Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

Abstract

We consider the linear stochastic heat equation on $\mathbb {R}^\ell$, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4}, \frac{1}{2}]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.

1 Introduction

The purpose of this paper is to study the stochastic heat equation

$$\begin{aligned} \frac{\partial u}{\partial t} =\frac{1}{2}\Delta u+u \diamond \frac{\partial ^{\ell +1}W}{\partial t\partial x_1\dots \partial x_\ell } \, , \end{aligned}$$

(1.1)

where $t\ge 0$, $x\in \mathbb {R}^\ell$ $(\ell \ge 1)$ and W is a centered Gaussian field, which is correlated in both temporal and spatial variables. We assume that the noise W is described by a centered Gaussian family $W=\{ W(\phi ), \phi \in \mathcal {S}(\mathbb {R}_+\times \mathbb {R}^\ell )\}$, with covariance

$$\begin{aligned} \mathbb {E}[W(\phi )W(\psi )]=\frac{1}{(2 \pi )^\ell } \int _0^\infty \int _0^\infty \int _{\mathbb {R}^\ell } \mathcal {F}\phi (s,\xi )\overline{\mathcal {F}\psi (r,\xi )}\gamma _0(s-r)\mu (d \xi )dsdr, \end{aligned}$$

(1.2)

where $\gamma _0$ is a nonnegative and nonnegative definite locally integrable function, $\mu$ is a tempered measure and $\mathcal {F}$ denotes the Fourier transform in the spatial variables. Throughout the paper, we denote by $|\cdot |$ the Euclidean norm in $\mathbb {R}^\ell$ and by $x\cdot y$ the usual inner product between two vectors x, y in $\mathbb {R}^\ell$. We are going to consider two types of spatial covariances:

• (H.1) $\ell =1$, the spectral measure $\mu$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb {R}$ with density f, that is $\mu (d \xi )=f(\xi )d \xi$, and f satisfies:

• (a) For all $\xi ,\eta$ in $\mathbb {R}$ and for some constant $\kappa _0>0$,

  $$\begin{aligned} f(\xi +\eta )\le \kappa _0 (f(\xi )+f(\eta )). \end{aligned}$$

  (1.3)
• (b)
  $$\begin{aligned} \int _\mathbb {R}\frac{f^2(\xi )}{1+|\xi |^2}d \xi <\infty \,. \end{aligned}$$

  (1.4)

To state the second type of covariance, we recall that the space of Schwartz functions is denoted by ${\mathcal {S}}(\mathbb {R}^\ell )$. The Fourier transform of a function $u \in {\mathcal {S}}(\mathbb {R}^\ell )$ is defined with the normalization

$$\begin{aligned} \mathcal {F}u ( \xi ) = \int _{\mathbb {R}^\ell } e^{- i \xi \cdot x } u ( x) d x, \end{aligned}$$

so that the inverse Fourier transform is given by $\mathcal {F}^{- 1} u ( \xi ) = ( 2 \pi )^{- \ell } \mathcal {F}u ( - \xi )$.

• (H.2) The inverse Fourier transform of $\mu$ is a nonnegative locally integrable function (or generalized function) denoted by $\gamma$

  $$\begin{aligned} \gamma (x)=\frac{1}{(2 \pi )^\ell }\int _{\mathbb {R}^\ell } e^{i \xi \cdot x}\mu (d \xi )\,, \end{aligned}$$

  (1.5)

  and $\mu$ satisfies Dalang’s condition

  $$\begin{aligned} \int _{ \mathbb {R}^\ell }\frac{\mu (d\xi ) }{1+ |\xi |^2} <\infty . \end{aligned}$$

  (1.6)

For the case (H.2), $\gamma$ is nonnegative definite and (1.2) can be written as

$$\begin{aligned} \mathbb {E}[W(\phi )W(\psi )]=\int _0^\infty \int _0^\infty \int _{\mathbb {R}^{2\ell }} \phi (s,x)\psi (r,y)\gamma _0(s-r)\gamma (x-y)dxdydsdr\,. \end{aligned}$$

(1.7)

Examples of covariance functions satisfying condition (H.2) are the Riesz kernel $\gamma (x)=|x|^{-\eta }$, with $0<\eta <2\wedge \ell$, the space-time white noise in dimension one, where $\gamma =\delta _0$, and the multidimensional fractional Brownian motion, where $\gamma (x)= \prod _{i=1}^\ell H_i (2H_i-1) |x_i|^{2H_i-2}$, assuming that $\sum _{i=1}^\ell H_i >\ell -1$ and $H_i \in (\frac{1}{2},1)$ for $i=1,\dots , \ell$.

In the case (H.1), the inverse Fourier transform of $\mu$ is at best a distribution and the expression (1.5) is only formal. The right-hand side of (1.7), however, makes sense by pairing between Schwartz functions and distributions. For our convenience, we will address $\gamma$ as a generalized covariance function if its Fourier transform is a (nonnegative) tempered measure. It also worths noting that by Jensen’s inequality, (1.4) implies Dalang’s condition (1.6). The basic example of a noise satisfying (H.1) is the rough fractional noise, where the spectral density is given by $f(\xi )=c_H |\xi |^{1-2H}$, with $H \in (\frac{1}{4}, \frac{1}{2}]$ and $c_H= \Gamma (2H+1) \sin (\pi H)$. In this case, the noise W has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2}]$ in the spatial variable. Condition (a) holds with $\kappa _0=1$ and condition (b) holds because of the restriction $H>\frac{1}{4}$.

These types of spatial covariances were introduced in our paper [10], where the noise is white in time. In [10] we proved the existence of a unique mild solution formulated using Itô-type stochastic integrals, we derived Feynman-Kac formulas for the moments of the solution using a family of independent Brownian bridges and we computed Lyapunov exponents and lower and upper exponential growth indices. The purpose of this paper is to carry out this program when the noise is not white in time. While the general methodology of the current article is similar to [10], the case of colored temporal noises has some distinct features and needs a different treatment. In particular the existence and estimation of Lyapunov exponents offer new difficulties that require techniques different from the white-time case.

After a section on preliminaries, Sect. 3 is devoted to show (see Theorem 3.2) the existence of a unique mild solution to Eq. (1.1), when the stochastic integral is understood in the Skorohod sense, and the initial condition satisfies the general integrability condition (3.2). We want to mention that the existence and uniqueness of a solution for the (H.2) type covariance in the case of a colored noise in time has been also obtained in the recent paper [1] by Balan and Chen. Then, in Sect. 4 we establish Feynman-Kac formulas for the moments of the solution in terms of independent Brownian motions or Brownian bridges (see Proposition 4.3 and Corollary 4.4). Section 5 is devoted to obtain Lyapunov exponents for exponential functionals of Brownian bridges, assuming that $\gamma _0(t)= |t|^{-\alpha _0}$ and $\gamma (cx)=c^{-\alpha } \gamma (x)$ for all $c>0$, where $\alpha _0\in (0,1)$ and $\alpha \in (0,2)$. The main result of this section is Theorem 5.2, whose proof is inspired by Theorem 1.1 of [2]. While Chen’s article [2] deals with exponential functionals of Brownian motions, we deal with exponential functionals of Brownian bridges. Another difference is that we allow noises which satisfy condition (H.1). In this case, the spatial covariance is generally a distribution and even if it is a function, it is not necessary non-negative and may switch signs. The former issue is solved by an appropriate approximation procedure. For the later issue, the compact folding argument in [2] is no longer applicable here. Instead, we use a moment comparison between Brownian motions and Ornstein-Uhlenbeck processes, which is observed by Donsker and Varadhan [7]. We refer to [3] for related results on temporal asymptotics for the fractional parabolic Anderson model.

Finally, in Sect. 6 we study the speed of propagation of intermittent peaks. The propagation of the farthest high peaks was first considered by Conus and Khoshnevisan [6] for a one-dimensional heat equation driven by a space-time white noise with compactly supported initial condition, where it is shown that there are intermittency fronts that move linearly with time as $\lambda t$. More precisely, they defined the lower and upper exponential growth indices as follows:

$$\begin{aligned} \lambda _*(n)=\sup \left\{ \lambda>0:\liminf _{t\rightarrow \infty }\frac{1}{t} \sup _{|x|\ge \lambda t}\log \mathbb {E}|u(t,x)|^n>0\right\} \, \end{aligned}$$

and

$$\begin{aligned} \lambda ^*(n)=\inf \left\{ \lambda >0:\limsup _{t\rightarrow \infty }\frac{1}{t} \sup _{|x|\ge \lambda t}\log \mathbb {E}|u(t,x)|^n<0\right\} \,. \end{aligned}$$

Generalizing previous results by Chen and Dalang [4], we proved in [10] that, assuming that $u_0$ is nonnegative,

$$\begin{aligned} \sqrt{\frac{2\mathcal {E}_n(\gamma )}{n}} \le \lambda _*(n)\le \lambda ^*(n) \le \inf _{\beta :\int _{\mathbb {R}^\ell } e^{\beta |y|}u_0(y)dy<\infty }\left( \frac{\beta }{2}+\frac{\mathcal {E}_n(\gamma )}{n \beta }\right) , \end{aligned}$$

(1.8)

where $\mathcal {E}_n(\gamma )$ is the nth Lyapunov exponent. In particular, If $u_0$ is nontrivial and supported on a compact set, then

$$\begin{aligned} \lambda ^*(n)=\lambda _*(n)=\sqrt{\frac{2\mathcal {E}_n(\gamma )}{n}}\,. \end{aligned}$$

(1.9)

In the reference [8], using the Feynman-Kac formula for the moments of the solution established in [9], the authors have obtained lower and upper bounds for the exponential growth indices when the noise has a general covariance of the form $\mathbb {E}[\dot{W}(t,x) \dot{W}(s,y)]=\gamma _0(t-s) \gamma (x-y)$, where $\gamma _0$ is locally integrable and the spatial covariance $\gamma$ satisfies (H.2). Here $\dot{W}(t,x)$ stands for $\frac{\partial ^{\ell +1}W}{\partial t \partial x_1 \cdots \partial x_\ell }$. In this general situation, to obtain non-trivial limits the factor $t^{-1}$ and the set $\{|x| \ge \lambda t\}$ appearing in the definition of the exponential growth indices, need to be changed. In the particular case $\gamma _0(t)=|t|^{-\alpha _0}$ and $\gamma (x)= |x|^{-\alpha }$ we need to replace $t^{-1}$ by $t^{-a}$ and the set $\{|x| \ge \lambda t\}$ by $\{|x| \ge \lambda t^{\frac{a+1}{2}}\}$, where $a=\frac{4-\alpha -\alpha _0}{2-\alpha }.$ In the present paper, we complete this analysis with the methodology developed in [10], based on large deviations.

As a consequence of the large deviation results obtained in Sects. 5 and 6, under suitable scaling hypotheses on the covariance of the noise, we deduce the following results on the exponential growth indices, that should be compared with (1.8) and (1.9):

• (i) If the initial condition $u_0$ is a nonnegative function such that $\int _{\mathbb {R}^\ell } e^{\beta |y|^b}u_0(y)dy<\infty$, where $b= \frac{4- \alpha -2 \alpha _0}{3- \alpha - \alpha _0}$, then

  $$\begin{aligned} \lambda ^*(n) \le g_\beta ^{-1} \left( \left( \frac{n-1}{2} \right) ^{ \frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )\right) . \end{aligned}$$

  where $g_\beta$ is an increasing function on $(0,\infty )$ defined by Eq. (6.9), and ${\mathcal {E}}(\alpha _0,\gamma )$ is a variational quantity defined in (5.1).

• (ii) Suppose $u_0$ is bounded below in a ball of radius M, and for some technical reasons assume that the spatial covariance satisfies (H.2). Then,

  $$\begin{aligned} \lambda _*(n) \ge a^{\frac{a}{2}}(a+1)^{-\frac{a+1}{2}} \sqrt{2 \left( \frac{n-1}{2} \right) ^{ \frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma ) }\,, \end{aligned}$$

where $a=\frac{4-\alpha -2\alpha _0}{2-\alpha }$. Moreover, as $\beta$ tends to infinity, the function $g_\beta (x)$ converges to $\sqrt{2x}$ and in the compact support case, the two bounds above differ only on the constant $a^{\frac{a}{2}} (a+1) ^{-\frac{a+1}{2}}$.

2 Preliminaries

Let $\mathcal {H}$ be the completion of ${\mathcal {S}}(\mathbb {R}_+\times \mathbb {R}^\ell )$ endowed with the inner product

$$\begin{aligned} \langle \varphi , \psi \rangle _{\mathcal {H}} =\frac{1}{(2\pi )^{\ell }}\int _{\mathbb {R}_{+}^{2}\times \mathbb {R}^{\ell }} \mathcal {F}\varphi (s,\xi ) \overline{ \mathcal {F}\psi (t,\xi )}\gamma _0(s-t) \mu (d\xi ) \, dsdt. \end{aligned}$$

(2.1)

The mapping $\varphi \rightarrow W(\varphi )$ defined on ${\mathcal {S}}(\mathbb {R}_+\times \mathbb {R}^\ell )$ can be extended to a linear isometry between $\mathcal {H}$ and the Gaussian space spanned by W. We will denote this isometry by

$$\begin{aligned} W(\phi )=\int _0^{\infty }\int _{\mathbb {R}^\ell }\phi (t,x)W(dt,dx) \end{aligned}$$

for $\phi \in \mathcal {H}$. If $\mu$ satisfies (H.2), the righ-hand side of (2.1) can be written in Cartesian coordinates as $\int _{\mathbb {R}_+^2\times \mathbb {R}^{2\ell }}\varphi (s,x)\psi (t,y)\gamma _0(s-t)\gamma (x-y)dxdydsdt$. Hence, a standard approximation (still assuming (H.2)) shows that $\mathcal {H}$ contains the class of measurable functions $\phi$ on $\mathbb {R}_+\times \mathbb {R}^\ell$ such that

$$\begin{aligned} \int _{\mathbb {R}^2_+ \times \mathbb {R}^{2\ell }} |\phi (s,x)\phi (t,y)| \, \gamma _0(s-t)\gamma (x-y) \, dxdydsdt < \infty \,. \end{aligned}$$

(2.2)

2.1 Elements of Malliavin calculus

We denote by D the Malliavin derivative. That is, if F is a smooth and cylindrical random variable of the form

$$\begin{aligned} F=f(W(\phi _1),\dots ,W(\phi _n))\,, \end{aligned}$$

with $\phi _i \in \mathcal {H}$, $f \in C^{\infty }_p (\mathbb {R}^n)$ (that is, f and all its partial derivatives have polynomial growth), then DF is the $\mathcal {H}$-valued random variable defined by

$$\begin{aligned} DF=\sum _{j=1}^n\frac{\partial f}{\partial x_j}(W(\phi _1),\dots ,W(\phi _n))\phi _j\,. \end{aligned}$$

The operator D is closable from $L^2(\Omega )$ into $L^2(\Omega ; \mathcal {H})$, and we define the Sobolev space $\mathbb {D}^{1,2}$ as the closure of the space of smooth and cylindrical random variables under the norm

$$\begin{aligned} \Vert DF\Vert _{1,2}=\sqrt{\mathbb {E}[F^2]+\mathbb {E}[\Vert DF\Vert ^2_{\mathcal {H}}]}\,. \end{aligned}$$

We denote by $\delta$ the adjoint of the derivative operator given by the duality formula

$$\begin{aligned} \mathbb {E}\left[ \delta (u)F \right] =\mathbb {E}\left[ \langle DF,u \rangle _{\mathcal {H}}\right] , \end{aligned}$$

(2.3)

for any $F \in \mathbb {D}^{1,2}$ and any element $u \in L^2(\Omega ; \mathcal {H})$ in the domain of $\delta$. The operator $\delta$ is also called the Skorohod integral because in the case of the Brownian motion, it coincides with an extension of the Itô integral introduced by Skorohod.

If $F\in \mathbb {D}^{1,2}$ and h is an element of $\mathcal {H}$, then Fh is Skorohod integrable and, by definition, the Wick product equals the Skorohod integral of Fh, that is,

$$\begin{aligned} \delta (Fh)=F\diamond W(h). \end{aligned}$$

(2.4)

We refer to the book [14] of Nualart for a detailed account of the Malliavin calculus with respect to a Gaussian process.

When handling the stochastic heat equation in the Skorohod sense we will make use of chaos expansions, which we briefly describe in the following. For any integer $n\ge 0$ we denote by $\mathbf {H}_n$ the nth Wiener chaos of W. We observe that $\mathbf {H}_0$ is $\mathbb {R}$ and for $n\ge 1$, $\mathbf {H}_n$ is the closed linear subspace of $L^2(\Omega )$ generated by the family of random variables $\{ H_n(W(h)), h \in \mathcal {H}, \Vert h\Vert _{\mathcal {H}}=1 \}$. Here $H_n$ is the nth Hermite polynomial. For any $n\ge 1$, we denote by $\mathcal {H}^{\otimes n}$ (resp. $\mathcal {H}^{\odot n}$) the nth tensor product (resp. the nth symmetric tensor product) of $\mathcal {H}$. Then, the mapping $I_n(h^{\otimes n})= H_n(W(h))$ can be extended to a linear isometry between $\mathcal {H}^{\odot n}$, equipped with the modified norm $\sqrt{n!}\Vert \cdot \Vert _{\mathcal {H}^{\otimes n}}$, and $\mathbf {H}_n$.

Let us consider a random variable $F\in L^2(\Omega )$ which is measurable with respect to the $\sigma$-field $\mathcal {F}^W$ generated by W. This random variable can be expressed (called the Wiener-chaos expansion of F) as

$$\begin{aligned} F= \mathbb {E}\left[ F\right] + \sum _{n=1} ^\infty I_n(f_n), \end{aligned}$$

(2.5)

where the series converges in $L^2(\Omega )$, and the elements $f_n \in \mathcal {H}^{\odot n}$, $n\ge 1$, are determined by F.

The Skorohod integral (or the divergence) of a random field u can be computed using the Wiener chaos expansion. More precisely, suppose that $u=\{u(t,x), (t,x) \in \mathbb {R}_+ \times \mathbb {R}^\ell \}$ is a random field such that for each (t, x), u(t, x) is an $\mathcal {F}^W$-measurable and square integrable random variable. Then, for each (t, x), u(t, x) has the Wiener chaos expansion of the form

$$\begin{aligned} u(t,x)= \mathbb {E}\left[ u(t,x)\right] + \sum _{n=1}^\infty I_n (f_n(\cdot ,t,x)). \end{aligned}$$

(2.6)

Suppose additionally that the trajectories of u belong to $\mathcal {H}$ and $\mathbb {E}[\Vert u\Vert ^2_{\mathcal {H}}] <\infty$. Then, we can interpret u as a square integrable random function with values in $\mathcal {H}$ and the kernels $f_n$ in the expansion (2.6) are functions in $\mathcal {H} ^{\otimes (n+1)}$ which are symmetric in the first n time-space variables. In this situation, u belongs to the domain of the divergence (that is, u is Skorohod integrable with respect to W) if and only if the following series converges in $L^2(\Omega )$

$$\begin{aligned} \delta (u)= \int _0 ^\infty \int _{\mathbb {R}^\ell } u(t,x) \delta W(t,x)= W(\mathbb {E}[u]) + \sum _{n=1}^\infty I_{n+1} ({\widetilde{f}}_{n}), \end{aligned}$$

(2.7)

where ${\widetilde{f}}_{n}$ denotes the symmetrization of $f_n$ in all its $n+1$ time-space variables.

2.2 Brownian bridges

Let $\{B(s), s\ge 0\}$ be an $\ell$-dimensional Brownian motion starting at 0. For every fixed time $t>0$ and $x,y\in \mathbb {R}^\ell$, the process

$$\begin{aligned} \left\{ \widetilde{B}(s)=x+B(s)-\frac{s}{t}(B(t)+x-y), 0\le s\le t\right\} \end{aligned}$$

is an $\ell$-dimensional Brownian bridge from x to y, i.e. $\widetilde{B}(0)=x$ and $\widetilde{B}(t)=y$. Away from the terminal time t, the law of Brownian bridge admits a density with respect to Brownian motion. Indeed, it is shown in [13, Lemma 3.1] that for every bounded measurable function F,

$$\begin{aligned}&\mathbb {E}\left[ F(\{\widetilde{B}(s),0\le s\le \lambda t\})\right] \nonumber \\&\quad =(1- \lambda )^{-\frac{\ell }{2}} \mathbb {E}\left[ \exp \left\{ -\frac{|y-x-B(\lambda t)|^2}{2t(1- \lambda )}+\frac{|y-x|^2}{2t} \right\} F(\{ B(s),0\le s\le \lambda t\})\right] .\nonumber \\ \end{aligned}$$

(2.8)

Throughout the paper, we denote by $\{B_{0,t}(s),0\le s\le t\}$ an $\ell$-dimensional Brownian bridge which starts and ends at the origin. A Brownian bridge from x to y can be expressed as

$$\begin{aligned} \left\{ B_{0,t}(s)+\frac{s}{t} y+(1-\frac{s}{t})x,0\le s\le t\right\} \,. \end{aligned}$$

3 Existence and uniqueness of a solution via chaos expansions

We denote by $p_{t}(x)$ the $\ell$-dimensional heat kernel $p_{t}(x)=(2\pi t)^{-\ell /2}e^{-|x|^2/2t}$, for any $t > 0$, $x \in \mathbb {R}^\ell$. For each $t\ge 0$ let $\mathcal {F}_t$ be the $\sigma$-field generated by the random variables $W(\varphi )$, where $\varphi$ has support in $[0,t ]\times \mathbb {R}^\ell$. We say that a random field $u=\{u({t,x}), (t,x) \in \mathbb {R}_+\times \mathbb {R}^\ell \}$ is adapted if for each (t, x) the random variable $u_{t,x}$ is $\mathcal {F}_t$-measurable.

We assume that the initial condition $u_0$ is a measurable function satisfying the condition

$$\begin{aligned} (p_t *|u_0|)(x) <\infty \text{ for } \text{ all } t>0 \text{ and } x\in \mathbb {R}^\ell \,, \end{aligned}$$

(3.1)

where $p_t *|u_0|$ denotes the convolution of the heat kernel $p_t$ and the function $|u_0|$. This condition is equivalent to

$$\begin{aligned} \int _{\mathbb {R}^\ell } e^{-\kappa |x|^2} |u_0(x) |dx <\infty , \end{aligned}$$

(3.2)

for all $\kappa >0$.

We define the solution of Eq. (1.1) as follows.

Definition 3.1

An adapted random field $u=\{u({t,x}), t \ge 0, x \in \mathbb {R}^\ell \}$ such that $\mathbb {E}u^2({t,x}) < \infty$ for all (t, x) is a mild solution to Eq. (1.1) with initial condition $u_0$ satisfying (3.2), if for any $(t,x) \in [0, \infty )\times \mathbb {R}^\ell$, the process $\{p_{t-s}(x-y)u({s,y})\mathbf{1}_{[0,t)}(s) , s \ge 0, y \in \mathbb {R}^\ell \}$ is Skorohod integrable, and the following equation holds

$$\begin{aligned} u(t,x)=p_t *u_0(x)+\int _0^t\int _{\mathbb {R}^\ell }p_{t-s}(x-y)u({s,y}) \, \delta W_{s,y}. \end{aligned}$$

(3.3)

Suppose now that $u=\{u({t,x}), t\ge 0, x \in \mathbb {R}^\ell \}$ is a mild solution to Eq. (3.3). Then according to (2.5), for any fixed (t, x) the random variable u(t, x) admits the following Wiener chaos expansion

$$\begin{aligned} u({t,x})=\sum _{n=0}^{\infty }I_n(f_n(\cdot ,t,x))\,, \end{aligned}$$

(3.4)

where for each (t, x), $f_n(\cdot ,t,x)$ is a symmetric element in $\mathcal {H}^{\otimes n}$. Thanks to (2.7) and using an iteration procedure, one can then find an explicit formula for the kernels $f_n$ for $n \ge 1$

$$\begin{aligned} f_n(s_1,y_1,\dots ,s_n,y_n,t,x)= & {} \frac{1}{n!}p_{t-s_{\sigma (n)}}(x-y_{\sigma (n)})\cdots \\&p_{s_{\sigma (2)}-s_{\sigma (1)}}(y_{\sigma (2)}-y_{\sigma (1)}) p_{s_{\sigma (1)}}*u_0(y_{\sigma (1)})\,, \end{aligned}$$

where $\sigma$ denotes the permutation of $\{1,2,\dots ,n\}$ such that $0<s_{\sigma (1)}<\cdots<s_{\sigma (n)}<t$ (see, for instance, equation (4.4) in [11], where this formula is established in the case of a noise which is white in space). Then, to show the existence and uniqueness of the solution it suffices to show that for all (t, x) we have

$$\begin{aligned} \sum _{n=0}^{\infty }n!\Vert f_n(\cdot ,t,x)\Vert ^2_{\mathcal {H}^{\otimes n}}< \infty \,. \end{aligned}$$

(3.5)

Theorem 3.2

Suppose that the spatial covariance satisfies (H.1) or (H.2). Then relation (3.5) holds for each $(t,x) \in (0,\infty )\times \mathbb {R}^{\ell }$. Consequently, Eq. (1.1) admits a unique mild solution in the sense of Definition 3.1.

Proof

Notice that the kernel $f_n$ can be written as

$$\begin{aligned} f_n(s,y,t,x)= \int _{\mathbb {R}^\ell } g_n(s,y,t,z) u_0(z) dz, \end{aligned}$$

where $s=(s_1, \dots , s_n)$, $y=(y_1,\dots , y_n)$ and

$$\begin{aligned} g_n(s,y,t,z)=\frac{1}{n!}p_{t-s_{\sigma (n)}}(x-y_{\sigma (n)})\cdots p_{s_{\sigma (2)}-s_{\sigma (1)}}(y_{\sigma (2)}-y_{\sigma (1)})p_{s_{\sigma (1)}}(y_{\sigma (1)}-z)\,. \end{aligned}$$

(3.6)

Then

$$\begin{aligned} n! \Vert f_n(\cdot ,t,x)\Vert ^2_{{\mathcal {H}}^{\otimes n}}= & {} \frac{n!}{(2\pi )^{n\ell }} \int _{[0,t]^{2n}} \int _{(\mathbb {R}^{\ell })^n} \Phi (s,\xi ) \overline{\Phi (r,\xi )} \mu (d\xi ) \prod _{j=1}^n \gamma _0(s_j-r_j) dsdr \\\le & {} \frac{n!}{(2\pi )^{n\ell }} \int _{[0,t]^{2n}} \left( \int _{(\mathbb {R}^{\ell })^n} |\Phi (s,\xi )|^2 \mu (d\xi ) \right) ^{\frac{1}{2}} \\&\times \left( \int _{\mathbb {R}^{n\ell }} |\Phi (r,\xi )| ^2 \mu (d\xi ) \right) ^{\frac{1}{2}} \prod _{j=1}^n \gamma _0(s_j-r_j) dsdr, \end{aligned}$$

where $\xi =(\xi ^1, \dots , \xi ^n)$, $\mu (d\xi )= \prod _{i=1}^n \mu (d\xi ^i)$,

$$\begin{aligned} \Phi (s,\xi )=\int _{\mathbb {R}^{\ell }} \mathcal {F}g_n(s,\cdot ,t,z)(\xi ) u_0(z)dz, \end{aligned}$$

$ds=ds_1\cdots ds_n$ and $dr=dr_1\cdots dr _n$. Using the inequality $ab \le \frac{1}{2} (a^2+b^2)$ and the fact that $\gamma _0$ is locally integrable, we obtain

$$\begin{aligned} n! \Vert f_n(\cdot ,t,x)\Vert ^2_{{\mathcal {H}}^{\otimes n}} \le C^n n! \int _{[0,t]^{n}} \int _{\mathbb {R}^{n\ell }} |\Phi (s,\xi )|^2 \mu (d\xi ) ds. \end{aligned}$$

By symmetry, this leads to

$$\begin{aligned} n! \Vert f_n(\cdot ,t,x)\Vert ^2_{{\mathcal {H}}^{\otimes n}} \le C^n ( n!)^2 \int _{[0,t]_<^{n}} \int _{\mathbb {R}^{n\ell }} |\Phi (s,\xi )|^2 \mu (d\xi ) ds, \end{aligned}$$

(3.7)

where for each $n\ge 2$, we denote

$$\begin{aligned}{}[0,t]^n_< := \{(t_1,\dots ,t_n): 0< t_1< \cdots< t_n < t\}. \end{aligned}$$

(3.8)

Fix $0<s_2<s_2< \cdots< s_n <t$. Notice that $(y,z) \mapsto n! g_n(s,y,t,z)$ is the joint density of the random vector $(B_{s_1}, B_{s_2}, \dots , B_{s_n}, B_t)$ at the point $(y_1-z, y_2-z, \dots , y_n-z,x-z)$ where $B=\{B_t, t\ge 0\}$ is an $\ell$-dimensional Brownian motion. Therefore, $n! g_n(s,\cdot ,t,z) /p_t(x-z)$ is the conditional density of $(B_{s_1}, B_{s_2}, \dots , B_{s_n})$ given $B_t=x-z$, which coincides with the law of the random vector

$$\begin{aligned} Z= \left( B_{s_1} -\frac{s_1}{t} B_t+ \frac{s_1}{t} (x-z), \dots , B_{s_n} - \frac{s_n}{t} B_t+ \frac{s_n}{t} (x-z)\right) . \end{aligned}$$

The characteristic function of this vector is given by

$$\begin{aligned} \mathbb {E}[e^{ i \xi \cdot Z }]= \exp \left( -\frac{1}{2} \mathbb {E}\left[ \left| \sum _{j=1} ^n \xi ^j \cdot B_{0,t}(s_j) \right| ^2 \right] \right) e^{i\frac{s_1 + \cdots +s_n}{t} (x-z) \cdot \xi }, \end{aligned}$$

where we recall that $\{B_{0,t}(s), s\in [0,t]\}$ denotes an $\ell$-dimensional Brownian bridge from zero to zero. This implies

$$\begin{aligned} |\Phi (s,\xi )| \le \frac{1}{n!} |p_t *u_0(x)| \exp \left( -\frac{1}{2} \mathrm{Var} \left( \sum _{j=1}^n \xi ^j \cdot B_{0,t}(s_j) \right) \right) . \end{aligned}$$

Substituting the previous estimate into (3.7) yields

$$\begin{aligned} n! \Vert f_n(\cdot ,t,x)\Vert ^2_{{\mathcal {H}}^{\otimes n}} \le C^n |p_t* u_0(x)|^2 \int _{[0,t]_<^{n}} \int _{\mathbb {R}^{n\ell }} \exp \left( -\mathrm{Var} \left( \sum _{j=1}^n \xi ^j \cdot B_{0,t}(s_j) \right) \right) \mu (d\xi ) ds. \end{aligned}$$

(3.9)

Finally, from Lemmas 9.1 and 9.4 of [10] we conclude that (3.5) holds. $\square$

4 Feynman-Kac formulas for the moments of the solution

For any $\varepsilon >0$, we define $\gamma _\varepsilon$ by

$$\begin{aligned} \gamma _ \varepsilon (x)=\frac{1}{(2 \pi )^\ell }\int _{\mathbb {R}^\ell } e^{- \varepsilon |\xi |^2} e^{i \xi \cdot x}\mu (d \xi )\,. \end{aligned}$$

(4.1)

Notice that for each $\varepsilon >0$, the spectral measure of $\gamma _ \varepsilon$ is $\mu _\varepsilon (d\xi ):= e^{-\varepsilon |\xi |^2}\mu (d \xi )$, which has finite total mass because $\mu$ is a tempered measure. Thus, $\gamma _ \varepsilon$ is a bounded positive definite function. The next proposition is the key ingredient in the proof of the Feynman-Kac formula for the moments of the solution to Eq. (1.1) using Brownian bridges.

Proposition 4.1

Suppose that the spatial covariance satisfies (H.1) or (H.2). Let $\kappa$ be a real number. Let $\{B^j_{0,t}(s), s\in [0,t]\}$, $j=1\dots , n$, be independent $\ell$-dimensional Brownian bridges from 0 to 0. Then for each $\varepsilon >0$, the function $F_\varepsilon : (\mathbb {R}^\ell )^n \rightarrow \mathbb {R}$ given by

$$\begin{aligned} F_ \varepsilon (x^1,\dots ,x^n)=\mathbb {E}\exp \left\{ \kappa \sum _{1\le j<k\le n} \int _0^t \int _0^t \gamma _0(s-r) \gamma _ \varepsilon (B_{0,t}^j(s)- B^k_{0,t}(r)+x^j-x^k)dr ds\right\} \end{aligned}$$

is well-defined and continuous. Moreover, as $\varepsilon \downarrow 0$, $F_ \varepsilon$ converges uniformly to a limit function denoted by

$$\begin{aligned} \mathbb {E}\exp \left\{ \kappa \sum _{1\le j<k\le n} \int _0^t \int _0^t \gamma _0(s-r) \gamma ( B_{0,t}^j(s)- B^k_{0,t}(r)+x^j-x^k)drds\right\} \,. \end{aligned}$$

(4.2)

Remark 4.2

Actually, for each $1 \le j<k \le n$, the integral

$$\begin{aligned} \int _0^t \int _0^t \gamma _0(s-r) \gamma _{\varepsilon }( B_{0,t}^j(s)- B^k_{0,t}(r)+x^j-x^k)drds \end{aligned}$$

converges in $L^p(\Omega )$ as $\varepsilon$ tends to zero, for each $p\ge 1$, and we can also denote the limit as

$$\begin{aligned} \frac{1}{(2\pi )^{\ell }} \int _0^t\int _0^t\int _{\mathbb {R}^\ell } \gamma _0(s-r) e^{i \xi \cdot (B_{0,t}^j(s)- B^k_{0,t}(r)+x^j-x^k)}\mu (d \xi )drds. \end{aligned}$$

Proof

We claim that for every $\kappa \in \mathbb {R}$

$$\begin{aligned} \sup _{\varepsilon >0} \mathbb {E}\exp \left\{ \kappa \sum _{1\le j<k\le n} \int _0^t \int _0^t \gamma _0(s-r) \gamma _ \varepsilon (B_{0,t}^j(s)- B^k_{0,t}(r))drds\right\} <\infty \,. \end{aligned}$$

(4.3)

By Hölder inequality, it suffices to show the previous inequality for $n=2$. For every $d \in \mathbb {N}$, we have

$$\begin{aligned}&\mathbb {E}\left[ \int _0^t \int _0^t \gamma _ \varepsilon (B_{0,t}^1(s)- B_{0,t}^2(r))\gamma _0(s-r) drds \right] ^d \\&\quad =\mathbb {E}\left[ \frac{1}{(2\pi )^{\ell }}\int _0^t \int _0^t \int _{\mathbb {R}^\ell } e^{i \xi \cdot (B_{0,t}^1(s)- B_{0,t}^2(r))} \gamma _0(s-r) \mu _{\varepsilon }(d\xi ) drds\right] ^d\\&\quad = \frac{1}{(2\pi )^{\ell d}}\int _{[0,t]^{2d}} \int _{(\mathbb {R}^\ell )^d} \mathbb {E}\exp \left\{ i \sum _{k=1}^d \xi ^k \cdot \left( B_{0,t}^1(s_k)- B_{0,t}^2(r_k) \right) \right\} \\&\quad \prod _{k=1}^d \gamma _0(s_k-r_k) \mu _\varepsilon (d\xi ) drds\,, \end{aligned}$$

where we use the notation $\mu _\varepsilon (d\xi ) = \prod _{k=1}^d e^{-\varepsilon |\xi ^k|^2}\mu (d\xi ^k)$ and $ds = ds_1 \cdots ds_d$. Using the independence of $B^1$ and $B^2$, Cauchy-Schwarz inequality, the inequality $ab \le \frac{1}{2}(a^2+b^2)$ and the fact that $\gamma _0$ is locally integrable, we obtain

$$\begin{aligned}&\mathbb {E}\left[ \int _0^t \int _0^t \gamma _ \varepsilon (B_{0,t}^1(s)- B_{0,t}^2(r))\gamma _0(s-r) drds \right] ^d \nonumber \\&\quad \le C^d \int _{[0,t]^d} \int _{(\mathbb {R}^\ell )^d} \left| \mathbb {E}\exp \left\{ i \sum _{k=1}^d \xi ^k \cdot B^1_{0,t}(s_k) \right\} \right| ^2 \mu _\varepsilon (d\xi ) ds\nonumber \\&\quad \le C^d d!\int _{[0,t]^d_<} \int _{(\mathbb {R}^\ell )^d} \exp \left\{ -\mathrm{Var} \left( \sum _{k=1}^d \xi ^k \cdot B^1_{0,t}(s_k) \right) \right\} \mu _\varepsilon (d\xi ) ds, \end{aligned}$$

(4.4)

where $[0,t]^d_<$ is defined in (3.8). Then (4.3) follows from the Taylor expansion of $e^x$ and Lemmas 9.1 and 9.4 in [10]. Finally, the proof of the uniform convergence of $F_ \varepsilon$ as $\varepsilon$ tends to zero can be done by the same arguments as in the proof of Proposition 4.2 in [10]. Notice that Lemma 4.1 in [10] has to be replaced by the inequality

$$\begin{aligned} \mathbb {E}\exp \left\{ \int _{[0,t]^2} \sum _{1\le j<k\le n} \kappa \gamma _\varepsilon ( G_s^j- G_r^k+ y_{s,r}^{jk})drds\right\} \nonumber \\ \le \mathbb {E}\exp \left\{ \int _{[0,t]^2}\sum _{1\le j<k\le n} |\kappa | \gamma _\varepsilon ( G_s^j- G_r^k)dr ds\right\} , \end{aligned}$$

(4.5)

where $\kappa \in \mathbb {R}$, $G=(G^1,\dots ,G^n)\in (\mathbb {R}^\ell )^n$ is a centered Gaussian process indexed by [0, t] and $y=(y^{jk})_{1\le j<k\le n} :[0,t]^2\rightarrow (\mathbb {R}^\ell )^{n(n-1)/2}$ is a measurable matrix-valued function. $\square$

As an application, we have the following Feynman-Kac formula based on Brownian bridges.

Proposition 4.3

Suppose that the spatial covariance satisfies (H.1) or (H.2). Suppose that $\{B_{0,t}^j(s),s\in [0,t]\}$, $j=1,\dots , n$ are $\ell$-dimensional independent Brownian bridges from zero to zero. Then for every $x^1,\dots ,x^n\in \mathbb {R}^\ell$,

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^n u(t,x^j)\right]&=\int _{(\mathbb {R}^\ell )^n} \mathbb {E}\exp \left\{ \int _{[0,t]^2} \right. \nonumber \\&\quad \sum _{1\le j<k\le n} \gamma \left( B_{0,t}^j(s)-B_{0,t}^k(r)+x^j-x^k+\frac{s}{t} y^j- \frac{r}{t} y^k\right) \nonumber \\&\quad \left. \times \gamma _0(s-r) drds \right\} \prod _{j=1}^n [u_0(x^j+y^j)p_t(y^j)]d y^1\cdots d y^n\,. \end{aligned}$$

(4.6)

Proof

For any $\varepsilon >0$ we denote by $u_\varepsilon (t,x)$ the solution to the stochastic heat equation

$$\begin{aligned} \frac{\partial u_\varepsilon }{\partial t }=\frac{1}{2} \Delta u_{\varepsilon } +u_\varepsilon \dot{W_\varepsilon }\,,\quad u(0,\cdot )=u_0(\cdot )\,, \end{aligned}$$

where $\dot{W}_\varepsilon$ is a Gaussian centered noise with covariance

$$\begin{aligned} \mathbb {E}[\dot{W}_\varepsilon (s,y) \dot{W}_\varepsilon (t,x)]= \gamma _0(s-t) \gamma _\varepsilon (x-y). \end{aligned}$$

From the results of Hu, Huang, Nualart and Tindel [9] we have the following Feynman-Kac formula for the moments of $u_\varepsilon$

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^n u_\varepsilon (t,x^j) \right]= & {} \mathbb {E}\Bigg ( \prod _{j=1}^n u_0(B^j(t)+x^j) \exp \Big \{\sum _{1\le j<k\le n}\nonumber \\&\quad \int _{[0,t]^2} \gamma _\varepsilon (B^j(s)-B^k(r)+(x^j-x^k) )\nonumber \\&\times \gamma _0(s-r) drds \Big \}\Bigg ), \end{aligned}$$

(4.7)

where $\{B^j, j=1,\dots , n\}$ are independent $\ell$-dimensional standard Brownian motions. We remark that in [9] it is required that $\gamma$ is a non-negative function, which is not necessarily true for $\gamma _\varepsilon$. However, $\gamma _\varepsilon$ is bounded, and, in this case, it is not difficult to show that (4.7) still holds. Also, [9] assumes that $u_0$ is bounded, but it is not difficult to show that (4.7) still holds assuming (3.2).

For each $j=1,\dots ,n$ and every fixed $t>0$, the Brownian motion $B^j$ admits the following decomposition

$$\begin{aligned} B^j(s)=B_{0,t}^j(s)+\frac{s}{t} B^j(t), \end{aligned}$$

(4.8)

where $\{B_{0,t}^j(s),s\in [0,t]\}$, $j=1,\dots ,n$, are Brownian bridges on $\mathbb {R}^\ell$ independent from $\{B^j(t), 1\le j \le n\}$ and from each other. Thus, identity (4.7) can be written as

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^n u_\varepsilon (t,x^j)\right]= & {} \int _{(\mathbb {R}^\ell )^n} \mathbb {E}\exp \Big \{\int _{[0,t]^2} \sum _{1\le j<k\le n} \gamma _\varepsilon \left( B_{0,t}^j(s)\right. \nonumber \\&\left. -B_{0,t}^k(r)+x^j-x^k+\frac{s}{t} y^j- \frac{r}{t} y^k\right) \nonumber \\&\times \gamma _0(s-r) drds \Big \} \prod _{j=1}^n [u_0(x^j+y^j)p_t(y^j)]d y^1\cdots d y^n\,.\nonumber \\ \end{aligned}$$

(4.9)

From Proposition 4.1 and the dominated convergence theorem, the right-hand side of (4.9) converges to the right-hand side of (4.6). From the Wiener chaos expansion of the solution and the computations in the proof of Theorem 3.2, it follows easily that $u_\varepsilon (t,x)$ converges in $L^2(\Omega )$ to u(t, x). On the other hand, from (4.9) it follows that the moments of all orders of $u_\varepsilon (t,x)$ are uniformly bounded in $\varepsilon$. As a consequence, the left-hand side of (4.9) converges to the left-hand side of (4.6). This completes the proof. $\square$

Corollary 4.4

Under the assumptions of Proposition 4.3 we have, for any $x\in \mathbb {R}^\ell$

$$\begin{aligned} \mathbb {E}\left[ u(t,x)^n\right]= & {} \mathbb {E}\Bigg ( \prod _{j=1}^n u_0(B^j(t)+x) \exp \Big \{\sum _{1\le j<k\le n}\int _{[0,t]^2} \gamma (B^j(s)\nonumber \\&-B^k(r)) \gamma _0(s-r) drds \Big \}\Bigg ), \end{aligned}$$

(4.10)

where $B^j$, $j=1,\dots , n$, are independent $\ell$-dimensional Brownian motions.

Remark 4.5

If the initial condition $u_0$ is nonnegative, one can show that $u(t,x) \ge 0$ a.s., for all $t\ge 0$ and $x\in \mathbb {R}^\ell$. This follows from the fact that $u_\varepsilon (t,x)$ is nonnegative for any $\varepsilon$, where $u_\varepsilon$ is the random field introduced in the proof of Proposition 4.3.

5 Lyapunov exponents of Brownian bridges

The following variational formula occurs frequently in our considerations,

$$\begin{aligned} {\mathcal {E}}(\alpha _0,\gamma )= & {} \sup _{g \in \mathcal {A}_\ell } \left\{ \int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }} \frac{\gamma (x-y)}{|s-r|^{\alpha _0}} g^2(s,x) g^2(r,y) dx dy dr ds\right. \nonumber \\&\left. - \frac{1}{2} \int _0^1 \int _{\mathbb {R}^\ell } |\nabla _x g(s,x)|^2 dx ds \right\} , \end{aligned}$$

(5.1)

where $\mathcal {A}_\ell$ is the class of functions defined by

$$\begin{aligned} \mathcal {A}_{\ell } = \left\{ g: g(s,\cdot ) \in W^{1,2}(\mathbb {R}^\ell ) \ \text {and}\ \int _{\mathbb {R}^{\ell }} g^2(s,x)dx=1, \, \mathrm{for \,\, all} \, 0 \le s \le 1 \right\} \,, \end{aligned}$$

(5.2)

where $\alpha _0 \in [0,1)$ and $\gamma$ is a generalized covariance function.

In general, if $\eta _0$ is a covariance function (nonnegative and nonnegative definite locally integrable function) on $\mathbb {R}$ and $\eta$ is a generalized covariance function on $\mathbb {R}^\ell$ with spectral measure $\nu$, we can define the variational quantity

$$\begin{aligned} \mathcal {E}(\eta _0,\eta )= & {} \sup _{g \in \mathcal {A}_\ell } \left\{ \int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }}\eta (x-y)\eta _0(s-r) g^2(s,x) g^2(r,y) dx dy dr ds \right. \nonumber \\&\left. - \frac{1}{2} \int _0^1 \int _{\mathbb {R}^\ell } |\nabla _x g(s,x)|^2 dx ds \right\} \,. \end{aligned}$$

(5.3)

It is evident that $\mathcal {E}(\alpha _0,\gamma )=\mathcal {E}(|\cdot |^{-\alpha _0},\gamma )$. The first integration in (5.3) is defined through Fourier transforms,

$$\begin{aligned}&\int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }}\eta (x-y)\eta _0(s-r) g^2(s,x) g^2(r,y) dx dy dr ds\nonumber \\&\quad = \frac{1}{(2\pi )^{\ell }}\int _{[0,1]^2} \int _{\mathbb {R}^\ell } \mathcal {F} g^2(s,\cdot )(\xi ) \overline{\mathcal {F}g^2(r,\cdot )}(\xi ) \nu (d\xi ) \eta _0(s-r) drds\,. \end{aligned}$$

(5.4)

A priori, $\mathcal {E}(\eta _0,\eta )$ can be infinite. However, if $\eta _0$ belongs to $L^1([-1,1])$ and $\eta$ satisfies the Dalang’s condition (1.6) (as in all cases in the current article), then $\mathcal {E}(\eta _0,\eta )$ is finite. Indeed, applying Cauchy-Schwarz inequality, we have

$$\begin{aligned}&\int _{[0,1]^2} \int _{\mathbb {R}^\ell } \mathcal {F} g^2(s,\cdot )(\xi ) \overline{\mathcal {F}g^2(r,\cdot )}(\xi ) \nu (d\xi ) \eta _0(s-r) drds\\&\quad \le \int _0^1\int _0^1\left[ \int _{\mathbb {R}^\ell }|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi ) \right] ^{\frac{1}{2}}\left[ \int _{\mathbb {R}^\ell }|\mathcal {F}g^2(r,\cdot )(\xi )|^2 \nu (d \xi ) \right] ^{\frac{1}{2}}\\&\quad \eta _0(s-r)dsdr\\&\quad \le \frac{1}{2}\int _0^1\int _0^1 \int _{\mathbb {R}^\ell }|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi )\eta _0(s-r)dsdr\\&\qquad +\frac{1}{2}\int _0^1\int _0^1 \int _{\mathbb {R}^\ell }|\mathcal {F}g^2(r,\cdot )(\xi )|^2 \nu (d \xi )\eta _0(s-r)dsdr\\&\quad \le \Vert \eta _0\Vert _{L^1([-1,1])}\int _0^1\int _{\mathbb {R}^\ell }|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi )ds\,. \end{aligned}$$

Moreover, for each $s\in [0,1]$, $| \mathcal {F} g^2(s,\cdot )(\xi )|$ is bounded by 1 and by $\frac{2\sqrt{\ell }}{|\xi |} \Vert \nabla _x g(s,\cdot )\Vert _{L^2(\mathbb {R}^\ell )}$. In fact, integrating by parts, we have

$$\begin{aligned} | \mathcal {F} g^2(s,\cdot )(\xi )|&\le \min _{1\le j \le \ell } \frac{1}{|\xi _j|} \int _{\mathbb {R}^\ell } \left| \frac{\partial g^2(s,x)}{\partial x_j} \right| dx\\&{=} { \min _{1\le j \le \ell } \frac{1}{|\xi _j|} \left\| \frac{\partial g^2(s,\cdot )}{\partial x_j} \right\| _{L^1(\mathbb {R}^\ell )} \le \frac{2\sqrt{\ell }}{|\xi |} \Vert \nabla _x g(s,\cdot )\Vert _{L^2(\mathbb {R}^\ell )}. } \end{aligned}$$

It follows that for every $R>0$,

$$\begin{aligned}&\int _0^1\int _{\mathbb {R}^\ell }|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi )ds\\&\quad =\int _0^1\int _{|\xi |\le R}|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi )ds+\int _0^1\int _{|\xi |>R}|\mathcal {F}g^2(s,\cdot )(\xi )|^2 \nu (d \xi )ds\\&\quad \le \int _{|\xi |\le R}\nu (d \xi )+4\ell \int _{|\xi |>R}\frac{\nu (d \xi )}{|\xi |^2}\int _0^1\int _{\mathbb {R}^\ell }|\nabla _x g(s,x)|^2dxds\,. \end{aligned}$$

Since $\nu$ satisfies Dalang’s condition $\int _{\mathbb {R}^\ell }\frac{\nu (d \xi )}{1+|\xi |^2}<\infty$, we can choose $R>0$ such that

$$\begin{aligned} {\Vert \eta _0\Vert _{L^1([-1,1])} } (2 \pi )^{-\ell }4\ell \int _{|\xi |>R}\frac{\nu (d \xi )}{|\xi |^2}<\frac{1}{2}. \end{aligned}$$

This implies that the right-hand side of (5.3) is at most ${\Vert \eta _0\Vert _{L^1([-1,1])} }(2 \pi )^{-\ell }\int _{|\xi |<R}\nu (d \xi )$, which is also an upper bound for $\mathcal {E}(\eta _0,\eta )$.

To conclude our discussion on basic properties of $\mathcal {E}(\eta _0,\eta )$, we describe a useful comparison principle. Suppose $\eta _0,\tilde{\eta }_0$ are covariance functions on $\mathbb {R}$ and $\eta ,\tilde{\eta }$ are generalized covariance functions on $\mathbb {R}^\ell$ such that the spectral measures of $\eta _0, \eta$ are less than the spectral measures of $\tilde{\eta }_0, \tilde{\eta }$ respectively. In other words, $\eta _0\le \tilde{\eta }_0$ and $\eta \le \tilde{\eta }$ in quadratic sense. Then

$$\begin{aligned} \mathcal {E}(\eta _0,\eta )\le \mathcal {E}(\tilde{\eta }_0,\tilde{\eta })\,. \end{aligned}$$

(5.5)

This is immediate from (5.3).

In the remaining of the article, we consider the following scaling condition on the noise:

• (S) There exist $\alpha _0\in (0,1)$ and $\alpha \in (0,2)$ such that $\gamma _0(t)=|t|^{-\alpha _0}$ and $\gamma (cx)=c^{-\alpha } \gamma (x)$ for all $t,c>0$ and $x\in \mathbb {R}^\ell$.

Under the scaling assumption (S), it is easy to check that for every $\theta >0$,

$$\begin{aligned} {\mathcal {E}}(\alpha _0,\theta \gamma )=\theta ^{\frac{2}{2- \alpha }}{\mathcal {E}}(\alpha _0,\gamma )\,. \end{aligned}$$

(5.6)

Proposition 5.1

Let K and $\psi$ be symmetric functions in $L^2(\mathbb {R}^\ell )$ and $L^2(\mathbb {R})$ respectively. We assume in addition that $\psi$ is nonnegative and $\psi '$ exists and belongs to $L^2(\mathbb {R})$. The functions $\eta _0=\psi *\psi$ and $\eta =K*K$ are bounded and nonnegative definite functions. Then for every $\theta >0$ and every integer $n\ge 1$,

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}\exp \left\{ \frac{\theta }{(n-1)t}\sum _{1\le j\ne k\le n}\int _0^t\int _0^t \eta (B^j(s)-B^k(r))\eta _0\left( \frac{s-r}{t}\right) dsdr \right\} \nonumber \\&\quad \le n \mathcal {E}(\theta \eta _0,\eta )\,, \end{aligned}$$

(5.7)

where $\mathcal {E}(\eta _0,\eta )$ is the variational quantity defined in (5.3).

Before giving the proof, let us explain our contribution. This result, together with Theorem 5.2 below, extends the result of Chen in [2, Section 4] , where $\eta$ is assumed to be nonnegative. In the aforementioned paper, the author uses a compact folding argument. When $\eta$ switches signs, this argument no longer works. In particular, [2, inequality (4.15)] fails. Here, we replace the compact folding argument by a moment comparison between Brownian motions and Ornstein-Uhlenbeck processes, which was observed earlier by Donsker and Varadhan [7] [see (5.9) below]. Unlike Brownian motions, the Ornstein-Uhlenbeck processes are ergodic. This makes the essential arguments of [2] carry through. Lastly, although the occupation times of Ornstein-Uhlenbeck processes satisfy (strong) large deviation principles, it cannot be applied here due to the time-dependent structure (namely $\eta _0$).

Proof of Proposition 5.1

We first observe that

$$\begin{aligned}&\sum _{1\le j\ne k\le n}\int _0^t\int _0^t \eta (B^j(s)-B^k(r))\eta _0\left( \frac{s-r}{t}\right) dsdr\\&\quad =\int _{\mathbb {R}^{\ell +1}}\left[ \sum _{j=1}^n\int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B^j(s))ds \right] ^2dudx\\&\qquad -\sum _{j=1}^n\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B^j(s))ds \right] ^2dudx\\&\quad \le (n-1)\sum _{j=1}^n\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B^j(s))ds \right] ^2dudx\,. \end{aligned}$$

In conjunction with the independence of the Brownian motions, we see that the left-hand side of (5.7) is at most

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{n}{t}\log \mathbb {E}\exp \left\{ \frac{\theta }{t}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx \right\} \,. \end{aligned}$$

Hence, it suffices to show

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}\exp \left\{ \frac{\theta }{t}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx \right\} \le \mathcal {E}(\theta \eta _0,\eta )\,. \end{aligned}$$

(5.8)

The proof is now divided into several steps.

Step 1. For each $\kappa >0$, let $\mathbb {P}_\kappa$ be the law of an Ornstein-Uhlenbeck process in $\mathbb {R}^\ell$ starting from 0 with generator $\frac{1}{2} \Delta - \kappa x\cdot \nabla$. Let $\mathbb {E}_\kappa$ denote the expectation with respect to $\mathbb {P}_\kappa$. We will show that

$$\begin{aligned}&\mathbb {E}\exp \left\{ \frac{\theta }{t}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx \right\} \nonumber \\&\quad \le \mathbb {E}_ \kappa \exp \left\{ \frac{\theta }{t}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx \right\} \,. \end{aligned}$$

(5.9)

We note that

$$\begin{aligned} \int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi (u-\frac{s}{t})K(x-B(s))ds \right] ^2dudx=\int _0^t\int _0^t \eta (B(s)-B(r))\eta _0(s-r)dsdr\,. \end{aligned}$$

Hence, it suffices to check that for each integer $d\ge 1$

$$\begin{aligned} \mathbb {E}\left[ \int _0^t\int _0^t \eta (B(s)-B(r))\eta _0(s-r)dsdr\right] ^d\le \mathbb {E}_{\kappa }\left[ \int _0^t\int _0^t \eta (B(s)-B(r))\eta _0(s-r)dsdr\right] ^d. \end{aligned}$$

Since $\eta _0$ is nonnegative, this amounts to show

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^d \eta (B(s_j)-B(r_j)) \right] \le \mathbb {E}_\kappa \left[ \prod _{j=1}^d \eta (B(s_j)-B(r_j)) \right] \end{aligned}$$

(5.10)

for arbitrary times $s_1,r_1,\dots ,s_d,r_d$ in [0, t]. By writing $\eta (z)=(2 \pi )^{-\ell }\int _{\mathbb {R}^\ell }e^{i \xi \cdot z}|\mathcal {F}K(\xi )|^2 d \xi$, we see that

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^d \eta (B(s_j)-B(r_j)) \right]&=(2 \pi )^{-\ell d}\int _{\mathbb {R}^{\ell d}}\mathbb {E}e^{i\sum _{j=1}^d \xi _j\cdot (B(s_j)-B(r_j))}\prod _{j=1}^d|\mathcal {F}K(\xi _j)|^2 d \xi _j\\&=(2 \pi )^{-\ell d}\int _{\mathbb {R}^{\ell d}}e^{-\frac{1}{2}\mathbb {E}\left[ \left( \sum _{j=1}^d \xi _j\cdot (B(s_j)-B(r_j))\right) ^2\right] }\prod _{j=1}^d|\mathcal {F}K(\xi _j)|^2 d \xi _j\,. \end{aligned}$$

Hence, (5.10) is evident provided that

$$\begin{aligned} \mathbb {E}\left[ \left( \sum _{j=1}^d \xi _j\cdot (B(s_j)-B(r_j))\right) ^2\right] \ge \mathbb {E}_\kappa \left[ \left( \sum _{j=1}^d \xi _j\cdot (B(s_j)-B(r_j))\right) ^2\right] \,. \end{aligned}$$

An observation made by Donsker-Varadhan [7, proof of Lemma 3.10] is that $\mathbb {E}[B(s)\otimes B(r)]\ge \mathbb {E}_\kappa [B(s)\otimes B(r)]$ in quadratic sense. This fact implies the above inequality.

Step 2. As a consequence, (5.8) is reduced to showing

$$\begin{aligned}&\limsup _{\kappa \downarrow 0}\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa \exp \left\{ \frac{\theta }{t}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx \right\} \nonumber \\&\quad \le \mathcal {E}(\theta \eta _0,\eta )\,. \end{aligned}$$

(5.11)

For each $t>0$ and each path B, we denote

$$\begin{aligned} Z_{t,B}(u,x) =\frac{1}{t}\int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \end{aligned}$$

and observe that

$$\begin{aligned} \int _{\mathbb {R}^{\ell +1}}|Z_{t,B}(u,x)|^2dudx&=\frac{1}{t^2}\int _{\mathbb {R}^{\ell +1}}\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds \right] ^2dudx\\&=\frac{1}{t^2}\int _0^t\int _0^t \eta (B(s)-B(r))\eta _0\left( \frac{s-r}{t}\right) dsdr\,. \end{aligned}$$

In particular $Z_{t,B}$ belongs to $L^2(\mathbb {R}^{\ell +1})$ and

$$\begin{aligned} \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}\le \eta (0)\eta _0(0)\,. \end{aligned}$$

(5.12)

Let N be a fixed positive number and denote $\Omega _{t,N}=\{B:\frac{1}{t}\int _0^t|B(s)|ds\le N \}$. The only advantage of $\mathbb {P}_\kappa$ over $\mathbb {P}$, for which we need, is the following inequality

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {P}_\kappa (\Omega _{t,N}^c)\le -N+\frac{1}{2 \kappa ^2}+\frac{\ell }{2} \kappa \,. \end{aligned}$$

(5.13)

In fact, by Girsanov’s theorem we have

$$\begin{aligned} \frac{d\mathbb {P}_\kappa }{d\mathbb {P}}\bigg |_{[0,t]}&=\exp \left\{ -\kappa \int _0^t B(s)\cdot dB(s)-\frac{1}{2} \kappa ^2\int _0^t|B(s)|^2ds \right\} \nonumber \\&=\exp \left\{ -\frac{1}{2}\kappa |B(t)|^2+\frac{\ell }{2} \kappa t-\frac{1}{2} \kappa ^2\int _0^t|B(s)|^2ds \right\} \,. \end{aligned}$$

(5.14)

It follows that

$$\begin{aligned} \mathbb {E}_\kappa \left\{ \exp \int _0^t|B(s)|ds\right\}&\le \mathbb {E}\exp \left\{ \int _0^t\left( |B(s)|-\frac{1}{2} \kappa ^2|B(s)|^2\right) ds+\frac{\ell }{2} \kappa t \right\} \\&\le \exp \left\{ \frac{1}{2 \kappa ^2}t+\frac{\ell }{2} \kappa t \right\} \end{aligned}$$

where the last inequality is a consequence of a Cauchy-Schwarz inequality. Hence, in conjunction with Chebyshev’s inequality, we obtain

$$\begin{aligned} \mathbb {P}_\kappa \left( \frac{1}{t}\int _0^t|B(s)|ds>N \right) \le e^{-Nt}\mathbb {E}_\kappa e^{\int _0^t|B(s)|ds}\le e^{-Nt+\frac{1}{2 \kappa ^2}t+\frac{\ell }{2} \kappa t }\,. \end{aligned}$$

The estimate (5.13) is directly derived from here.

The set $M=\{Z_{t,B}\}_{B\in \Omega _{t,N},t>0}$ is then a subset of $L^2(\mathbb {R}^{\ell +1})$. We will show that M is relatively compact in $L^2(\mathbb {R}^{\ell +1})$. Indeed, we verify that $\mathcal {F}M=\{\mathcal {F}Z_{t,B}\}_{B\in \Omega _{t,N},t>0}$ satisfies the Kolmogorov-Riesz’s compactness criterion in $L^2(\mathbb {R}^{\ell +1})$ (cf. [12, Theorem 5]). More precisely, we check that

$$\begin{aligned}&\sup _{B\in \Omega _{t,N},t>0} \Vert Z_{t,B}\Vert _{L^2(\mathbb {R}^{\ell +1})}<\infty \,, \end{aligned}$$

(5.15)

$$\begin{aligned}&\lim _{\rho \rightarrow \infty }\sup _{B\in \Omega _{t,N},t>0}\int _{|(\eta ,\xi )|>\rho }|\mathcal {F}Z_{t,B}(\eta ,\xi )|^2d \eta d \xi =0\,, \end{aligned}$$

(5.16)

$$\begin{aligned}&\lim _{\rho \downarrow 0}\sup _{|(\tau ',\xi ')|<\rho } \sup _{B\in \Omega _{t,N},t>0}\int _{\mathbb {R}^{\ell +1}}|\mathcal {F}Z_{t,B}(\tau +\tau ',\xi +\xi ')-\mathcal {F}Z_{t,B}(\tau ,\xi )|^2d \tau d \xi =0.\nonumber \\ \end{aligned}$$

(5.17)

Notice that (5.15) is evident from (5.12). We can easily compute the Fourier transform of $Z_{t,B}$

$$\begin{aligned} \mathcal {F}Z_{t,B}(\tau ,\xi )=\mathcal {F}\psi (\tau )\mathcal {F}K(\xi ) \frac{1}{t}\int _0^t e^{-i \tau \frac{s}{t}-i \xi \cdot B(s)}ds\,. \end{aligned}$$

Hence,

$$\begin{aligned} \sup _{B\in \Omega _{t,N},t>0}\int _{|(\tau ,\xi )|>\rho }|\mathcal {F}Z_{t,B}(\tau ,\xi )|^2d \tau d \xi \le \int _{|(\tau ,\xi )|>\rho }|\mathcal {F}\psi (\tau )|^2|\mathcal {F}K(\xi )|^2 d \tau d \xi \,, \end{aligned}$$

which implies (5.16). To show (5.17), let us first fix $\varepsilon >0$ and choose a function g in $C_c^\infty (\mathbb {R}^{\ell +1})$ such that $\Vert \mathcal {F}\psi \otimes \mathcal {F}K-g\Vert _{L^2(\mathbb {R}^{\ell +1})}< \varepsilon$. We denote $Y_{t,B}(\tau ,\xi )=g(\tau ,\xi )\frac{1}{t}\int _0^t e^{-i \tau \frac{s}{t}-i \xi \cdot B(s)}ds$ and observe that for every path B in $\Omega _{t,N}$ and $|(\tau ',\xi ')|<\rho$, we have

$$\begin{aligned}&|Y_{t,B}(\tau +\tau ',\xi +\xi ')- Y_{t,B}(\tau ,\xi )|\\&\quad \le \left| g(\tau +\tau ',\xi +\xi ')-g(\tau ,\xi )\right| +|g(\tau ,\xi )|\left| \frac{1}{t}\int _0^te^{-i \tau \frac{s}{t}-i \xi \cdot B(s)}(e^{-i \tau '\frac{s}{t}-i \xi '\cdot B(s)}-1)ds \right| \\&\quad \le \left| g(\tau +\tau ',\xi +\xi ')-g(\tau ,\xi )\right| +2\left| g(\tau ,\xi )\right| \left( |\tau '|+ |\xi '|\frac{1}{t}\int _0^t|B(s)|ds\right) \,.\\&\quad \le |g(\tau +\tau ',\xi +\xi ')-g(\tau ,\xi )|+2\rho (N+1)|g(\tau ,\xi )|. \end{aligned}$$

It follows that

$$\begin{aligned}&\lim _{\rho \downarrow 0}\sup _{|(\tau ',\xi ')|<\rho } \sup _{B\in \Omega _{t,N},t>0}\int _{\mathbb {R}^{\ell +1}}|\mathcal {F}Z_{t,B}(\tau +\tau ',\xi +\xi ')-\mathcal {F}Z_{t,B}(\tau ,\xi )|^2 d \tau d \xi \\&\quad \le 4 \varepsilon ^2+\lim _{\rho \downarrow 0}\sup _{|(\tau ',\xi ')|<\rho } \sup _{B\in \Omega _{t,N},t>0}\int _{\mathbb {R}^{\ell +1}}| Y_{t,B}(\tau +\tau ',\xi +\xi ')- Y_{t,B}(\tau ,\xi )|^2 d \tau d \xi \\&\quad \le 4 \varepsilon ^2+\lim _{\rho \downarrow 0}\sup _{|(\tau ',\xi ')|<\rho }\int _{\mathbb {R}^{\ell +1}}| g(\tau +\tau ',\xi +\xi ')- g(\tau ,\xi )|^2 d \tau d \xi \,. \end{aligned}$$

Since g is uniformly continuous, the last limit above vanishes. Hence,

$$\begin{aligned} \lim _{\rho \downarrow 0}\sup _{|(\tau ',\xi ')|<\rho } \sup _{B\in \Omega _{t,N},t>0}\int _{\mathbb {R}^{\ell +1}}|\mathcal {F}Z_{t,B}(\tau +\tau ',\xi +\xi ')-\mathcal {F}Z_{t,B}(\tau ,\xi )|^2d \tau d \xi \le 4 \varepsilon ^2 \end{aligned}$$

for every $\varepsilon >0$. This in turn implies (5.17).

Step 3. Applying (5.12),

$$\begin{aligned} \mathbb {E}_\kappa e^{t\theta \Vert Z_{t,B}\Vert ^2_{L(\mathbb {R}^{\ell +1})}}&\le \mathbb {E}_\kappa \left[ \mathbf{1}_{\Omega _{t,N}} e^{t \theta \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}}\right] +\mathbb {P}_\kappa (\Omega _{t,N}^c)e^{t \theta \eta (0)\eta _0(0)}\,. \end{aligned}$$

Together with (5.13), the previous estimate yields

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa e^{t\theta \Vert Z_{t,B}\Vert ^2_{L(\mathbb {R}^{\ell +1})}}\le (\theta \eta (0)\eta _0(0)-N+\frac{1}{2 \kappa ^2}+\frac{\ell }{2} \kappa )\vee \nonumber \\&\quad \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa \left[ \mathbf{1}_{\Omega _{t,N}} e^{t \theta \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}}\right] \,. \end{aligned}$$

(5.18)

To deal with the limit on the right-hand side above, we adopt an argument from [2]. Let $\varepsilon$ be a fixed positive number and define

$$\begin{aligned} {\mathcal {O}}_h=\{g\in L^2(\mathbb {R}^{\ell +1}):\Vert g\Vert ^2<-\Vert h\Vert ^2+2\langle g,h\rangle +\varepsilon \}\,. \end{aligned}$$

The collection $\{{\mathcal {O}}_h\}_{h\in M}$ forms an open cover of M in $L^2(\mathbb {R}^{\ell +1})$. Since M is relatively compact, we can find deterministic functions $h_1,\dots ,h_m\in M$ such that $M\subset \cup _{j=1}^m {\mathcal {O}}_{h_j}$. It follows that for every $t>0$ and $B\in \Omega _{t,N}$,

$$\begin{aligned} \Vert Z_{t,B}\Vert _{L^2(\mathbb {R}^{\ell +1})}^2<\max _{j=1,\dots ,m}\left( -\Vert h_j\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}+2 \langle h_j,Z_{t,B}\rangle _{L^2(\mathbb {R}^{\ell +1})}+\varepsilon \right) \end{aligned}$$

and hence,

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa \left[ \mathbf{1}_{\Omega _{t,N}} e^{t \theta \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}}\right] \nonumber \\&\quad \le \max _{j=1,\dots ,m}\left( -\theta \Vert h_j\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}+\varepsilon \theta + \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa e^{2t \theta \langle h_j,Z_{t,B}\rangle _{L^2(\mathbb {R}^{\ell +1})}}\right) \,.\nonumber \\ \end{aligned}$$

(5.19)

We note that

$$\begin{aligned} \langle h_j,Z_{t,B}\rangle _{L^2(\mathbb {R}^{\ell +1})}&=\frac{1}{t}\int _{\mathbb {R}^{\ell +1}}h_j(u,x)\left[ \int _0^t \psi \left( u-\frac{s}{t}\right) K(x-B(s))ds\right] dudx\\&=\frac{1}{t}\int _0^t{\bar{h}}_j\left( \frac{s}{t},B(s)\right) ds\,, \end{aligned}$$

where

$$\begin{aligned} {\bar{h}}_j(s,z)=\int _{\mathbb {R}^{\ell +1}}h_j(u,x) \psi (u-s)K(x-z)dudx=h_j*(\psi \otimes K)(s,z) \,. \end{aligned}$$

Since ${\bar{h}}_j$ is the convolution of $L^2$-functions, it is continuous and bounded. Moreover, since $\psi '$ belongs to $L^2(\mathbb {R})$, $\partial _s{\bar{h}}_j$ exists and $\Vert \partial _s{\bar{h}}_j\Vert _{L^\infty }\le \Vert h_j\Vert _{L^2}\Vert \partial _s \psi \otimes K\Vert _{L^2}$. In particular, ${\bar{h}}_j$ satisfies the hypothesis of [5, Proposition 3.1]. We also note that from (5.14), $\frac{d\mathbb {P}_k}{d\mathbb {P}}\big |_{[0,t]}\le e^{\frac{\ell }{2} \kappa t}$. In conjunction with [5, Proposition 3.1], it follows that

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_{\kappa }e^{2t \theta \langle h_j,Z_{t,B}\rangle _{L^2(\mathbb {R}^{\ell +1})}}\\&\quad \le \frac{\ell }{2} \kappa +\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}\exp \left\{ 2\theta \int _0^t{\bar{h}}_j\left( \frac{s}{t},B(s)\right) ds \right\} \\&\quad \le \frac{\ell }{2} \kappa +\sup _{g\in \mathcal {A}_\ell }\left\{ 2 \theta \int _0^1\int _{\mathbb {R}^{\ell }}{\bar{h}}_j(s,x)g^2(s,x)dxds-\frac{1}{2}\int _0^1\int _{\mathbb {R}^\ell }|\nabla _xg(s,x)|^2dxds \right\} \\&\quad = \frac{\ell }{2} \kappa +\sup _{g\in \mathcal {A}_\ell }\left\{ 2 \theta \langle h_j,(\psi \otimes K)* g^2\rangle _{L^2(\mathbb {R}^{\ell +1})} -\frac{1}{2}\int _0^1\int _{\mathbb {R}^\ell }|\nabla _xg(s,x)|^2dxds \right\} \,, \end{aligned}$$

where each for $g\in \mathcal {A}_\ell$ we conventionally set $g(s,x)=0$ if $s\notin [0,1]$. Gluing together our argument since (5.19), we obtain

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa \left[ \mathbf{1}_{\Omega _{t,N}}e^{t \theta \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}} \right] \\&\quad \le \max _{j=1,\dots ,m}\sup _{g\in \mathcal {A}_\ell }\left\{ -\theta \Vert h_j\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}+2 \theta \langle h_j,(\psi \otimes K)*g^2\rangle _{L^2(\mathbb {R}^{\ell +1})}\right. \\&\quad \quad -\left. \frac{1}{2}\int _0^1\int _{\mathbb {R}^\ell }|\nabla _xg(s,x)|^2dxds \right\} + \varepsilon \theta +\frac{\ell }{2} \kappa . \end{aligned}$$

Applying the Cauchy-Schwarz inequality $-\Vert h_j\Vert ^2_{L^2}+2 \langle h_j,(\psi \otimes K)*g^2\rangle _{L^2}\le \Vert (\psi \otimes K)*g^2\Vert ^2_{L^2}$, we further have

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}_\kappa \left[ \mathbf{1}_{\Omega _{t,N}}e^{t \theta \Vert Z_{t,B}\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}} \right] \\&\quad \le \sup _{g\in \mathcal {A}_\ell }\left\{ \theta \Vert (\psi \otimes K)*g^2\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}-\frac{1}{2}\int _0^1\int _{\mathbb {R}^\ell }|\nabla _xg(s,x)|^2dxds \right\} +\varepsilon \theta +\frac{3}{2} \kappa \ell \,. \end{aligned}$$

Together with (5.9), (5.18) and the fact that

$$\begin{aligned} \Vert (\psi \otimes K)*g^2\Vert ^2_{L^2(\mathbb {R}^{\ell +1})}=\int _0^1\int _0^1\int _{\mathbb {R}^\ell \times \mathbb {R}^\ell }\eta (x-y)\eta _0(s-r)g^2(s,x)g^2(r,y)dxdydsdr \end{aligned}$$

we see that the left-hand side of (5.8) is at most

$$\begin{aligned} \left( \theta \eta (0)\eta _0(0)-N+\frac{1}{2 \kappa ^2}+\frac{\ell }{2} \kappa \right) \vee \left( \mathcal {E}(\theta \eta _0,\eta )+\varepsilon \theta +\frac{\ell }{2} \kappa \right) \,. \end{aligned}$$

We now send $N\rightarrow \infty$, $\kappa \downarrow 0$ and $\varepsilon \downarrow 0$ to obtain (5.8) and complete the proof. $\square$

The following result provides an upper bound for the Lyapunov exponents of Brownian bridges.

Theorem 5.2

Suppose that the covariance of the noise satisfies condition (H.1) or (H.2) and also (S). Assume that the spectral density $f(\xi )$ exists. Suppose that $\{B^j_{0,t} (s), s\in [0,t]\}$, $j=1\dots , n$, are independent $\ell$-dimensional Brownian bridges from zero to zero. Then,

$$\begin{aligned}&\limsup _{t \rightarrow \infty } t^{-a}\log \mathbb {E}\exp \left\{ \sum _{0\le j< k\le n} \int _{[0,t]^2}\frac{\gamma (B_{0,t}^j(s)-B_{0,t}^k (r))}{|s-r|^{\alpha _0}} dr ds \right\} \nonumber \\&\quad \le n \left( \frac{n-1}{2} \right) ^{\frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )\,, \end{aligned}$$

(5.20)

where we recall that $a=\frac{4-\alpha - 2\alpha _0}{2-\alpha }$.

Proof

For suitable distributions $\eta _0,\eta$, we are going to make use of the notation

$$\begin{aligned} Q_t(\eta _0,\eta ):= \sum _{0\le j<k \le n} \int _{[0,t]^2} \eta (B_{0,t}^j(s)-B_{0,t}^k (r))\eta _0(\frac{s-r}{t})drds\,. \end{aligned}$$

Let $\gamma _0(s)=|s|^{-\alpha _0}$ denote the temporal covariance. With these notation, the expectation in (5.20) can be written as $\mathbb {E}\exp \left\{ t^{-\alpha _0}Q_t(\gamma _0,\gamma ) \right\}$. We note that $\gamma _0=\psi *\psi$ where $\psi (s)=c(\alpha _0) |s|^{-\frac{1+\alpha _0}{2}}$ with some suitable constant $c(\alpha _0)$. For each $\delta >0$ we set $\psi _\delta =p_{\delta /2}* \psi$ and $\gamma _{0,\delta }=\psi _\delta *\psi _\delta$. To prove (5.20), the main ideas are first approximate the singular covariances $\gamma _0,\gamma$ by regular covariances $\gamma _{0,\delta },\gamma _\varepsilon$; then upper bound the exponential functional of Brownian bridges by that of Brownian motions. At the final stage, we will apply Proposition 5.1. This procedure will be carried out in detail in several steps below. For the moment, let us put $t_n=(n-1)^{\frac{2}{2- \alpha }}t^a$ and observe that by making change of variables $s\rightarrow \frac{t}{t_n}s$, $r\rightarrow \frac{t}{t_n}r$ and using the scaling properties of $\gamma$ and of Brownian bridges (i.e. (S) and $\{B_{0,t}(\lambda s),s\le t/ \lambda \}\mathop {=}\limits ^{d} \sqrt{\lambda } \{B_{0,\frac{t}{\lambda }}(s),s\le t/\lambda \}$ for any $\lambda >0$) we have

$$\begin{aligned} \mathbb {E}\exp \left\{ t^{-\alpha _0} Q_t(\gamma _0, \gamma )\right\} = \mathbb {E}\exp \left\{ \frac{1}{(n-1)t_n} Q_{t_n} (\gamma _0,\gamma ) \right\} . \end{aligned}$$

(5.21)

Therefore, in conjunction with (5.6), (5.20) is equivalent to

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}\exp \left\{ \frac{1}{(n-1)t} Q_{t} (\gamma _0,\gamma ) \right\} \le n \mathcal {E}(\frac{1}{2}\gamma _0,\gamma )\,. \end{aligned}$$

(5.22)

Step 1. Fix $\varepsilon >0$. For any $p,q>1$, $\frac{1}{p} + \frac{2}{q} =1$, applying Hölder inequality, we have

$$\begin{aligned} \log \mathbb {E}e^{\frac{1}{(n-1)t} Q_t(\gamma _0, \gamma )}&\le \frac{1}{p}\log \mathbb {E}e^{ \frac{p}{(n-1)t}Q_t\left( \gamma _{0,\delta },\gamma _\varepsilon \right) }\nonumber \\&\quad +\frac{1}{q}\log \mathbb {E}e^{\frac{q}{(n-1)t}Q_t \left( \gamma _0,\gamma -\gamma _\varepsilon \right) } +\frac{1}{q}\log \mathbb {E}e^{ \frac{q}{(n-1)t}Q_t \left( \gamma _0- \gamma _{0,\delta },\gamma _\varepsilon \right) }\,. \end{aligned}$$

(5.23)

We claim that

$$\begin{aligned} \limsup _{\varepsilon \downarrow 0} \limsup _{t\rightarrow \infty } \frac{1}{t} \log \mathbb {E}e^{\frac{q}{(n-1)t}Q_t \left( \gamma _0,\gamma -\gamma _\varepsilon \right) } \le 0 \end{aligned}$$

(5.24)

and

$$\begin{aligned} {\limsup _{\varepsilon \downarrow 0}}\limsup _{\delta \downarrow 0}\limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}e^{ \frac{q}{(n-1)t}Q_t \left( \gamma _0- \gamma _{0,\delta },\gamma _\varepsilon \right) }\le 0\,. \end{aligned}$$

(5.25)

Let us focus on (5.24). By Hölder’s inequality, it suffices to show that for any $\kappa \in \mathbb {R}$

$$\begin{aligned} {\limsup _{\varepsilon \downarrow 0}} \limsup _{t\rightarrow \infty } \frac{1}{t} \log \mathbb {E}\exp \left\{ \kappa t^{\alpha _{0}-1} \int _{[0,t]^2}\frac{(\gamma -\gamma _\varepsilon )(B_{0,t}^1(s)-B_{0,t}^2 (r))}{|s-r|^{\alpha _0}} dr ds \right\} \le 0. \end{aligned}$$

(5.26)

For each integer $d\ge 1$, we can write

$$\begin{aligned}&\mathbb {E}\left[ \int _{[0,t]^2}\frac{(\gamma -\gamma _\varepsilon )(B_{0,t}^1(s)-B_{0,t}^2 (r))}{|s-r|^{\alpha _0}} dr ds \right] ^d\\&\quad = \frac{1}{(2\pi )^{\ell d}}\int _{[0,t]^{2d} } \int _{(\mathbb {R}^\ell )^d} \mathbb {E}e^{i}\sum _{j=1}^d \xi ^j \cdot (B^1_{0,t} (s_j)- B^2_{0,t}(r_j) \\&\quad \prod _{j=1}^d |s_j-r_j|^{-\alpha _0} (1- e^{-\varepsilon |\xi ^j|^2}) \mu (d\xi ) drds. \end{aligned}$$

Then, using Cauchy-Schwarz inequality and the inequality $ab \le \frac{1}{2}(a^2+b^2)$, we obtain

$$\begin{aligned}&\mathbb {E}\left[ \kappa t^{\alpha _0-1} \int _{[0,t]^2}\frac{(\gamma -\gamma _\varepsilon )(B_{0,t}^1(s)-B_{0,t}^2 (r))}{|s-r|^{\alpha _0}} dr ds \right] ^d\\&\quad \le C^d \int _{[0,t]^{d} } \int _{(\mathbb {R}^\ell )^d} \left| \mathbb {E}e^{ i \sum _{j=1}^d \xi ^j \cdot B^1_{0,t} (s_j)} \right| ^2 \prod _{j=1}^d (1- e^{-\varepsilon |\xi ^j|^2}) \mu (d\xi ) ds\\&\quad = \mathbb {E}\left[ C \int _0^t(\gamma -\gamma _\varepsilon )(B_{0,t}^1(s) -B^2_{0,t}(s))ds \right] ^d, \end{aligned}$$

for some constant C depending only on $\kappa$. Therefore, the claim (5.26) is reduced to

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \limsup _{t\rightarrow \infty } \frac{1}{t} \log \mathbb {E}\exp \left\{ C \int _0^t (\gamma -\gamma _\varepsilon )(B_{0,t}^1(s)-B^2_{0,t} (s)) ds \right\} \le 0, \end{aligned}$$

which follows from Lemma 5.3 in [10]. This completes the proof of (5.24).

To show (5.25), we use the estimate

$$\begin{aligned} Q_t(\gamma _0- \gamma _{0,\delta },\gamma _\varepsilon ) \le \frac{n(n-1)t^2}{2}\Vert \gamma _\varepsilon \Vert _{L^\infty (\mathbb {R}^\ell )} \Vert \gamma _{0}-\gamma _{0,\delta }\Vert _{L^1([0,1])} \end{aligned}$$

to obtain

$$\begin{aligned} \frac{1}{t}\log \mathbb {E}e^{\frac{q}{(n-1)t}Q_t(\gamma _0- \gamma _{0,\delta },\gamma _\varepsilon )}\le \frac{n}{2}\Vert \gamma _\varepsilon \Vert _{L^\infty (\mathbb {R}^\ell )} \Vert \gamma _{0}-\gamma _{0,\delta }\Vert _{L^1([0,1])}\,. \end{aligned}$$

This implies (5.25) since $\gamma _0\in L^1([0,1])$ and $\lim _{\delta \downarrow 0}\gamma _{0,\delta }=\gamma _0$ in $L^1([0,1])$.

Step 2. We claim that

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t} \log \mathbb {E}\exp \left\{ \frac{p}{(n-1)t} Q_{t}\left( \gamma _{0,\delta }, \gamma _\varepsilon \right) \right\} \le n {\mathcal {E}}(\frac{p}{2}\gamma _{0,\delta },\gamma _\varepsilon ). \end{aligned}$$

(5.27)

Notice that the function $\gamma _\varepsilon$ is bounded and can be expressed in the form $\gamma _\varepsilon =K_\varepsilon * K_\varepsilon$, where the function $K_\varepsilon$, defined by

$$\begin{aligned} K_{\varepsilon }(x)=\frac{1}{(2\pi )^{\ell }} \int _{\mathbb {R}^{\ell }} e^{i\xi \cdot x-\frac{\varepsilon }{2}|\xi |^2} \sqrt{f(\xi )}d\xi , \end{aligned}$$

(5.28)

is bounded with bounded first derivatives, symmetric and $K_\varepsilon \in L^2 (\mathbb {R}^\ell )$. Let $\lambda$ be a fixed number in (0, 1) and set $\rho _{\lambda }=\int _{[0,1]^2\setminus [0,\lambda ]^2} \gamma _{0,\delta }(s-r)dsdr$. For each $j\ne k$, we use the estimate

$$\begin{aligned}&\int _0^t\int _0^t \gamma _\varepsilon (B^j_{0,t}(s)-B^k_{0,t}(r))\gamma _{0,\delta }\left( \frac{s-r}{t}\right) dsdr\\&\quad \le \int _0^{\lambda t}\int _0^{\lambda t}\gamma _\varepsilon (B^j_{0,t}(s)-B^k_{0,t}(r))\gamma _{0,\delta }\left( \frac{s-r}{t}\right) dsdr +\Vert \gamma _\varepsilon \Vert _\infty \rho _\lambda t^2 \end{aligned}$$

together with (2.8) to obtain

$$\begin{aligned}&\mathbb {E}e^{\frac{p}{(n-1)t}Q_t(\gamma _{0,\delta },\gamma _\varepsilon )}\\&\quad \le e^{\frac{n}{2} p\Vert \gamma _\varepsilon \Vert _\infty \rho _\lambda t} \mathbb {E}\exp \left\{ \frac{p}{(n-1)t}\sum _{0\le j<k \le n}\int _{[0,\lambda t]^2}\gamma _\varepsilon ({B_{0,t}^j(s)-B_{0,t}^k(r)})\gamma _{0,\delta }\left( \frac{s-r}{t}\right) drds \right\} \\&\quad \le \frac{e^{\frac{n}{2} p\Vert \gamma _\varepsilon \Vert _\infty \rho _\lambda t}}{(1- \lambda )^{\ell /2}} \mathbb {E}\exp \left\{ \frac{p}{(n-1)t}\sum _{0\le j<k \le n}\int _{[0,\lambda t]^2}\gamma _\varepsilon (B^j(s)-B^k(r))\gamma _{0,\delta }\left( \frac{s-r}{t}\right) drds \right\} \,. \end{aligned}$$

At this point, we apply Proposition 5.1 to get

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t} \log \mathbb {E}e^{\frac{p}{(n-1)t}Q_t(\gamma _{0,\delta },\gamma _\varepsilon )} \le \frac{n}{2} p\Vert \gamma _\varepsilon \Vert _\infty \rho _\lambda +\lambda n\mathcal {E}\left( \frac{p \lambda }{2} \gamma _{0,\delta },\gamma _ \varepsilon \right) . \end{aligned}$$

Passing through the limit $\lambda \uparrow 1$, noting that $\rho _\lambda \rightarrow 0$, we obtain (5.27).

Step 3. We combine (5.23), (5.24), (5.25) and (5.27) to get

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}e^{\frac{1}{(n-1)t}Q_t(\gamma _0,\gamma )} \le \limsup _{\varepsilon \downarrow 0}\limsup _{\delta \downarrow 0} \frac{n}{p}\mathcal {E}\left( \frac{p}{2} \gamma _{0,\delta },\gamma _\varepsilon \right) \end{aligned}$$

Note that the order of the limits $\delta \downarrow 0$ and $\varepsilon \downarrow 0$ can not be interchanged. It is evident to check that $\gamma _{0,\delta }\le \gamma _0$ and $\gamma _\varepsilon \le \gamma$ in quadratic sense. Hence, using (5.5) we have

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \mathbb {E}e^{\frac{1}{(n-1)t}Q_t(\gamma _0,\gamma )} \le \frac{n}{p}\mathcal {E}(\frac{p}{2} \gamma _{0},\gamma )\,. \end{aligned}$$

Finally, letting $p\downarrow 1$, we obtain (5.22), which completes the proof. $\square$

Remark 5.3

The case of time-independent noises corresponds to $\alpha _0=0$. In this case, the function $\gamma _0\equiv 1$ can not be written as a convolution of a function with itself. Thus the proof of Proposition 5.1 does not work in this case.

Corollary 5.4

Suppose that the covariance of the noise satisfies (H.1) or (H.2), condition (S) holds and the spectral measure $\mu$ is absolutely continuous. Let u(t, x) be the solution to (1.1) with nonnegative initial condition $u_0$ satisfying condition (3.1). Then for any integer $n \ge 2$,

$$\begin{aligned} \limsup _{t \rightarrow \infty } t^{-\frac{4-\alpha -2\alpha _0}{2-\alpha }} \log \sup _{(x^1, \dots , x^n) \in (\mathbb {R}^\ell )^n} \frac{\mathbb {E}\left[ \prod _{j=1}^n u(t,x^j)\right] }{ \prod _{j=1}^n p_t* u_0(x^j)} \le n \left( \frac{n-1}{2}\right) ^{\frac{2}{2-\alpha }}{\mathcal {E}}(\alpha _0,\gamma )\,. \end{aligned}$$

(5.29)

Proof

Let $\{B_{0,t}^j(s), s \in [0,t], j=1,\dots , n\}$, be $\ell$-dimensional Brownian bridges from zero to zero. Using the moment formula for the solution (4.6) proved in Proposition 4.3, we have

$$\begin{aligned} \mathbb {E}\left[ \prod _{j=1}^n u(t,x^j) \right]= & {} \mathbb {E}\bigg (\int _{(\mathbb {R}^{\ell })^n}\prod _{j=1}^n u_0(x^j + y^j) p_t(y^j)\\&\times \exp \left\{ \frac{1}{2} \sum _{j\ne k}^n \int _{[0,t]^2}\frac{\gamma (B_{0,t}^j(s)-B_{0,t}^k(r)+\frac{s}{t}y^j-\frac{r}{t}y^k +x^j -x^k)}{|s-r|^{\alpha _0}} dr ds \right\} dy\bigg )\\\le & {} \prod _{j=1}^n p_t * u_0(x^j)\mathbb {E}\exp \left\{ \frac{1}{2} \sum _{j\ne k}^n \int _0^t \int _0^t \frac{\gamma (B_{0,t}^j(s)-B_{0,t}^k(r))}{|s-r|^{\alpha _0}} dr ds \right\} \,, \end{aligned}$$

where the last inequality follows from (4.5). Then, the upper bound is a consequence of Theorem 5.2. $\square$

Remark 5.5

Using the approach developed in [2], we can also show the corresponding lower bound in (5.20), assuming  (H.1) or (H.2) and (S) (but not necessarily the absolute continuity of $\mu$). However, a lower bound similar to that proved in Corollary  5.4 cannot be obtained. For this reason, the proof of a lower bound for the exponential growth indices needs a direct approach as it is done in the next section.

6 Exponential growth indices

In this section we denote by u(t, x) the solution to (1.1) with nonnegative initial condition $u_0$ satisfying condition (3.1). The exponential growth indices are defined as follows:

$$\begin{aligned} \lambda _*(n)=\sup \left\{ \lambda>0:\liminf _{t\rightarrow \infty } t^{-a} \sup _{|x|\ge \lambda t^{ \frac{a+1}{2}} }\log \mathbb {E}u^n(t,x)>0\right\} \, \end{aligned}$$

(6.1)

and

$$\begin{aligned} \lambda ^*(n)=\inf \left\{ \lambda >0:\limsup _{t\rightarrow \infty } t^{-a} \sup _{|x|\ge \lambda t^{ \frac{a+1}{2}} }\log \mathbb {E}u^n(t,x)<0\right\} \,, \end{aligned}$$

(6.2)

where we recall that $a=\frac{4-\alpha -2\alpha _0}{2-\alpha }$.

6.1 Upper bound for $\lambda ^*(n)$

Set

$$\begin{aligned} b=\frac{2a}{a+1}=\frac{4- \alpha -2 \alpha _0}{3- \alpha - \alpha _0}. \end{aligned}$$

(6.3)

It may be helpful to note that $a,b\in (1,2)$. For each positive number $\beta$, we define two auxiliary functions $\psi _ \beta$ and $\phi _ \beta$. The function $\psi _ \beta :(0,\infty )\rightarrow (0,\infty )$ is defined by

$$\begin{aligned} \psi _ \beta (w)=\frac{1}{2} \beta ^2 b^2 w^{2b-2}+ \beta w^{b} \end{aligned}$$

(6.4)

and $\phi _ \beta :(0,\infty )\rightarrow (0,\infty )$ is uniquely defined by the relation

$$\begin{aligned} \beta b(\phi _ \beta (x))^{b-1}=x- \phi _ \beta (x)\,,\quad \forall x>0\,. \end{aligned}$$

(6.5)

For every fixed $x>0$, $\phi _ \beta (x)$ can be recognized as the unique minimizer of the function

$$\begin{aligned} y\mapsto f_{\beta ,x}(y):=\frac{1}{2}(y-x)^2+ \beta y^b \end{aligned}$$

(6.6)

on $(0,\infty )$. Together with (6.5), it follows that

$$\begin{aligned} f_{\beta ,x}(y)\ge \psi _ \beta (\phi _ \beta (x)) \end{aligned}$$

(6.7)

for every $\beta ,x>0$. Relation (6.5) implies that $\phi _ \beta$ is strictly increasing.

Theorem 6.1

Assume conditions (H.1) or (H.2), condition (S) and the absolute continuity of $\mu$. Suppose that $u_0$ satisfies

$$\begin{aligned} \int _{\mathbb {R}^\ell } e^{\beta |y|^b}u_0(y)dy<\infty \end{aligned}$$

(6.8)

for some $\beta >0$, then

$$\begin{aligned} \lambda ^*(n) \le g_\beta ^{-1} \left( \left( \frac{n-1}{2} \right) ^{ \frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )\right) , \end{aligned}$$

where the function $g_\beta (\lambda )=\psi _ \beta (\phi _ \beta (\lambda ))$ is given by

$$\begin{aligned} g_\beta (\lambda )= \frac{1}{2} \beta ^2 b^2 \phi _\beta (\lambda )^{2b-2} + \beta \phi _\beta (\lambda )^b, \end{aligned}$$

(6.9)

and $\phi _\beta$ is characterized by (6.5).

Proof

It suffices to show that for any $\lambda >0$,

$$\begin{aligned} \limsup _{t\rightarrow \infty }t^{-a}\log \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}}\mathbb {E}u^n(t,x)\le n\left( \frac{n-1}{2}\right) ^{\frac{2}{2- \alpha }}\mathcal {E}(\alpha _0,\gamma )-n \psi _ \beta (\phi _ \beta (\lambda )). \end{aligned}$$

We write

$$\begin{aligned} \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}}\mathbb {E}u^n(t,x) \le \left( \sup _{y\in \mathbb {R}^\ell }\mathbb {E}\left( \frac{u(t,y)}{p_t*u_0(y)}\right) ^n \right) \left( \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}}p_t*u_0(x) \right) ^n\,. \end{aligned}$$

Together with Corollary 5.4, it suffices to show the inequality

$$\begin{aligned} \limsup _{t\rightarrow \infty }t^{-a} \log \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}}p_t*u_0(x) \le -\psi _ \beta (\phi _ \beta (\lambda )) \,. \end{aligned}$$

(6.10)

We observe that by the triangle inequality,

$$\begin{aligned} \frac{1}{2t}|y-x|^2+\beta |y|^b\ge t^a f_{\beta ,|x|t^{-\frac{a+1}{2}}}(|y|t^{-\frac{a+1}{2}})\,. \end{aligned}$$

Hence, together with (6.7), we see that for every $|x|\ge \lambda t^{\frac{a+1}{2}}$

$$\begin{aligned} \frac{1}{2t}|y-x|^2+\beta |y|^b\ge t^a \psi _ \beta (\phi _ \beta (\lambda ))\,. \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}}\int _{\mathbb {R}^\ell }e^{-\frac{1}{2t}|y-x|^2}u_0(y)dy\le e^{-t^a \psi _ \beta (\phi _ \beta (\lambda ))}\int _{\mathbb {R}^\ell }e^{\beta |y|^b}u_0(y)dy\,. \end{aligned}$$

which implies (6.10). $\square$

As $\beta$ tends to infinity, $\phi _\beta (\lambda )$ tends to zero and it behaves as $\left( \lambda / b\beta \right) ^{\frac{1}{b-1}}$. Therefore, $g_\beta (\lambda )$ behaves as $\frac{1}{2} \lambda ^2$. These facts lead to the following result.

Corollary 6.2

Under the assumptions of Theorem  6.1, if $u_0$ satisfies $\int _{\mathbb {R}^\ell } e^{\beta |y|^b}u_0(y)dy<\infty$ for all $\beta >0$, then

$$\begin{aligned} \lambda ^*(n) \le \sqrt{ 2 \left( \frac{n-1}{2} \right) ^{ \frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma ) }. \end{aligned}$$

6.2 Lower bound for $\lambda _*(n)$

The main result is the following.

Theorem 6.3

Assume conditions (H.2) and (S). Suppose that $u_0$ is non-trivial and non-negative. In addition, we assume that

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t^a}\left| \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}} \log p_t*u_0(x) \right| <\infty \end{aligned}$$

for any $\lambda >0$. Then,

$$\begin{aligned} \lambda _*(n) \ge a^{\frac{a}{2}} (a+1) ^{-\frac{a+1}{2}} \sqrt{ 2 \left( \frac{n-1}{2} \right) ^{ \frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma ) }. \end{aligned}$$

Proof

Set

$$\begin{aligned} I(t):= \frac{1}{t^a} \sup _{|x| \ge \lambda t^{\frac{ a+1}{2}}} \log \mathbb {E}u^n (t,x) . \end{aligned}$$

To derive a lower bound for $I_t$ we proceed as follows. We will make use of the notation

$$\begin{aligned} Q_t\gamma (y):= \sum _{0\le j<k \le n} \int _{[0,t]^2} \gamma (B^j_{0,t}(s)- B^k_{0,t}(r) + \frac{s}{t} y^j -\frac{r}{t} y^k) {|s-r|^{-\alpha _0}} drds. \end{aligned}$$

Then, by the Feynman-Kac formula for the moments of the solution in terms of Brownian bridges proved in Proposition 4.3, we have

$$\begin{aligned} \mathbb {E}u^n(t,x) =\int _{(\mathbb {R}^\ell )^n}\mathbb {E}\prod _{j=1}^n u_0(x+y^j) p_t(y^j) \exp \left\{ Q_t\gamma (y) \right\} dy. \end{aligned}$$

(6.11)

For each $\varepsilon >0$, $p>1$, applying Hölder inequality, we see that

$$\begin{aligned} \mathbb {E}u^n(t,x)&\ge \left( \int _{(\mathbb {R}^\ell )^n}\mathbb {E}\prod _{j=1}^n u_0(x+y^j) p_t(y^j) \exp \left\{ \frac{1}{p} Q_t\gamma _\varepsilon (y) \right\} dy \right) ^p\nonumber \\&\quad \times \left( \int _{(\mathbb {R}^{\ell })^n}\prod _{j=1}^n u_0(x+y^j) p_t(y^j) \mathbb {E}\exp \left\{ \frac{1}{p-1} Q_t( \gamma _\varepsilon - \gamma )(y) \right\} dy\right) ^{1-p}. \end{aligned}$$

(6.12)

Notice that, from (4.5) we can write

$$\begin{aligned} \mathbb {E}\exp \left\{ \frac{1}{p-1} Q_t( \gamma _\varepsilon - \gamma )(y) \right\} \le \mathbb {E}\exp \left\{ \frac{1}{p-1} Q_t( \gamma - \gamma _\varepsilon )(0) \right\} . \end{aligned}$$

(6.13)

Substituting (6.13) into (6.12) yields

$$\begin{aligned} I(t)\ge & {} - \frac{p-1}{t^a} \log \mathbb {E}\exp \left\{ \frac{1}{p-1} Q_t( \gamma - \gamma _\varepsilon )(0) \right\} \nonumber \\&-n\frac{p-1}{t^a}\left| \sup _{|x|\ge \lambda t^\frac{a+1}{2}} \log (p_t*u_0(x))\right| \nonumber \\&+ \frac{p}{t^a} \sup _{|x|\ge \lambda t^\frac{a+1}{2}} \log \int _{(\mathbb {R}^\ell )^n} \prod _{j=1}^n u_0(x+y^j) p_t(y^j) \mathbb {E}\exp \left\{ \frac{1}{p}Q_t\gamma _\varepsilon (y) \right\} dy\nonumber \\:= & {} I_1(t)+I_2(t) + I_3(t). \end{aligned}$$

(6.14)

Choosing $\varepsilon =\varepsilon (t)=\delta t^{1-a}$ with $\delta >0$, from (5.24) we obtain

$$\begin{aligned} \lim _{\delta \rightarrow 0} \limsup _{t\rightarrow \infty } \frac{1}{t^a} \log \mathbb {E}\exp \left\{ \frac{q}{p} Q_t(\gamma - \gamma _{\delta t^{1-a}})(0) \right\} \le 0. \end{aligned}$$

In addition, from our assumption,

$$\begin{aligned} \lim _{p\rightarrow 1}\limsup _{t\rightarrow \infty }I_2(t)=0\,. \end{aligned}$$

In other words, $I_1(t)$ and $I_2(t)$ are negligible in the limits $t\rightarrow \infty$, $\delta \rightarrow 0$ and $p\rightarrow 1$.

We now consider $I_3$. It can be written as

$$\begin{aligned} I_3(t)= \frac{p}{t^a} \sup _{|x|\ge \lambda t^{\frac{a+1}{2}}} \log \mathbb {E}(u_{\varepsilon ,p}^n(t,x)), \end{aligned}$$

where $u_{\varepsilon ,p}(t,x)$ denotes the solution of Eq. (1.1) with initial condition $u_0$ and spatial covariance $\frac{1}{p}\gamma _{\varepsilon (t)}$, where $\varepsilon =\varepsilon (t) = \delta t^{1-a}$. Define $\mathcal {H}_{p,\varepsilon }$ as in (2.1), but with $\mu (d\xi )$ replaced by $\frac{1}{p}e^{-\varepsilon |\xi |^2}\mu (d\xi )$.

For every $\phi$ in $\mathcal {H}_{p,\varepsilon }$, we denote by $Z(\phi )$ the (Wick) exponential functional

$$\begin{aligned} Z(\phi )=\exp \left\{ W(\phi )-\frac{1}{2}\Vert \phi \Vert _{\mathcal {H}_{p,\varepsilon }}^2 \right\} \,. \end{aligned}$$

By the Feynman-Kac formula for the solution of Eq. (1.1), when the spatial covariance is bounded, we obtain

$$\begin{aligned} u_{\varepsilon ,p}(t,x) =\mathbb {E}_B \int _{\mathbb {R}^\ell } u_0 (x+y) p_t(y) Z(\psi _{x,y}) dy, \end{aligned}$$

where $\psi _{x,y}(s,z)= \delta (B_{0,t} (t-s) +x + \frac{t-s}{t} y -z)\mathbf{1}_{[0,t]}(s)$ and

$$\begin{aligned} \Vert \psi _{x,y}\Vert ^2_{{\mathcal {H}_{p,\varepsilon }}} =\int _{[0,t]^2} { \frac{1}{p}} \gamma _\varepsilon \left( B_{0,t}(s) -B_{0,t}(r) +\frac{s-r}{t} y\right) |s-r|^{-\alpha _0} dsdr . \end{aligned}$$

Let q be such that $\frac{1}{n}+\frac{1}{q}=1$. Using Hölder inequality, for any $\phi \in \mathcal {H}_{p,\varepsilon }$ we have

$$\begin{aligned} \mathbb {E}(u_{\varepsilon ,p}^n(t,x))= & {} \mathbb {E}_W \left( \mathbb {E}_B \int _{\mathbb {R}^\ell } u_0(x+y) p_t(y) Z(\psi _{x,y}) dy \right) ^n\nonumber \\\ge & {} \Vert Z(\phi )\Vert _{L^q(\Omega )}^{-n} \left( \mathbb {E}_W \left( Z(\phi ) \mathbb {E}_{B} \int _{\mathbb {R}^\ell } u_0(x+y) p_t(y) Z(\psi _{x,y}) dy \right) \right) ^n\nonumber \\= & {} \exp \left\{ -\frac{n}{2(n-1)}\Vert \phi \Vert ^2_{\mathcal {H}_{p,\varepsilon }} \right\} \nonumber \\&\times \left( \int _{\mathbb {R}^\ell } u_0(x+y) p_t(y) \mathbb {E}_B [\exp \{ \langle \phi , \psi _{x,y} \rangle _{\mathcal {H}_{p,\varepsilon }} \} ] dy \right) ^n. \end{aligned}$$

(6.15)

We are going to choose an element $\phi$, which depends on t and x.

Our next step is the computation of the inner product $\langle \phi , \psi _{x,y} \rangle _{\mathcal {H}_{p,\varepsilon }}$. We can write

$$\begin{aligned} \langle \phi , \psi _{x,y} \rangle _{\mathcal {H}_{p,\varepsilon }}= & {} \frac{1}{p}\int _0^t \int _0^t |s-r|^{-\alpha _0} \int _{\mathbb {R}^\ell } \phi (r,z) \gamma _\varepsilon (B_{0,t}(t-s) +x \\&+\frac{t-s}{t} y-z)dzdsdr\\= & {} \frac{1}{p} \int _0^t \int _0^t |s-r|^{-\alpha _0} \int _{\mathbb {R}^\ell } \phi (t-r,z+x) \gamma _\varepsilon (B_{0,t}(s) \\&+\frac{s}{t} y-z)dzdsdr. \end{aligned}$$

Set

$$\begin{aligned} t_n=c t^a, \end{aligned}$$

where $a=\frac{4-\alpha -2\alpha _0}{2-\alpha }$ and $c=(n-1)^{\frac{2}{2-\alpha }}$. Making the change of variables $s \rightarrow \frac{t}{t_n} s$ and $r \rightarrow tr$ and using the scaling property for Brownian bridge, we obtain that

$$\begin{aligned} \langle \phi , \psi _{x,y} \rangle _{\mathcal {H}_{p,\varepsilon }}= & {} \frac{1}{p} c^{\frac{\alpha }{2}-1} \int _0^1 \int _0^{t_n} \left| \frac{s}{t_n}-r\right| ^{-\alpha _0} \int _{\mathbb {R}^\ell } \phi (t-rt,z+x) \\&\times \gamma _{\varepsilon \frac{t_n}{t}}\left( B_{0,t_n}(s) +\frac{s}{\sqrt{t_nt}} y- \sqrt{\frac{t_n}{t}}z\right) dzdsdr. \end{aligned}$$

Finally, the change of variables $z\rightarrow \sqrt{\frac{t}{t_n}}z$ yields

$$\begin{aligned} \langle \phi , \psi _{x,y} \rangle _{\mathcal {H}_{p,\varepsilon }}= & {} \frac{1}{p} \left( \frac{t}{t_n} \right) ^{\frac{\ell }{2}} c^{\frac{\alpha }{2}-1} \int _0^1 \int _0^{t_n} \left| \frac{s}{t_n}-r\right| ^{-\alpha _0} \int _{\mathbb {R}^\ell } \phi (t-rt, \sqrt{\frac{t}{t_n}} z+x) \\&\times \gamma _{ c\delta }\left( B_{0,t_n}(s) +\frac{s}{\sqrt{t_nt}} y- z\right) dzdsdr. \end{aligned}$$

Choosing $\phi$ of the form

$$\begin{aligned} \phi (r,z)= \left( \frac{t_n}{t} \right) ^{\frac{\ell }{2}} c^{1-\frac{\alpha }{2}} \widehat{\phi } \left( \frac{t-r}{t}, \sqrt{\frac{t_n}{t}}(z - x)\right) \mathbf {1}_{[0,t]}(r), \end{aligned}$$

where $\widehat{\phi }$ satisfies

$$\begin{aligned} \sup _{r\in [0,1]} \int _{\mathbb {R}^\ell } |\widehat{\phi }(r,y)|dy <\infty , \end{aligned}$$

(6.16)

we can write

$$\begin{aligned} \langle \phi , \psi _{x,y}\rangle _{\mathcal {H}_{p,\varepsilon }}= & {} \frac{1}{p} \int _0^1 \int _0^{t_n} \left| \frac{s}{t_n}-r\right| ^{-\alpha _0}\\&\int _{\mathbb {R}^\ell } \widehat{\phi }(r, z) \gamma _{ c\delta }\left( B_{0,t_n}(s) +\frac{s}{\sqrt{t_nt}} y- z\right) dzdsdr. \end{aligned}$$

Set

$$\begin{aligned} f(s, w) = \frac{1}{p}\int _0^1 \int _{\mathbb {R}^\ell } \frac{\widehat{\phi }(r,z) \gamma _{c\delta } (w-z)}{| s -r |^{\alpha _0}} dzdr. \end{aligned}$$

Then, we obtain

$$\begin{aligned} \langle \phi , \psi _{x,y}\rangle _{\mathcal {H}_{p,\varepsilon }}=\int _0^{t_n} f\left( \frac{s}{t_n}, B_{0,t_n}(s) +\frac{s}{\sqrt{t_nt}} y\right) ds. \end{aligned}$$

(6.17)

On the other hand, for this choice of $\phi$, we obtain

$$\begin{aligned} \Vert \phi \Vert _{\mathcal {H}_{p,\varepsilon }} ^2= & {} \frac{1}{p}\left( \frac{t_n}{t} \right) ^{\ell } c^{2-\alpha } \int _0^t \int _0^t |s-r|^{-\alpha _0} \\&\times \int _{(\mathbb {R}^\ell )^2} \widehat{\phi } \left( \frac{t-r}{t}, \sqrt{\frac{t_n}{t}} (z- x)\right) \widehat{\phi } \left( \frac{t-s}{t}, \sqrt{\frac{t_n}{t}}{w - x}\right) \\&\gamma _{\varepsilon } (z-w) dzdwdrds. \end{aligned}$$

The change of variables $s\rightarrow t-ts$, $r\rightarrow t-tr$, $z\rightarrow \sqrt{\frac{t}{t_n }} z + x$ and $w\rightarrow \sqrt{\frac{t}{t_n }} w + x$ leads to

$$\begin{aligned} \Vert \phi \Vert _{\mathcal {H}_{p,\varepsilon }} ^2 = \frac{1}{p} t^a c^{2-\frac{\alpha }{2}} \int _0^1 \int _0^1 \int _{(\mathbb {R}^\ell )^2} \frac{ \widehat{\phi } (r,z) \widehat{\phi } (s,w)}{ |s-r|^{-\alpha _0} } \gamma _{c\delta } (z-w) dzdwdrds. \end{aligned}$$

(6.18)

Substituting (6.17) and (6.18) into (6.15), we get

$$\begin{aligned} \frac{p}{t^a} \log \mathbb {E}(u_{\varepsilon ,p}^n(t,x))\ge & {} -\frac{n}{2(n-1)} c^{2-\frac{\alpha }{2}} \int _0^1 \int _0^1 \int _{(\mathbb {R}^\ell )^2} \frac{ \widehat{\phi } (r,z) \widehat{\phi } (s,w)}{|s-r|^{\alpha _0}} \gamma _{c\delta } (z-w) dzdwdrds \\&+\frac{np}{t^a} \log \int _{\mathbb {R}^\ell } u_0 (x+y) p_t(y) \mathbb {E}_B \exp \left\{ \int _0^{t_n} f(\frac{s}{t_n}, B_{0,t_n}(s) +\frac{s}{\sqrt{t_nt}}y)ds \right\} dy. \end{aligned}$$

This together with (6.14) leads to the inequality, for any $K>0$,

$$\begin{aligned} I_3(t) \ge I_{3,1} + I_{3,2}(t) + I_{3,3}(t) , \end{aligned}$$

where

$$\begin{aligned} I_{3,1}= & {} -\frac{n}{2(n-1)} c^{2-\frac{\alpha }{2}} \int _0^1 \int _0^1 \int _{(\mathbb {R}^\ell )^2} \frac{ \widehat{\phi } (r,z) \widehat{\phi } (s,w)}{|s-r|^{\alpha _0}} \gamma _{c\delta } (z-w) dzdwdrds,\\ I_{3,2}({t,x})= & {} \frac{np}{t^a} \log \int _{|y| \le K\sqrt{tt_n}} u_0(x+y) p_t(y)dy \end{aligned}$$

and

$$\begin{aligned} I_{3,3}(t)=\frac{np}{t^a} \inf _{|y| \le Kt_n} \log \mathbb {E}_B \exp \left\{ \int _0^{t_n} f\left( \frac{s}{t_n}, B_{0,t_n}(s) +\frac{s}{t_n}y\right) ds\right\} . \end{aligned}$$

We are going to analyze these three terms and this will be done in several steps.

Step 1. Using the properties of the initial condition, we claim that if $\lambda < K\sqrt{c}$, then

$$\begin{aligned} \liminf _{t\rightarrow \infty } I_{3,2}(t) \ge -\frac{np}{2} K^2c. \end{aligned}$$

(6.19)

Notice first that $\sqrt{tt_n}= \sqrt{c} t^{\frac{a+1}{2}}$. Recall that $u_0$ is non-trivial, there exists $M>0$ such that $\int _{|y|\le M}u_0(y)dy>0$. For t large enough, $\lambda t^{\frac{ a+1}{2}} +M \le K\sqrt{c} t^{\frac{a+1}{2}}$. Therefore, choosing $x_0$ such that $|x_0| =\lambda t^{\frac{ a+1}{2}}$ implies that

$$\begin{aligned} \{y: |x_0+ y|\le M \} \subset \{y: |y|\le K\sqrt{c} t^{\frac{a+1}{2}} \}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \liminf _{t\rightarrow \infty } I_{3,2}(t) \ge \liminf _{t\rightarrow \infty } \frac{np}{t^a} \log \int _{|x_0 +y| \le M} e^{-\frac{K^2 c t^a}{2}} u_0(x_0+y) dy=-\frac{np}{2} K ^2c, \end{aligned}$$

which is (6.19).

Step 2. We can write

$$\begin{aligned} \liminf _{t\rightarrow \infty } I_{3,3}(t) =\liminf _{t\rightarrow \infty } \frac{npc}{t} \inf _{|y| \le Kt } \log \mathbb {E}_B \exp \left\{ \int _0^{t} f\left( \frac{s}{t}, B_{0,t}(s) +\frac{s}{t} y\right) ds\right\} \end{aligned}$$

For any $\rho \in (0,1)$, we can write

$$\begin{aligned} \mathbb {E}_B \exp \left\{ \int _0^{t} f\left( \frac{s}{t}, B_{0,t}(s) +\frac{s}{t} y\right) ds\right\} \ge \mathbb {E}_B \exp \left\{ \int _0^{\rho t} f\left( \frac{s}{t}, B_{0,t}(s) +\frac{s}{t} y\right) ds\right\} . \end{aligned}$$

From (2.8), we get

$$\begin{aligned}&\mathbb {E}_B \exp \left\{ \int _0^{\rho t} f\left( \frac{s}{t}, B_{0,t}(s) +\frac{s}{t}y\right) ds\right\} \nonumber \\&\quad \ge (1-\rho )^{-\frac{\ell }{2}} \mathbb {E}_B \left( \mathbf{1}_{ A_R} \exp \left\{ \int _0^{\rho t} f\left( \frac{s}{t}, B(s)\right) ds +\frac{|y|^2}{2t} -\frac{|y-B(\rho t)|^2}{2t(1-\rho )} \right\} \right) ,\nonumber \\ \end{aligned}$$

(6.20)

where $A_R =\{ \sup _{0\le s\le \rho t} |B(s)| \le R\}$ for $R >0$. Notice that, if $|y| \le Kt$, on the set $A_R$ we have

$$\begin{aligned} \frac{|y|^2}{2t} -\frac{|y-B(\rho t)|^2}{2t(1-\rho )} \ge - \frac{\rho }{1-\rho } \frac{K^2}{2} t -\frac{KR}{(1-\rho )} -\frac{R^2}{2t(1-\rho )}. \end{aligned}$$

(6.21)

On the other hand, by Proposition 3.1 of [5] we obtain

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{1}{t} \log \mathbb {E}_B \left( 1_{ A_R} \exp \left\{ \int _0^{\rho t} f\left( \frac{s}{t}, B(s)\right) ds \right\} \right) =\rho \int _0^1 \Lambda _R(f(\rho s, \cdot ))ds, \end{aligned}$$

where

$$\begin{aligned} \Lambda _R(f(\rho s, \cdot )) =\sup _{g\in \mathcal {F}_\ell (B_R)} \left\{ \int _{B_R} f(\rho s,x) g^2(x) dx -\frac{1}{2} \int _{B_R} | \nabla g(x)|^2dx\right\} , \end{aligned}$$

and $\mathcal {F}_\ell (B_R)$ is the set of smooth functions on $B_R:=\{ x: |x| \le R\}$ with $\Vert g\Vert _{L^2(B_R)}=1$ and $g(\partial B_R)=\{0\}$. For this result we need that for each $0\le s\le 1$, the function $f(\rho s, \cdot )$ is bounded and continuous and the family of functions $\{s\rightarrow f(\rho s, x), x\in \mathbb {R}^\ell \}$ is equicontinuous in [0, 1]. These properties are a consequence of assumption (6.16). In conclusion, we have proved that

$$\begin{aligned} \liminf _{t\rightarrow \infty } I_{3,3}(t) \ge -npc \frac{\rho }{1-\rho } \frac{K^2}{2}+cnp \rho \int _0^1 \Lambda _R(f(\rho s, \cdot ))ds. \end{aligned}$$

(6.22)

From (6.22), (6.21) and (6.19), letting $K\downarrow \lambda /\sqrt{c}$ and $R\uparrow \infty$, we obtain

$$\begin{aligned}&\liminf _{t\rightarrow \infty } (I_{3,2}(t)+ I_{3,3}(t) ) \ge -\frac{np}{2(1-\rho )} \lambda ^2 \nonumber \\&\quad + nc \rho p \left( \int _0^1 \int _{\mathbb {R}^\ell } f(s\rho , x) g^2(s,x) dxds - \frac{1}{2} \int _0^1 \int _{\mathbb {R}^\ell } |\nabla g(s,x)|^2 dxds \right) ,\qquad \qquad \end{aligned}$$

(6.23)

for any function g(s, x) in $\mathcal {A}_\ell$, where $\mathcal {A}_\ell$ has been defined in (5.2). We can write

$$\begin{aligned} \int _0^1 \int _{\mathbb {R}^\ell } f(s\rho , x) g^2(s,x) dsdx =\frac{1}{p}\int _0^1 \int _0^1 \int _{\mathbb {R}^{2\ell }} \frac{ \widehat{\phi } (r,y) g^2(s,x) }{|\rho s -r|^{\alpha _0} } \gamma _{c\delta } (x-y) dx dy dsdr. \end{aligned}$$

Making the change of variables $s\rho \rightarrow s$, yields

$$\begin{aligned} \int _0^1 \int _{\mathbb {R}^\ell } f(s\rho , x) g^2(s,x) dsdx= & {} \frac{1}{p\rho }\int _0^1 \int _0^\rho \int _{\mathbb {R}^{2\ell }} \frac{\widehat{\phi } (r,y) g^2( s / \rho ,x) }{|s -r|^{\alpha _0} }\\&\gamma _{c\delta } (x-y) dx dy dsdr. \end{aligned}$$

Now choose the function $\widehat{\phi }$ of the form $\widehat{\phi } (r,x)= g^2(\frac{r}{\rho }, x) \mathbf{1}_{[0,\rho ]} (r)$. With this choice we obtain

$$\begin{aligned} \int _0^1 \int _{\mathbb {R}^\ell } f(s\rho , x) g^2(s,x) dsdx\ge & {} \frac{1}{p\rho }\int _0^\rho \int _0^\rho \int _{\mathbb {R}^{2\ell }} \frac{ g^2 (r / \rho , y) g^2( s/ \rho ,x) }{|s -r|^{\alpha _0} } \gamma _{c\delta } (x-y) dx dy dsdr \\= & {} \frac{1}{p}\rho ^{1-\alpha _0} \int _0^1 \int _0^1 \int _{\mathbb {R}^{2\ell }} \frac{ g^2 (r, y) g^2( s ,x) }{|s -r|^{\alpha _0} } \gamma _{c\delta } (x-y) dx dy dsdr. \end{aligned}$$

Step 3. With the above choice for $\widehat{\phi }$ and letting $p\rightarrow 1$, the term $I_{3,1}$ can be written as

$$\begin{aligned} I_{3,1}= & {} -\frac{n}{2(n-1)} c^{2-\frac{\alpha }{2}} \int _0^\rho \int _0^\rho \int _{\mathbb {R}^{2\ell }} \frac{g^2 ( r/ \rho , y) g^2(s/ \rho ,x) }{ |s-r|^{\alpha _0} } \gamma _{c\delta } (z-x) dxdydrds\nonumber \\= & {} -\frac{nc}{2} \rho ^{2-\alpha _0} \int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }} \frac{g^2 ( r, y) g^2(s ,x)}{ |s-r|^{\alpha _0} } \gamma _{c\delta } (z-x) dxdydrds. \end{aligned}$$

(6.24)

Finally, from (6.23) and (6.24), we obtain

$$\begin{aligned} \lim _{p \downarrow 1}\liminf _{t\rightarrow \infty } I_3(t)\ge & {} -\frac{n}{2(1-\rho )} \lambda ^2 \\&+\,nc \rho \left( \frac{\rho ^{1-\alpha _0}}{2}\int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }} \frac{ g^2 (r, y) g^2( s ,x) }{|s -r|^{\alpha _0} }\gamma _{c\delta } (x-y)dxdydrds\right. \\&\left. - \frac{1}{2} \int _0^1 \int _{\mathbb {R}^\ell } |\nabla g(s,x)|^2 dxds \right) , \end{aligned}$$

Letting $\delta \downarrow 0$, we obtain

$$\begin{aligned} \lim _{\delta \downarrow 0,p \downarrow 1}\liminf _{t\rightarrow \infty } I_3(t)\ge & {} -\frac{n}{2(1-\rho )} \lambda ^2 \\&+\, nc \rho \left( \frac{\rho ^{1-\alpha _0}}{2}\int _{[0,1]^2} \int _{\mathbb {R}^{2\ell }} \frac{ g^2 (r, y) g^2( s ,x) }{|s -r|^{\alpha _0} }\gamma (x-y)dxdydrds\right. \\&\left. - \frac{1}{2} \int _0^1 \int _{\mathbb {R}^\ell } |\nabla g(s,x)|^2 dxds \right) , \end{aligned}$$

Now we write $\widehat{g}(r,x)= \sqrt{\varkappa } g(r,\varkappa x)$ where $\varkappa$ is a constant whose value will be determined very soon, and we obtain, using the scaling properties of $\gamma$,

$$\begin{aligned}&\frac{\rho ^{1-\alpha _0}}{2} \int _{[0,1]^2}\int _{\mathbb {R}^{2\ell }} \frac{ \widehat{ g}^2(r,y) \widehat{g}^2(s,x)}{|s-r|^{\alpha _0}} \gamma (x-y)dxdydrds-\frac{1}{2} \int _0^1 \Vert \nabla \hat{g}(s,\cdot )\Vert ^2_{L^2(\mathbb {R}^\ell )}ds \\&\quad =\frac{\varkappa ^\alpha \rho ^{1-\alpha _0} }{2} \int _{[0,1]^2}\int _{\mathbb {R}^{2\ell }} \frac{ g^2(r,y) g^2(s,x)}{|s-r|^{\alpha _0}} \gamma (x-y)dxdydrds\\&\quad -\frac{\varkappa ^2}{2} \int _0^1 \Vert \nabla g(s,\cdot )\Vert ^2_{L^2(\mathbb {R}^\ell )}ds \end{aligned}$$

Finally, choosing $\varkappa =2^{ \frac{1}{\alpha -2}} \rho ^{\frac{1-\alpha _0}{2-\alpha }}$ and taking the supremum over g, we obtain

$$\begin{aligned} \liminf _{t\rightarrow \infty } I_3(t) \ge -\frac{n}{2(1-\rho )} \lambda ^2 + n \rho ^{a} \left( \frac{n-1}{2} \right) ^{\frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma ). \end{aligned}$$

Optimizing in $\rho$, this produces the lower bound

$$\begin{aligned} \lambda _*(n) \ge a^{\frac{a}{2}} (a+1) ^{-\frac{a+1}{2}} \sqrt{2 \left( \frac{n-1}{2} \right) ^{\frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )}. \end{aligned}$$

The proof is now complete. $\square$

Remark 6.4

Putting together the results from Corollary 6.2 and Theorem 6.3 we obtain, for a nontrivial $u_0$ with compact support and assuming a covariance satisfying conditions (H.2), (S) and the absolute continuity of $\mu$,

$$\begin{aligned}&a^{\frac{a}{2}} (a+1) ^{-\frac{a+1}{2}} \sqrt{2 \left( \frac{n-1}{2} \right) ^{\frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )}\le \lambda _*(n) \le \lambda ^*(n) \\&\quad \le \sqrt{2 \left( \frac{n-1}{2} \right) ^{\frac{2}{2-\alpha }} {\mathcal {E}}(\alpha _0,\gamma )}. \end{aligned}$$

Notice that when $\alpha _0 \uparrow 1$ the constant a converges to 1 and the above factor converges to $\frac{1}{2}$. In this sense, in comparison with (1.9), our result is not optimal. We conjecture that the constant in the left-hand side should be 1, but our techniques do not allow to show this.