simplifies to the condition that \( \Vert Q\Vert _q <+\,\infty \) where \(q=+\,\infty \) if \(d=1\) and \(2<q< \frac{2d}{d-2}\) if \(d\ge 2\). This is the same as Assumption 2 in [2].
Condition (2.5) also guarantees that the heat equation is well-posed in the more general case that the eigenfunctions are not equibounded (see for example [4, Hypothesis 1]). We will prove that the same conditions on Q that imply the well-posedness of the stochastoc heat equation imply the well-posedness of the stochastic wave equation as well as the validity of the Smoluchowski–Kramers approximation.
Remark 2.4
Because the \(\alpha _k \sim k^{2/d}\) and \(\inf _k |e_k|_{L^\infty (D)}>0\), the condition \(\Vert (-A)^{-1}\Vert _\beta <+\,\infty \) requires \(\beta >1\) unless the spatial dimension \(d=1\). This means that q could only possibly be \(+\,\infty \) if \(d=1\). On the other hand, \(|e_k|_{L^\infty (D)}\) can not grow arbitrarily quickly. There must exist some \(\rho >0\) such that \(|e_k|_{L^\infty (D)} \le C \alpha _k^\rho \le C k^{2\rho /d}\) (see for example [17, Theorem 6.3.5]). This means that there always exists some \(\beta <+\,\infty \) such that \(\Vert (-A)^{-1}\Vert _\beta <+\,\infty \), and therefore one can always choose \(q>2\).
For \(\delta \in {\mathbb {R}}\), define the Hilbert spaces \(H^\delta \) to be the completion of \(C_0^\infty (D)\) under the norm
For \(\delta >0\), these spaces are equivalent to the fractional Sobolev spaces \(W^{\delta ,2}_0(D)\) [16].
It is helpful to study the wave equation as a system in an appropriate phase space,
Define the phase spaces \({\mathcal {H}}_\delta := H^\delta \times H^{\delta -1}\). We also use the notation \({\mathcal {H}}:= {\mathcal {H}}_0\). Define the linear operator \(A_\mu : D(A_\mu )={\mathcal {H}}_{\delta -1} \rightarrow {\mathcal {H}}_\delta \) by
The operator \({\mathcal {A}_\mu }\) generates a \(C_0\) semigroup \({\mathcal {S}_\mu }(t): {\mathcal {H}}_\delta \rightarrow {\mathcal {H}}_\delta \).
Define the composition mapping \(B:[0,+\,\infty )\times H \rightarrow H\) by, for any \(t\ge 0\) and \(u \in H\)
Define the composition operator \(G: [0,+\,\infty ) \times H \rightarrow {\mathscr {L}}(L^\infty (D):H)\) by, for any \(t\ge 0\), \(u \in H\), and \(h \in L^\infty (D)\),
Note that for \(u \in H\), G(t, u) is also well-defined as a bounded linear mapping from H to \(L^1(D)\) by Hölder inequality. Because of Assumption 2.1, B and G are Lipschitz continuous in the second variable.
Define \(\varPi _1: {\mathcal {H}}_\delta \rightarrow H^\delta \) is the projection onto the first component and \(\varPi _2: {\mathcal {H}}_\delta \rightarrow H^{\delta -1}\) is the projection onto the second component. That is, for any \((u,v) \in {\mathcal {H}}^\delta \),
Define \({\mathcal {I}}_\mu : H^\delta \rightarrow {\mathcal {H}}_\delta \) such that
The Eq. (2.7) can be rewritten in the abstract formulation where \(z^\mu (t) = (u^\mu (t),v^\mu (t))\)
Definition 2.5
The mild solution to (2.13) is defined to be the solution of the integral equation.
where \(z_0 = (u_0,v_0)\). Then \(u^\mu (t) = \varPi _1 z^\mu (t)\) is the mild solution to (1.1).
For any \(T>0\) the function spaces C([0, T] : H) and \(C([0,T]:{\mathcal {H}})\) are the Banach spaces of H (resp. \({\mathcal {H}}\))-valued continuous functions on [0, T]. They are endowed with the supremeum norm
Let \((\varOmega , \mathcal {F}, {\mathbb {P}})\) be a probability space. For any Banach space E, the space \(L^p(\varOmega :E)\) is the set of all E-valued random variables with the property that \({\mathbb {E}}|\varphi |_E^p<+\,\infty \). \(L^p(\varOmega :E)\) is a Banach space. In this paper we are most interested in the case where \(E = C([0,T]:H)\) or \(E= C([0,T]:{\mathcal {H}})\).
Throughout this paper, the letter C refers to an arbitrary positive constant whose value can change from line to line.
3 Heat equation
In this section we recall some of the well-posedness results for the heat equation (1.2). Using the notation of Sect. 2, (1.2) can be written in the abstract formulation in H
The mild solution for the heat equation is the solution to the integral equation
where S(t) is the heat equation semigroup, which satisfies \(S(t)e_k = e^{-\alpha _k t} e_k\). All of the results of this section can be found in [2,3,4].
Denote the heat equation’s stochastic convolution by
where we will set \(\varPhi (t) = G(t,\varphi (s))\) or \(\varPhi (t) = (G(t,\varphi (t)) - G(t,\psi (t)))\).
By the factorization formula of [15, Chapter 5.3.1],
where
We collect some results that we will use later in the paper.
Lemma 3.1
Let \(q,\beta \) satisfy (2.5)–(2.6). For any \(\alpha \in (0,1/2)\) satisfying \(0<2\alpha <1- \frac{\beta (q-2)}{q}\), \(p > \frac{1}{\alpha }\), and any \(T>0\), there exists \(C=C(T,p,\alpha )>0\) such that for any \(t \in [0,T]\),
For more information about the proof of this Lemma see Lemma 3.3 of [2] or Lemma 4.1 of [4].
Lemma 3.2
Let \(q,\beta \) satisfy (2.5)–(2.6). For \(\alpha \in (0,1/2)\) satisfying \(0<2\alpha <1- \frac{\beta (q-2)}{q}\) and \(p \ge \frac{1}{\alpha }\),
Lemma 3.3
Let \(P_N\) be the projection onto \(\text {span}\{e_k\}_{k=1}^N\). Let \(\varPhi \) fixed progressively measurable \({\mathscr {L}}(L^\infty (D),H)\) valued process satisfying
Then for any fixed \(\alpha >0\) satisfying the conditions of Lemma 3.2,
Proof
This is an immediate consequence of the dominated convergence theorem. \(\square \)
The following Theorem is presented in [4, Proposition 4.2] and we state it without proof.
Theorem 3.4
(Proposition 4.2 of [4]) Assume that Assumptions 2.1 and 2.2 hold. For any initial condition \(u_0 \in H\), there exists a unique solution \(u\in L^p(\varOmega : C([0,T]:H))\) to (3.2) where \(p\ge 2\) satisfies the conditions of Lemma 3.2.
The proof is based on the well-posedness of the stochastic convolutions and a fixed point argument.
4 Main results
The first main result of this paper is that the mild solutions \(z^\mu \) solving (2.14) are well defined.
Theorem 4.1
Assume that Assumptions 2.1 and 2.2 hold. For any initial conditions \((u_0,v_0) \in {\mathcal {H}}\) and \(\mu >0\), there exists a unique mild solution \(z^\mu \in L^p(\varOmega : C([0,T]:{\mathcal {H}}))\) to (2.14).
The proof of Theorem 4.1 is given in Sect. 7. The proof requires careful analysis of the Fourier decomposition of the wave equation semigroup and the stochastic convolution, which can be found in Sects. 5 and 6.
The next main result is that the Smoluchowski–Kramers approximation is valid for these wave equations with multiplicative noise in any spatial dimension. The convergence of \(u^\mu \) to u is in \(L^p(\varOmega :C([0,T]:H))\), which is an improvement over previous results, which were known to converge in probability. Furthermore, this result is true in any spatial dimension \(d\ge 1\).
Theorem 4.2
(Smoluchowski–Kramers approximation) Assume that Assumptions 2.1 and 2.2 hold. Let u be the mild solution of (3.2) with initial condition \(u_0 \in H\) and \(u^\mu = \varPi _1 z^\mu \) be the mild solution of (2.14) with the same initial position \(u_0 \in H\) and any fixed initial velocity \(v_0 \in H^{-1}\). There exists \(p \ge 2\) such that for any \(T>0\),
5 Estimates on the wave equation semigroup \({\mathcal {S}_\mu }(t)\)
In this section we investigate the properties of the semigroup \({\mathcal {S}_\mu }(t)\). The exact form of the semigroup can be found in [5, Proposition 2.2]. We briefly recall some of the main observations about this semigroup and then we introduce some new analysis. Because A is diagonalized by the orthonormal basis \(\{e_k\}\), for any \(k \in \mathbb {N}\) the operator \({\mathcal {A}_\mu }\) is invariant on the two dimensional linear span in \({\mathcal {H}}\) of the form \(\{(u_k e_k, v_k e_k): u_k,v_k \in {\mathbb {R}}\}\) . The semigroup \({\mathcal {S}_\mu }(t)\) is also invariant on each of these two-dimensional spans.
Let \(u\in H\) and \(v \in H^{-1}\). Set \(u_k = \langle u,e_k\rangle _H\), \(v_k = \langle v,e_k\rangle _H\), and let
and
Then
By the definition of \({\mathcal {A}_\mu }\), \(g_k^\mu (t;u_k,v_k) = (f_k^\mu )'(t;u_k,v_k)\) and \(f_k^\mu (t,u_k,v_k)\) solves
To study the stochastic convolution, we will be particularly interested in the case where \(u_k=0\) and \(v_k = 1\). According to [5, Proposition 2.2],
We use the notation that when \(1-4\mu \alpha _k <0\), \(\sqrt{1-4\mu \alpha _k} := i \sqrt{4\mu \alpha _k -1}\). When \(1-4\mu \alpha _k = 0\), \(f_k^\mu (t;0,1):= t e^{-\frac{t}{2\mu }}\). We see that that the solutions to (5.3) feature different behaviors depending on whether \(1-4\mu \alpha _k \ge 0\) or \(1-4\mu \alpha _k <0\). When \(1-4\mu \alpha _k\ge 0\), the behavior is dominated by the exponential term \(\exp \left( -t\left( \frac{1-\sqrt{1-4\mu \alpha _k}}{2\mu } \right) \right) \). This exponent is bounded by \(-\alpha _kt\) because
Consequently, for any fixed \(\mu >0\) there are a finite number of \(k \in \mathbb {N}\) satisfying \(1-4\mu \alpha _k\ge 0\), and for this finite number of Fourier modes, \(f_k^\mu (t;0,1)\) can be bounded by terms that behave like \(\mu e^{-\alpha _k t}\).
On the other hand, for the infinite number of modes satisfying \(1-4\mu \alpha _k<0\),
In this regime, the functions no longer behave like their parabolic analogue. They behave approximately as \(\sqrt{\frac{\mu }{\alpha _k}}\exp \left( -\frac{t}{2\mu }\right) \). These observations are verified in the next sequence of lemmas.
Lemma 5.1
Assume that \(f_k^\mu (t;u,v)\) solves (5.2) for \(u,v \in {\mathbb {R}}\).
-
1.
If \(u=0\) and \(1-4\mu \alpha _k \ge 0\), then
$$\begin{aligned} |f_k^\mu (t;0,v)| \le 4 \mu |v| e^{-\alpha _k t} \end{aligned}$$(5.5)and
$$\begin{aligned} |(f_k^\mu )'(t;0,v)| \le 2|v|e^{-\alpha _k t}. \end{aligned}$$(5.6) -
2.
If \(u=0\) and \(1-4\mu \alpha _k < 0\), then
$$\begin{aligned} |f_k^\mu (t;0,v)| \le \frac{\sqrt{4\mu }|v|}{\sqrt{\alpha _k}} e^{-\frac{t}{4\mu }} \end{aligned}$$(5.7)and
$$\begin{aligned} |(f_k^\mu )'(t;0,v)| \le 2 |v| e^{-\frac{t}{4\mu }} \end{aligned}$$(5.8) -
3.
For any \(k \in \mathbb {N}\), \(\mu >0\) and \(u,v \in {\mathbb {R}}\),
$$\begin{aligned} \mu |(f_k^\mu )'(t;u,v)|^2 + \alpha _k |f_k^\mu (t;u,v)|^2 \le \mu |v|^2 + \alpha _k |u|^2. \end{aligned}$$(5.9)
Remark 5.2
An immediate consequence of (5.9) is that if \(v =0\) and \(u \in {\mathbb {R}}\), then for any \(k \in \mathbb {N}\),
Proof
For the simplicity of notation, we let \(f(t) = f_k^\mu (t;u,v)\) and specify k, \(\mu \), u, and v throughout the proof. Let \(\gamma \ge 0\) and define \(h(t) = e^{\gamma t} f(t)\). We will set \(\gamma \) to be either \(\alpha _k\) or \(\frac{1}{4\mu }\) depending on the relationship between \(\alpha _k\) and \(\mu \). h solves the equation
We calculate two energy estimates. First, by multiplying (5.11) by \(h'(t)\),
Therefore, by integrating the above expression and multiplying by 2,
We derive a second energy estimate based on the fact that
The last equality is a consequence of (5.11). Integrating both sides,
If \(1-4\mu \alpha _k \ge 0\), we set \(\gamma =\alpha _k\). This choice guarantees that the coefficients in (5.11) are positive. Specifically,
Then according to (5.12), if \(u=0\)
and by the triangle inequality, (5.13), and the previous display,
Then by (5.14),
We chose \(h(t) = e^{\alpha _k t} f(t)\). It follows that \(|f(t)| \le 4 \mu |v| e^{-\alpha _k t}\) which is (5.5). Similarly, \(h'(t) = \alpha _k f(t)e^{\alpha _k t} + f'(t) e^{\alpha _k t}\). Therefore,
In this regime \(4\mu \alpha _k \le 1\) so we can conclude that (5.6) holds.
Now we study the case where \(1 - 4\mu \alpha _k < 0\). In this case we set \(\gamma = \frac{1}{4\mu }\). Then
because \(\frac{3}{16\mu } \le \frac{3\alpha _k}{4}\). If \(u =0\), then by (5.12),
and
Therefore by (5.15),
and
because \(4\mu \alpha _k >1\). This proves (5.7) and (5.8).
Finally, (5.9) is a consequence of (5.12) with \(\gamma =0\). \(\square \)
Lemma 5.3
For any \(t\ge 0\) and \(\mu >0\) it holds that
Proof
This is an immediate consequence of (5.5) and (5.7). By (5.1),
\(\varPi _1 S_\mu (t) \mathcal {I}_\mu e_k = f_k^\mu (t;0,1/\mu ) e_k\). The \(e_k\) are a complete orthonormal basis of H and are eigenfunctions of \(\varPi _1 S_\mu (T)\mathcal {I}_\mu \) and therefore
For k satisfying \(1- 4\mu \alpha _k \ge 0\), (5.5) implies that \(|f_k(t;0,1/\mu )| \le 4\). For k satisfying \(1-4\mu \alpha _k < 0\), (5.7) implies that \(|f_k^\mu (t;0,1/\mu )| \le \frac{\sqrt{4}}{\sqrt{\mu \alpha _k}}\). For these k, \(\mu \alpha _k > \frac{1}{4}\) and we can conclude that
\(\square \)
Lemma 5.4
For any \(\mu >0\) and \(t\ge 0\),
Proof
This is an immediate consequence of (5.10) because
\(\square \)
Lemma 5.5
Let \(N_\mu = \max \{ k \in \mathbb {N} : 1-4\mu \alpha _k \ge 0\}\) and let \(P_{N_\mu }\) be the projection onto the span of \(\{e_k\}_{k \le N_\mu }\). Then for any \(t\ge 0\),
Proof
By (5.5),
\(\square \)
Lemma 5.6
Let \(N_\mu = \max \{k \in \mathbb {N}: 1 - 4\mu \alpha _k \ge 0\}\) and let \(P_{N_\mu }\) be the projection onto the span of \(\{e_k\}_{k \le N_\mu }\). Then for any \(t\ge 0\),
Proof
Because of the presence of the \((I-P_{N_\mu })\) projection and of the fact that \(\varPi _1 S_\mu (t)\mathcal {I}_1 e_k = f_k^\mu (t;0,1)e_k\),
Notice that the \(\sqrt{\alpha _k}\) is included because this is considered as a linear map from \(H^{-1} \rightarrow H\). By (5.7),
\(\square \)
Lemma 5.7
For any \(\mu \in (0,1)\) and \(t\ge 0\), it holds that
Proof
Because \(\mu \in (0,1)\) and the definition of \({\mathcal {H}}\), for any \((u,v) \in {\mathcal {H}}\) and \(t \ge 0\),
By the Fourier decomposition (5.1), right-hand side of the above expression equals
where \(u_k = \langle u,e_k\rangle _H\) and \(v_k = \langle v,e_k \rangle _H\). It follows from (5.9) that the above expression is bounded by
Because \(\mu \in (0,1)\), this implies
Therefore, for any \((u,v) \in {\mathcal {H}}\),
proving the result. \(\square \)
Now we study the convergence of the Fourier coefficients \(f_k^\mu (t;u,v)\) as \(\mu \rightarrow 0\).
Theorem 5.8
(Convergence) Let \(f_k^\mu (t;u,v)\) solve (5.2).
-
1.
For any \(k \in \mathbb {N}\), \(T>0\), and \(u \in {\mathbb {R}}\),
$$\begin{aligned} \lim _{\mu \rightarrow 0} \sup _{t \in [0,T]}|f_k^\mu (t;u,0) - u e^{-\alpha _k t}| = 0. \end{aligned}$$(5.21) -
2.
For any \( k \in \mathbb {N}\), \(T>0\)\(t_0 \in (0,T]\), and \(v \in {\mathbb {R}}\),
$$\begin{aligned} \lim _{\mu \rightarrow 0} \sup _{t \in [t_0,T]}| f_k^\mu (t;0,v/\mu ) - ve^{-\alpha _k t}|=0. \end{aligned}$$(5.22) -
3.
For any \(k \in \mathbb {N}\), \(T>0\), \(t_0 \in (0,T]\), and \(v \in {\mathbb {R}}\),
$$\begin{aligned} \lim _{\mu \rightarrow 0}\sup _{t \in [t_0,T]} |(f_k^\mu )'(t;0,v)| = 0. \end{aligned}$$(5.23)
Proof
One can prove each of these directly from the explicit formulas in [5, Proposition 2.2]. Below we present an alternative proof based on some arguments from [18]. Let \(f^\mu _k(t) = f_k^\mu (t;u,v)\). Then because \(\mu (f^\mu _k)''(t) + (f^\mu _k)'(t) + \alpha _k f^\mu _k(t) = 0\),
Integrating both sides,
and
Integrating once more and changing the order of integration,
If \(v =0\) and a limit \(f_k^\mu (t) \rightarrow {\bar{f}}_k(t)\) exists, then the limit must solve
the unique solution of which is \({\bar{f}}_k(t) = u e^{-\alpha _k t}\). To prove that \(f_k^\mu (t)\) converges to \({\bar{f}}_k\), set \(g^\mu _k(t) = f^\mu _k(t) - {\bar{f}}_k(t)\). Then
A standard Grönwall along with the estimate (5.10) proves that
and consequently \(\sup _{t \in [0,T]}|g^\mu _k(t)| \rightarrow 0\) and (5.21) follows.
We can use a similar argument to show (5.22). If \(u = 0\) and \(v = \frac{1}{\mu }\) in (5.25), then
Let \({\bar{f}}(t) = e^{-\alpha _k t}\) and note that \({\bar{f}}(t) = 1 -\alpha _k \int _0^t {{\bar{f}}}(s)ds\). Setting \(g^\mu _k(t) = f^\mu _k(t) - {{\bar{f}}}(t)\), we see that \(g^\mu _k\) solves
If \(\mu >0\) is small enough that \(1-4\mu \alpha _k>0\), then (5.5) implies that for any \(t>0\)\(|f^\mu _k(t)| \le 4\). Therefore, \(\left| \int _0^t e^{-\frac{(t-s)}{\mu }} f^\mu _k(s)ds \right| \le 4\mu .\) By Grönwall’s inequality,
Therefore, for any \(0<t_0<T\),
and (5.22) follows for \(v=1\). For general \(v \in {\mathbb {R}}\), simply multiply both \(f^\mu _k\) and \({{\bar{f}}}\) by v.
Finally, we let \(u=0\) and \(v \in {\mathbb {R}}\) in (5.24). Then for \(t \in [t_0,T]\),
By (5.5), for \(\mu < \frac{1}{4\alpha _k}\), \(|f^\mu _k(s)|\le 4 \mu |v| e^{-\alpha _k s}\). Therefore,
which converges to zero uniformly over \(t \in [t_0,T]\) as \(\mu \rightarrow 0\). \(\square \)
6 Regularity of the stochastic convolution
Let G be the operator defined in (2.10) and let \(\varphi (t)\) and \(\psi (t)\) be some H-valued processes that are adapted to the natural filtration of w(t). In this section we study the stochastic convolution processes
and the differences
In order to study both of these objects at the same time and to simplify our notation, for the rest of this section we will let \(\varPhi (t)\) denote either \(G(\varphi (t))\) or \(G(\varphi (t)) - G(\psi (t))\).
Before establishing estimates on the stochastic convolution we discuss the properties of such a \(\varPhi \). For any \(t\ge 0\), \(\varPhi (t)\) is a bounded linear operator from \(L^\infty (D)\) to H. \(\varPhi (t)\) is also a bounded linear operator from H to \(L^1(D)\).
If \(\varphi (t) \in H\), and \(h \in L^\infty (D)\) then by the linear growth of g in Assumption 2.1,
If \(\varphi (t) \in H\) and \(h \in H\), then
Similarly, if \(\varPhi (t) = (G(t,\varphi (t)) - G(t, \psi (t)))\), and \(\varphi (t), \psi (t) \in H\) and \(h \in L^\infty (D)\),
and if \(h \in H\), then
Let \(\varPhi ^\star (t)\) denote the adjoint of \(\varPhi (t)\) in H in the sense that if \(h_1 \in L^\infty (D)\) and \(h_2 \in H = L^2(D)\) or \(h_1 \in H\) and \(h_2 \in L^\infty (D)\),
Notice that if \(\varPhi (t) = G(t,\varphi (t))\), \(h_1\in L^\infty (D)\) and \(h_2 \in H\),
In this way, \(\varPhi (t)\) is a self-adjoint \({\mathscr {L}}(L^\infty (D),H) \cap {\mathscr {L}}(H, L^1(D))\)-valued process that is adapted to the natural filtration of w(t). We define the stochastic convolution
By the stochastic factorization formula [15, Chapter 5.3.1], for \(0< \alpha < 1\) to be chosen later,
where
We begin with estimates on \(\varGamma _\alpha ^\mu \).
Remark 6.1
All of the proofs in the section are written for the case \(q<+\,\infty \) where q satisfies Assumption 2.2. A standard straightforward modification of the proofs is required if \(q=+\,\infty \).
Lemma 6.2
Let \(q,\beta \) satisfy (2.5)–(2.6). Let \(0<2\alpha < 1- \frac{\beta (q-2)}{q}\). Then for any \(p\ge 2\) and \(T>0\), there exists a constant \(C = C(\alpha ,p,T)\) independent of \(\mu \) such that for any \(t \in [0,T]\),
‘
Proof
By the Burkholder–Davis–Gundy inequality [15, Theorem 4.36],
where \(\{e_j\}\) is the complete orthonormal basis of H that diagonalizes Q and A in Assumption 2.2.
For the rest of the proof, it is enough to study the quadratic variation.
We expand this expression into a double sum
Notice that for any \(k, j \in \mathbb {N}\) and \(t\ge 0\)
where \(f_k^\mu (t) = f_k^\mu (t;0,1/\mu )\) solves (5.2) with \(u_k=0\) and \(v_k = 1/\mu \). Therefore, along with the fact that \(Q e_j = \lambda _j e_j\), the quadratic variation can be written as
Apply Hölder’s inequality with exponents \(\frac{q}{2}\) and \(\frac{q}{q-2}\) to the double sum where q is from Assumption 2.2,
The final inequality is a consequence of the fact that \(\varPhi (t) = \varPhi ^\star (t)\). Letting \(\Vert Q\Vert _{q} \) be defined as in (2.4),
We analyze the sum
by splitting it into two pieces. Let \(N_\mu = \max \{k: 1-4\mu \alpha _k \ge 0\}.\) Then by (5.5) and (5.7) with \(v=1/\mu \)
For any \(x,y\ge 0\) it follows that \((x+y)^{(q-2)/q} \le x^{(q-2)/q} + y^{(q-2)/q}\). Therefore, the above expression is bounded by
The finite sum \(J_1\) behaves like the eigenfunctions of the semigroup in the parabolic case considered in [2,3,4]. Let \(\beta >0\) be from (2.5) and (2.6). There exists a constant such that for all \(k \in \mathbb {N}\) and \(t>0\), \(e^{-\alpha _k t} \le C\frac{1}{\alpha _k^\beta t^\beta }\). It follows that
We show that the tail sum \(J_2\) is small.
It follows from (2.6) that \(\beta < \frac{q}{q-2}\) and it follows from the definition of \(N_\mu \) that \(\alpha _k \ge \frac{1}{4\mu }\) for all \(k\ge N_\mu + 1\). Therefore for all \(k \ge N_\mu +1\), \(\alpha _k^{\beta -q/(q-2)} \le (4\mu )^{q/(q-2)-\beta }\) and by (2.5),
This means that
Plugging (6.8) and (6.10) back into (6.7),
By a change of variables,
From these estimates we see that
We assumed that \(2\alpha < 1 - \frac{\beta (q-2)}{q}\). Therefore, there exists a constant \(C>0\) independent of \(\mu \in (0,1)\) such that
The result follows by the BDG inequality (6.6). \(\square \)
Now we analyze the second component of \(\varGamma _\alpha ^\mu (t)\). This will diverge as \(\mu \rightarrow 0\). It will be convenient to analyze the moments of \(\varGamma _\alpha ^\mu \) in two pieces. Let \(N_\mu = \max \{k: 1-4\mu \alpha _k \ge 0\}\) as above. Let \(P_{N_\mu }\) be the projection in H onto the span of the modes \(\{e_1,.. e_{N_\mu }\}\).
Lemma 6.3
Let \(q,\beta \) satisfy (2.5)–(2.6). Let \(0<2\alpha < 1- \frac{\beta (q-2)}{q}\). Let \(\varGamma ^\mu _\alpha \) be given by (3.4). Then for any \(p\ge 2\) and \(T>0\), there exist constants \(C = C(\alpha ,p,T) >0\) and \(\zeta = \zeta (\alpha ,p,T)\in (0,p)\) such that
-
1.
For any \(t \in [0,T]\), and \(\mu \in (0,1)\),
$$\begin{aligned} {\mathbb {E}}\left| P_{N_\mu }\varPi _2\varGamma _\alpha ^\mu (t) \right| _H^p \le \frac{C}{\mu ^{p}} {\mathbb {E}}\sup _{s \in [0,t]} \Vert \varPhi (s)\Vert _{{\mathscr {L}}(L^\infty (D),H)}^{p}. \end{aligned}$$(6.12) -
2.
For any fixed \(t \in [0,T]\),
$$\begin{aligned} \lim _{\mu \rightarrow 0} \mu ^p{\mathbb {E}}| P_{N_\mu } \varPi _2 \varGamma _\alpha ^\mu (t)|_H^p = 0. \end{aligned}$$(6.13) -
3.
For any fixed \(t \in [0,T]\) and \(\mu \in (0,1)\),
$$\begin{aligned} {\mathbb {E}}\left| (I - P_{N_\mu })\varPi _2 \varGamma _\alpha ^\mu (t) \right| _{H^{-1}}^p \le \frac{C}{\mu ^{(p-\zeta )/2}} {\mathbb {E}}\sup _{s \in [0,t]} \Vert \varPhi (s)\Vert _{{\mathscr {L}}(L^\infty (D),H)}^p.\nonumber \\ \end{aligned}$$(6.14)
Proof
The proofs of this lemma are similar to the proof of Lemma 6.2. Let \(\varLambda _1(t)\) be the quadratic variation of \(P_{N_\mu }\varPi _2 \varGamma ^\mu _\alpha \).
The eigenvalues satisfy \(Qe_j = \lambda _j e_j\) and \({\mathcal {I}}_\mu ^\star {\mathcal {S}_\mu ^\star }(t-s)\varPi _2^\star e_k = (f^\mu _k)'(t-s)e_k\) where \(f^\mu _k\) solves (5.2) with \(u_k=0\) and \(v_k = 1/\mu \). Then
By (5.6) with \(v=\frac{1}{\mu }\), for \(k \in \{1, \ldots , N_\mu \}\)
Therefore,
By the Hölder inequality on the double sum and following the arguments of the proof of Lemma 6.2,
By the same reasoning that we used in (6.8),
By the BDG inequality,
and (6.12) follows.
All of the previous calculations allow us to use a dominated convergence theorem to prove (6.13). The upper bound for (6.15) using (5.6) was established above. Specifically, for \(k \in \{1,\ldots ,N_\mu \}\), \(j \in \mathbb {N}\), and \(s,t \in [0,T]\),
Notice that \(\mu (f_k^\mu )'(t,0,{1}/{\mu }) = (f_k^\mu )'(t;0,1)\). By (5.23), for each \(s>0\), \(k \le N_\mu \), and \(j \in \mathbb {N}\),
Therefore, by (6.15) and the dominated convergence theorem \(\varLambda _1(t) \rightarrow 0\) with probability 1. Then by using the BDG inequality and one more application of the dominated convergence theorem, (6.13) follows.
As for the higher modes, let
Expanding this to a double sum,
Recognize that for \(k,j \in \mathbb {N}\)
By (5.8),
Applying the Hölder inequality to (6.16),
By (6.11),
We chose \(\alpha \) so that \(2\alpha + \frac{\beta (q-1)}{q} <1\). This means that there exists \(\zeta >0\) such that
By the BDG inequality,
\(\square \)
Now we can establish a priori bounds on the supremum norm of the stochastic convolution.
Theorem 6.4
Let \(q,\beta \) satisfy (2.5)–(2.6). Let \(\varGamma ^\mu (t)\) be given by (6.2). For any \(p \ge \frac{1}{\alpha }\) where \(0<2\alpha <1-\frac{\beta (q-2)}{q}\) and \(T\ge 0\), there exists a constant \(C= C(\alpha ,p,T)\) such that for all \(\mu \in (0,1)\)
Notice that this constant is independent of \(\mu \in (0,1)\).
Proof
We use the stochastic convolution formula (6.3),
We divide \(\varGamma _\alpha ^\mu \) into three different pieces. Recall that \(\varPi _1, \varPi _2\) defined in (2.11) and \(P_{N_\mu }, (I-P_{N_\mu })\) defined above Lemma 6.3 are all projections. We can rewrite the stochastic convolution formula (6.3) as
Choose \(\alpha >0\) satisfying the assumptions of Lemmas 6.2 and 6.3. Let \(p> \frac{1}{\alpha }\).
Applying the Hölder inequality and using (5.17) and (6.5),
The previous line follows because \(p> \frac{1}{\alpha }\) implies \((\alpha -1)p/(p-1)>-1\).
By the same argument with (5.18) and (6.12),
Therefore the result follows. \(\square \)
Theorem 6.5
Let \(\varGamma ^\mu (t)\) be given by (6.2). For any \(p \ge \frac{1}{\alpha }\) where \(0<2\alpha < 1 - \frac{\beta (q-2)}{q}\), and \(T\ge 0\), there exists a constant \(C= C(p,T,\mu )\) such that
Proof
The proof is similar to the proof of Theorem 6.4, but it is less complicated because the constant is allowed to depend on \(\mu \). The main difference is that we use Lemma 5.7 instead of Lemmas 5.4–5.6 in the stochastic convolution argument. We omit further details. \(\square \)
7 Well-posedness of the stochastic wave equation: Proof of Theorem 4.1
Let \(\mu >0\). We show that for any \((u_0,v_0) \in {\mathcal {H}}\) there is a unique mild solution \(z^\mu \in C([0,T]:H)\) solving
We prove well-posedness with the contraction mapping principle. Let \({\mathscr {K}}^\mu : L^p(\varOmega :C([0,T]:{\mathcal {H}})) \rightarrow L^p(\varOmega :C([0,T]:{\mathcal {H}}))\) by
Well-posedness follows from proving that there exists a unique fixed point for \({\mathscr {K}}^\mu \).
For any \(\varphi _1, \varphi _2 \in L^p(\varOmega :C([0,T]:{\mathcal {H}}))\),
By Lemma 5.7, \(\sup _{t \ge 0} \Vert {\mathcal {S}_\mu }(t)\Vert _{{\mathscr {L}}({\mathcal {H}})} \le \mu ^{-1/2}\). By the Lipschitz continuity of B (Assumption 2.1), for any \(t \in [0,T]\),
For the stochastic term, Theorem 6.5 and (6.1) guarantee that
It follows from these two estimates that
Therefore, for small enough \(T_0>0\), \({\mathscr {K}}^\mu \) is a contraction on \(L^p(\varOmega :C([0,T_0]:{\mathcal {H}}))\). We can use standard methods to append solutions in the intervals \([0,T_0]\), \( [T_0,2T_0]\), \( [2T_0,3T_0],\)...to get a unique solution to (7.1) in \(L^p(\varOmega :C([0,T]:H))\) for any \(T>0\).
8 Convergence: Proof of Theorem 4.2
Before proving Theorem 4.2, we state two auxilliary results about the convergence of the stochastic convolutions and Lebesgue integral convolutions with the wave and heat semigroups. We state a result about the convergence of the stochastic convolutions where \(\varGamma ^\mu \) defined in (6.2) converge to \(\varGamma \) defined in (3.3).
Theorem 8.1
Let \(q,\beta \) satisfy (2.5)–(2.6). Let \(T>0\), let \(\alpha \in (0,1/2)\) satisfy \(0<2\alpha <1- \frac{\beta (q-2)}{q}\) and let \(p > \frac{1}{\alpha }\). For any self-adjoint, progressively measurable \(\varPhi \in L^p(\varOmega :L^\infty ([0,T]:{\mathscr {L}}(L^\infty (D),H)))\) let \(\varGamma ^\mu \) and \(\varGamma \) be given by (6.2) and (3.3) respectively. Then
Theorem 8.1 is really the most technical piece of this paper. We will delay its proof to Sect. 8.1. We will need a similar result about the Lebesgue integrals.
Theorem 8.2
For any \(T>0\) and \(\varphi \in L^\infty ([0,T]:H)\),
The proof is in Sect. 8.2.
We now prove the main convergence result via Theorems 8.1 and 8.2.
Proof
(Proof of Theorem4.2) We decompose the difference between the mild solutions (2.14) and (3.2) into the following pieces
Letting \(u_k = \langle u_0,e_k\rangle _H\) it follows from (5.1) that
The above expression converges to zero by the dominated convergence theorem and (5.21). Similarly, letting \(v_k = \langle v_0,e_k\rangle _H\), and \(N_\mu = \max \{k \in \mathbb {N}: 1-4\mu \alpha _k\ge 0\}\) it follows from (5.5) and (5.7) that
If \(k \le N_\mu \), then \(1-4\mu \alpha _k \ge 0\). In particular, \(\mu \le \frac{1}{4\alpha _k}\) and \(\mu ^2 \le \frac{\mu }{4\alpha _k}\). Applying this bound to the first sum in the above display, it follows that
These calculations show that
By Theorem 3.4, the unique solution to (3.2) is in \( L^p(\varOmega :C([0,T]:H))\). By the linear growth of B (see (2.2)), \(B(\cdot , u(\cdot )) \in L^p(\varOmega :C([0,T]:H)\) as well. It follows from Theorem 8.2 and the dominated convergence theorem that
By the Lipschitz continuity of B (2.1), there exists a constant \(C>0\) such that for all \(s \in [0,T]\), \(|B(s,u(s)) - B(s,u^\mu (s))|_H \le C |u(s) - u^\mu (s)|_H\). By Lemma 5.3 and a Hölder inequality,
From the linear growth of G (2.2) and the fact that \(u \in L^p(\varOmega : C([0,T]:H))\), it follows that \(G(\cdot , u(\cdot )) \in L^p(\varOmega : L^\infty ([0,T]:{\mathscr {L}}(L^\infty (D),H)))\). Theorem 8.1 implies that
By Theorem 6.4
By the Lipschitz continuity of G (6.1), there exists a constant independent of s and \(\mu \) such that \(\Vert G(s,u(s)) - G(s,u^\mu (s))\Vert _{{\mathscr {L}}(L^\infty (D),H)} \le C|u(s) - u^\mu (s)|_H\). It follows that
It now follows from (8.3), (8.6), and (8.8), that there exists an increasing \(C(T)>0\) such that for any \(T>0\)
By Grönwall’s inequality, for any \(T>0\),
Finally, we conclude that the above display converges to zero due to (8.4), (8.5), and (8.7). \(\square \)
8.1 Proof of Theorem 8.1
Lemma 8.3
Let \(\alpha \) satisfying \(0<2\alpha < 1 - \frac{\beta (q-2)}{q}\), \(p>\frac{1}{\alpha }\) and \(\varPhi \in L^p(\varOmega : L^\infty ([0,T]:{\mathscr {L}}(L^\infty (D),H)))\) satisfy the assumptions of Theorem 8.1. Let \(\varGamma _\alpha ^\mu \) be given by (6.4) and \(\varGamma _\alpha \) be given by (3.4). For any \(t>0\),
Proof
The scalar quadratic variation of \(\varPi _1\varGamma _\alpha ^\mu (t) - \varGamma _\alpha (t)\) is
Writing this expression as a double sum and using the fact that \(e_k\) are eigenfunctions for S(t), \(\varPi _1{\mathcal {S}_\mu }(t){\mathcal {I}}_\mu \) and Q,
For fixed \(k,j \in \mathbb {N}\) and \(s \in [0,t]\), this integrand is dominated by,
which is integrable by the arguments of Lemma 6.2 and [2, Section 3]. By (5.22) and the dominated convergence theorem, \(\varLambda (t) \rightarrow 0\). By the BDG inequality,
\(\square \)
Lemma 8.4
For any \(N \in \mathbb {N}\) and \(t\ge 0\),
Proof
Notice that because these operators are diagonalized by the orthonormal basis \(\{e_k\}\),
and the above expression converges to zero by (5.21). The limit will not be true without the projection onto a finite dimensional span. \(\square \)
Proof
(Proof of Theorem8.1) By the factorization method of [15, Chapter 5.3.1],
where \(\varGamma _\alpha \) and \(\varGamma ^\mu _\alpha \) are defined in (3.4) and (6.4).
We split up the difference into five pieces. Let \(N \in \mathbb {N}\) be chosen later. Let \(N_\mu = \sup \{k \in \mathbb {N}: 1-4\mu \alpha _k \ge 0\}\).
We also denote \(I^\mu _i(t) := I^\mu _{i,N}(t)\) for \(i = 3,4,5\) because these terms are independent of the choice of N.
By the Hölder inequality, for \(p>\frac{1}{\alpha }\) and \(N \in \mathbb {N}\),
By Lemma 8.4 and the dominated convergence theorem, for any fixed \(N \in \mathbb {N}\),
The dominated convergence is valid by Lemma 8.4, the well-known fact that the heat equation semigroup is uniformly bounded, and the fact that \(p> \frac{1}{\alpha }\) implies \(\frac{(\alpha -1)(p-1)}{p}>-1\).
Note that Lemma 3.1 implies that \( {\mathbb {E}}|\varGamma _\alpha (t)|_H^p \) is bounded uniformly in \(t \in [0,T]\). It follows that for any fixed \(N \in \mathbb {N}\),
Now we show that \(I^\mu _{2,N}\) converges to 0 as \(N\rightarrow +\,\infty \) independently of \(\mu >0\). By the Hölder inequality,
The first integral is uniformly bounded by Lemma 5.4 and the boundedness of the heat equation semigroup. Specifically, for any \(N \in \mathbb {N}\) and \(\mu \in (0,1)\),
For any fixed \(s\in [0,T]\), \({\mathbb {E}}|(I-P_N)\varGamma _\alpha (s)|_H^p\) converges to 0 as \(N \rightarrow +\,\infty \) by Lemma 3.3. Therefore,
For \(I^\mu _3\), we notice that
Lemma 5.4 guarantees that the first integral is uniformly bounded. Lemma 8.3 and the dominated convergence theorem guarantees that
The dominated convergence is valid due to Lemma 6.2.
For \(I^\mu _4\),
The first integral is bounded by Lemma 5.5. The second integral goes to zero as \(\mu \) goes to zero by (6.12), (6.13), and the dominated convergence theorem. Therefore,
Finally,
By Lemma 5.6, there exists \(C>0\) independent of \(\mu \) such that
By (6.14),
Therefore,
We can now complete the proof. Pick any arbitrary \(\eta >0\). There exists a constant \(C>0\) such that by (8.9),
Choose N large enough so that by (8.11), \({\mathbb {E}}\sup _{t \in [0,T]}|I^\mu _{2,N}(t)|_H^p<\frac{\eta }{5C}\). Then choose \(\mu _0>0\) small enough so that for any \(\mu \in (0,\mu _0)\), (8.10), (8.12), (8.13), and (8.14) guarantee that \({\mathbb {E}}\sup _{t \in [0,T]}|I^\mu _{i,N}(t)|_H^p<\frac{\eta }{5C}\) for \(i=1,3,4,5\). Then for \(\mu \in (0,\mu _0)\),
The result follows because \(\eta >0\) was arbitrary. \(\square \)
8.2 Proof of Theorem 8.2
Let \(P_N\) be the projection onto the finite dimensional span \(\{e_k\}_{k=1}^N\). The following lemma is a consequence of (5.22).
Lemma 8.5
For any \(0<t_0<T\) and \(N \in {\mathbb {N}}\),
Proof
Because for any fixed \(t>0\), the operators S(t) and \(\varPi _1{\mathcal {S}_\mu }(t) {\mathcal {I}}_\mu \) are both diagonalized by the orthonormal basis \(\{e_k\}\),
where \(f^\mu _k(t;0,1/\mu )\) solves (5.2). The result follows by (5.22) and the fact that we are only working with a finite number of modes at a time. \(\square \)
Proof
(Proof of Theorem8.2) Let \(T>0\) and \(\varphi \in L^\infty ([0,T]:H)\). For any \(N \in \mathbb {N}\),
The last inequality is due to the fact that by Lemma 5.3 for any \(t\ge 0\),
It follows from (8.16) that
By Lemma 8.5 and the dominated convergence theorem,
Finally, we recall that \(N \in \mathbb {N}\) was arbitrary and that the dominated convergence theorem guarantees that the limit of the right-hand side as \(N \rightarrow +\,\infty \) is 0. \(\square \)
References
Birrell, J., Hottovy, S., Volpe, G., Wehr, J.: Small mass limit of a Langevin equation on a manifold 18(2), 707–755 (2017)
Cerrai, S.: Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Relat. Fields 125(2), 271–304 (2003)
Cerrai, S.: Asymptotic behavior of systems of stochastic partial differential equations with multiplicative noise. In: Stochastic Partial Differential Equations and Applications VII, pp. 61–75 (2006)
Cerrai, S.: A Khasminskii type averaging principle for stochastic reaction–diffusion equations. Ann. Appl. Probab. 19(3), 899–948 (2009)
Cerrai, S., Freidlin, M.: On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom. Probab. Theory Relat. Fields 135(3), 363–394 (2006)
Cerrai, S., Freidlin, M.: Smoluchowski–Kramers approximation for a general class of SPDEs. J. Evol. Equ. 6(4), 657–689 (2006)
Cerrai, S., Freidlin, M.: Small mass asymptotics for a charged particle in a magnetic field and long-time influence of small perturbations. J. Stat. Phys. 144(1), 101–123 (2011)
Cerrai, S., Freidlin, M.: Large deviations for the Langevin equation with strong damping. J. Stat. Phys. 161(4), 859–875 (2015)
Cerrai, S., Glatt-Holtz, N.: On the convergence of stationary solutions in the Smoluchowski–Kramers approximation of infinite dimensional systems. arXiv preprint arXiv:1806.05319 (2018)
Cerrai, S., Salins, M.: Smoluchowski–Kramers approximation and large deviations for infinite dimensional gradient systems. Asymptot. Anal. 88(4), 201–215 (2014)
Cerrai, S., Salins, M.: On the Smoluchoski–Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field. Stoch. Process. Appl. 127(1), 273–303 (2016). https://doi.org/10.1016/j.spa.2016.06.008
Cerrai, S., Salins, M.: Smoluchowski–Kramers approximation and large deviations for infinite-dimensional nongradient systems wih applications to the exit problem. Ann. Prob. 44(4), 2591–2642 (2016)
Chen, G., Duan, J., Zhang, J.: Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition. SIAM J. Math. Anal. 45(5), 2790–2814 (2013)
Chen, Z., Freidlin, M.: Smoluchowski–Kramers approximation and exit problems. Stoch. Dyn. 5(04), 569–585 (2005)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhikers guide to the fractional sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Evans, L.: Partial Differential Equations. Graduate studies in mathematics. American Mathematical Society. https://books.google.com/books?id=Xnu0o_EJrCQC (2010)
Freidlin, M.: Some remarks on the Smoluchowski–Kramers approximation. J. Stat. Phys. 117(3), 617–634 (2004)
Freidlin, M., Hu, W.: Smoluchowski–Kramers approximation in the case of variable friction. J. Math. Sci. 179(1), 184–207 (2011)
Freidlin, M., Hu, W., Wentzell, A.: Small mass asymptotic for the motion with vanishing friction. Stoch. Process. Appl. 123(1), 45–75 (2013)
Herzog, D.P., Hottovy, S., Volpe, G.: The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction. J. Stat. Phys. 163(3), 659–673 (2016)
Hottovy, S., McDaniel, A., Volpe, G., Wehr, J.: The Smoluchowski–Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. 336(3), 1259–1283 (2015)
Hu, W., Spiliopoulos, K.: Hypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptotics. Electron. J. Probab. 22, 1–38 (2017)
Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304 (1940)
Lee, J.J.: Small mass asymptotics of a charged particle in a variable magnetic field. Asymptot. Anal. 86(2), 99–121 (2014)
Lv, Y., Roberts, A.: Large deviation principle for singularly perturbed stochastic damped wave equations. Stoch. Anal. Appl. 32(1), 50–60 (2014)
Lv, Y., Wang, W.: Limiting dynamics for stochastic wave equations. J. Differ. Equ. 244(1), 1–23 (2008)
von Smoluchowski, M.: Drei vortrage uber diffusion. Brownsche bewegung und koagulation von kolloidteilchen. Z. Phys. 17, 557–585 (1916)
Spiliopoulos, K.: A note on the Smoluchowski–Kramers approximation for the Langevin equation with reflection. Stoch. Dyn. 7(02), 141–152 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salins, M. Smoluchowski–Kramers approximation for the damped stochastic wave equation with multiplicative noise in any spatial dimension. Stoch PDE: Anal Comp 7, 86–122 (2019). https://doi.org/10.1007/s40072-018-0123-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40072-018-0123-z
Keywords
- Smoluchowski–Kramers approximation
- Stochastic wave equation
- Stochastic heat equation
- Stochastic partial Differential equations