Spatial asymptotic of the stochastic heat equation with compactly supported initial data

TL;DR

This paper studies the extreme growth behavior of solutions to the stochastic heat equation with localized initial data, revealing how the peaks evolve over large spatial scales under spatially correlated noise.

Contribution

It extends previous work by analyzing the spatial asymptotics of the stochastic heat equation with compactly supported initial data, including Dirac masses, under correlated noise.

Findings

Growth of peaks characterized over large radii

Results relate to previous constant initial data studies

Includes cases with Dirac delta initial conditions

Abstract

We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang's condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of Conus, Joseph and Khoshnevisan 2013 and X. Chen 2016, where constant initial data are considered.

Equations

$$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\dot{W},u(0,\cdot )=u_0(\cdot )$$

$$\mathbb{E}[W(\varphi )W(\psi )]=\frac{1}{(2\pi {)}^{\ell }}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^{\ell }}\mathcal{F}\varphi (s,\xi )\overline{\mathcal{F}\psi (s,\xi )}\mu (\xi )d\xi ds,$$

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

$$\mathbb{E}[W(\varphi )W(\psi )]={\int }_0^{\infty }{\iint }_{{\mathbb{R}}^{2\ell }}\varphi (s,x)\psi (s,y)\gamma (x-y)dxdyds.$$

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

$$c\le u_0(x)\le C,\text{mbox}forsomepositivenumbersc,C.$$

$$c_1\le \underset{R\to \infty }{\mathrm{limsup}}(\log R{)}^{-\frac{2}{3}}\log \underset{|x|\le R}{\sup }u(t,x)\le c_2.$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{3}}\log \underset{|x|\le R}{\sup }u(t,x)=\frac{3}{4}{\left(\frac{2t}{3}\right)}^{\frac{1}{3}}a.s.$$

$$\underset{m\to \infty }{\lim }{m}^{-3}\log \mathbb{E}u(t,x{)}^m=\frac{t}{24}.$$

$$\underset{t\to \infty }{\lim }\frac{1}{t}\log \mathbb{E}u(t,x{)}^m={\mathcal{E}}_m$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{1}{2}}\log \underset{|x|\le R}{\sup }u(t,x)=\sqrt{2\ell \gamma (0)t}a.s.,$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{4-\alpha }}\underset{|x|\le R}{\sup }\log u(t,x)=\frac{4-\alpha }{2}{\ell }^{\frac{2}{4-\alpha }}{\left(\frac{{\mathcal{E}}_H(\gamma )}{2-\alpha }t\right)}^{\frac{2-\alpha }{4-\alpha }}a.s.,$$

$$p_t\ast |u_0|(x)<\infty \forall t>0,x\in {\mathbb{R}}^{\ell }.$$

$$\underset{t\to \infty }{\lim }\frac{1}{t}\log \underset{x\in \mathbb{R}}{\sup }\mathbb{E}{\left(\frac{u(t,x)}{p_t\ast u_0(x)}\right)}^m={\mathcal{E}}_m,$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{1}{2}}\underset{|x|\le R}{\sup }\left(\log u(t,x)-\log p_t\ast u_0(x)\right)=\sqrt{2\ell \gamma (0)t}a.s.$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{4-\alpha }}\underset{|x|\le R}{\sup }\left(\log u(t,x)-\log p_t\ast u_0(x)\right)=\frac{4-\alpha }{2}{\ell }^{\frac{2}{4-\alpha }}{\left(\frac{{\mathcal{E}}_H(\gamma )}{2-\alpha }t\right)}^{\frac{2-\alpha }{4-\alpha }}a.s.$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{3}}\underset{|x|\le R}{\sup }(\log u(t,x)-\log p_t\ast u_0(x))=\frac{3}{4}{\left(\frac{2t}{3}\right)}^{\frac{1}{3}}a.s.$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{3}}\underset{|x|\le R}{\sup }\left(\log u(t,x)+\frac{(x-x_0{)}^2}{2t}\right)=\frac{3}{4}{\left(\frac{2t}{3}\right)}^{\frac{1}{3}}.$$

$$\bar{\alpha}=\left\{\begin{array}[]{ll}0&\mbox{if \ref{Cov:bounded} holds},\\ \alpha&\mbox{if \ref{Cov:Dalang} holds},\end{array}\right.\quad\mbox{and}\quad{\mathcal{E}}=\left\{\begin{array}[]{ll}\gamma(0)&\mbox{if \ref{Cov:bounded} holds},\\ {\mathcal{E}}_{H}(\gamma)&\mbox{if \ref{Cov:Dalang} holds},\end{array}\right.$$

$$\underset{R\to \infty }{\mathrm{limsup}}(\log R{)}^{-\frac{2}{4-\bar{\alpha }}}\underset{|x|\le R}{\sup }\left(\log u(t,x)-\log p_t\ast u_0(x)\right)\le \frac{4-\bar{\alpha }}{2}{\ell }^{\frac{2}{4-\bar{\alpha }}}{\left(\frac{\mathcal{E}}{2-\bar{\alpha }}t\right)}^{\frac{2-\bar{\alpha }}{4-\bar{\alpha }}}a.s.$$

$${\gamma }_{ϵ}(x)=(2\pi {)}^{-\ell }{\int }_{{\mathbb{R}}^{\ell }}{e}^{-2ϵ|\xi {|}^2}{e}^{i\xi \cdot x}\mu (\xi )d\xi ,$$

$$W_{ϵ}(\varphi )=W(p_{ϵ}\ast \varphi )$$

$$\mathbb{E}[W_{ϵ}(\varphi )W_{ϵ}(\psi )]$$

$$={\int }_0^{\infty }{\iint }_{{\mathbb{R}}^{2\ell }}\varphi (s,x)\psi (s,y){\gamma }_{ϵ}(x-y)dxdyds$$

$${\gamma }_{ϵ}(x)={ϵ}^{-\frac{\alpha }{2}}{\gamma }_1({ϵ}^{-\frac{1}{2}}x)\text{mbox}forallϵ>0,x\in {\mathbb{R}}^{\ell }.$$

$$\underset{R\to \infty }{\mathrm{limsup}}(\log R{)}^{-\frac{2}{4-\bar{\alpha }}}\underset{|x|\le R,ϵ\in (0,1)}{\sup }\left(\log u_{ϵ}(t,x)-\log p_t\ast u_0(x)\right)\le \frac{4-\bar{\alpha }}{2}{\ell }^{\frac{2}{4-\bar{\alpha }}}{\left(\frac{\mathcal{E}}{2-\bar{\alpha }}t\right)}^{\frac{2-\bar{\alpha }}{4-\bar{\alpha }}}a.s.$$

$$\underset{R\to \infty }{\lim }(\log R{)}^{-\frac{2}{4-\bar{\alpha }}}\underset{|x|\le R,ϵ\in (0,1)}{\sup }\left(\log u_{ϵ}(t,x)+\frac{(x-x_0{)}^2}{2t}\right)=\frac{4-\bar{\alpha }}{2}{\ell }^{\frac{2}{4-\bar{\alpha }}}{\left(\frac{\mathcal{E}}{2-\bar{\alpha }}t\right)}^{\frac{2-\bar{\alpha }}{4-\bar{\alpha }}}a.s.$$

$$\mathcal{F}g(\xi )={\int }_{{\mathbb{R}}^{\ell }}{e}^{-i\xi \cdot x}g(x)dx,$$

$${\int }_{{\mathbb{R}}^{\ell }}|f(x){|}^{2}dx=\frac{1}{(2\pi {)}^{\ell }}{\int }_{{\mathbb{R}}^{\ell }}|\mathcal{F}f(\xi ){|}^{2}d\xi .$$

$$⟨g,h{⟩}_{{\mathfrak{H}}_0}=\frac{1}{(2\pi {)}^{\ell }}{\int }_{{\mathbb{R}}^{\ell }}\mathcal{F}g(\xi )\overline{\mathcal{F}h(\xi )}\mu (\xi )d\xi .$$

$$⟨g,h{⟩}_{{\mathfrak{H}}_0}={\iint }_{{\mathbb{R}}^{\ell }\times {\mathbb{R}}^{\ell }}g(x)h(y)\gamma (x-y)dxdy.$$

Full text

This work is partially supported by NSF Grant no. 0932078000

1. Introduction

We consider the stochastic heat equation in $\mathbb{R}^{\ell}$

[TABLE]

where $t\geq 0$, $x\in\mathbb{R}^{\ell}$ $(\ell\geq 1)$ and $u_{0}$ is a Borel measure. Herein, $W$ is a centered Gaussian field, which is white in time and it has a correlated spatial covariance. More precisely, we assume that the noise $W$ is described by a centered Gaussian family $W=\{W(\phi),\phi\in C_{c}^{\infty}(\mathbb{R}_{+}\times\mathbb{R}^{\ell})\}$, with covariance

[TABLE]

where $\mu$ is non-negative measurable function and ${\mathcal{F}}$ denotes the Fourier transform in the spatial variables. To avoid trivial situations, we assume that $\mu$ is not identical to zero. The inverse Fourier transform of $\mu$ is in general a distribution defined formally by the expression

[TABLE]

If $\gamma$ is a locally integrable function, then it is non-negative definite and (1.2) can be written in Cartesian coordinates

[TABLE]

The following two distinct hypotheses on the spatial covariance of $W$ are considered throughout the paper.

1. (H.0)

$\mu$ is integrable, that is $\int_{\mathbb{R}^{\ell}}\mu(\xi)d\xi<\infty$. In this case, the inverse Fourier transform of $\mu(\xi)$ exists and is a bounded continuous function $\gamma$. Assume in addition that $\gamma$ is $\kappa$-Hölder continuous function at 0. 2. (H.0)

$\mu$ satisfies the following conditions:

1. (H.2a)

The inverse Fourier transform of $\mu(\xi)$ is either the Dirac delta mass at 0 or a nonnegative locally integrable function $\gamma$. 2. (H.2b)

[TABLE] 3. (H.2c)

(Scaling) There exists $\alpha\in(0,2)$ such that $\mu(c\xi)=c^{\alpha-\ell}\mu(\xi)$ for all positive numbers $c$.

Hereafter, 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 $\mathbb{R}^{\ell}$. Condition 2b is known as Dalang’s condition and is sufficient for existence and uniqueness of a random field solution. If $\gamma$ exists as a function, condition 2c induces the scaling relation $\gamma(cx)=c^{-\alpha}\gamma(x)$ for all $c>0$.

Equation (1.1) with noise satisfying condition 2 was introduced by Dalang in [Dal]. In [HLN15], for a large class of initial data, we show that equation (1.1) has a unique random field solution under the hypothesis 2. Under hypothesis 1, we note that $\gamma$ may be negative, but proceeding as in [Huang], a simple Picard iteration argument gives the existence and uniqueness of the solution. In addition, in both cases, the solution has finite moments of all positive orders. We give a few examples of covariance structures which are usually considered in literatures.

Example 1.1*.*

Covariance functions satisfying 2 includes the Riesz kernel $\gamma(x)=|x|^{-\eta}$, with $0<\eta<2\wedge\ell$, the space-time white noise in dimension one, where $\gamma=\delta_{0}$, the Dirac delta mass at 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 $\sum_{i=1}^{\ell}H_{i}>\ell-1$ and $H_{i}>\frac{1}{2}$ for $i=1,\dots,\ell$. Covariance functions satisfying 1 includes $e^{-|x|^{2}}$ and the inverse Fourier transform of $|\xi|^{2}e^{-|\xi|^{2}}$.

Suppose for the moment that $\dot{W}$ is a space-time white noise and $u_{0}$ is a function satisfying

[TABLE]

It is first noted in [MR3098071] that there exist positive constants $c_{1},c_{2}$ such that almost surely

[TABLE]

Later Xia Chen shows in [MR3474477] that indeed the precise almost sure limit can be computed, namely,

[TABLE]

One of the key ingredients in showing (1.8) is the following moment asymptotic result

[TABLE]

Thanks to the scaling property of the space-time white noise, Xia Chen has managed to derive (1.9) from the following long term asymptotic result

[TABLE]

where the constant ${\mathcal{E}}_{m}$ grows as $\frac{1}{24}m^{3}$ when $m\to\infty$.

Under condition (1.6), analogous results for other kinds of noises are also obtained in [MR3474477]. More precisely, for noises satisfying 1

[TABLE]

and for noises satisfying 2,

[TABLE]

where the variational quantity ${\mathcal{E}}_{H}(\gamma)$ is introduced in (3.39).

On the other hand, it is known that equation (1.1) has a unique random field solution under either 1 or 2 provided that $u_{0}$ satisfies

[TABLE]

Hence, condition (1.6) excludes other initial data of interests such as compactly supported measures. It is our purpose in the current paper to investigate the almost sure spatial asymptotic of the solutions corresponding to these initial data.

Upon reviewing the method in obtaining (1.8) described previously, one first seeks for an analogous result to (1.10) for general initial data. In fact, it is noted in [HLN15] that for every $u_{0}$ satisfying (1.13), one has

[TABLE]

where ${\mathcal{E}}_{m}$ is a constant whose asymptotic as $m\to\infty$ is known. It is suggestive from (1.14) that with a general initial datum, one should normalized $u(t,x)$ in (1.8) (and (1.9)) by the factor $p_{t}*u_{0}(x)$. Therefore, we anticipate the following almost sure spatial asymptotic result.

Conjecture 1.2**.**

Assume that $u_{0}$ satisfies (1.13). Under 1 we have

[TABLE]

Under 2, we have

[TABLE]

In the particular case of space-time white noise, we conjecture that

[TABLE]

In the case of space-time white noise, note that if $u_{0}$ satisfies the condition (1.6), (1.17) is no different than (1.8). On the other hand, if $u_{0}$ is a Dirac delta mass at $x_{0}$, (1.17) precisely describes the spatial asymptotic of $\log u(t,x)$: at large spatial sites, $\log u(t,x)$ is concentrated near a logarithmic perturbation of the parabola $-\frac{1}{2t}(x-x_{0})^{2}$. More precisely, (1.17) with this specific initial datum reduces to

[TABLE]

While a complete answer for Conjecture 1.2 (including (1.18)) is still undetermined, the current paper offers partial results, focusing on initial data with compact supports, especially Dirac masses. To unify the notation, we denote

[TABLE]

where the variational quantity ${\mathcal{E}}_{H}(\gamma)$ is introduced below in (3.39). For bounded covariance functions, we obtain the following result.

Theorem 1.3**.**

Assume that 1 holds and $u_{0}=\delta(\cdot-x_{0})$ for some $x_{0}\in\mathbb{R}^{\ell}$. Then (1.15) holds.

For noises satisfying 2, or for initial data with compact supports, the picture is less complete.

Theorem 1.4**.**

Assume that $u_{0}$ is a non-negative measure with compact support and either 1 or 2 holds. Then we have

[TABLE]

For initial data satisfying (1.6), the lower bound of (1.16) is proved in [MR3474477] using a localization argument initiated from [MR3098071]. In our situation, a technical difficulty arises in applying this localization procedure, which leads to the missing lower bound in Theorem 1.4. A detailed explanation is given at the beginning of Subsection 6.2. As an attempt to obtain the exact spatial asymptotics, we propose an alternative result which is described below. We need to introduce a few more notation. For each $\epsilon>0$, we denote

[TABLE]

which is a bounded non-negative definite function. Let $W_{\epsilon}$ be a centered Gaussian field defined by

[TABLE]

for all $\phi\in C_{c}^{\infty}(\mathbb{R}_{+}\times\mathbb{R}^{\ell})$. In the above, $p_{\epsilon}=(2\pi\epsilon)^{-\ell/2}e^{-|x|^{2}/(2\epsilon)}$ and $p_{\epsilon}*\phi$ is the convolution of $p_{\epsilon}$ with $\phi$ in the spatial variables. The covariance structure of $W_{\epsilon}$ is given by

[TABLE]

for all $\phi,\psi\in C_{c}^{\infty}(\mathbb{R}_{+}\times\mathbb{R}^{\ell})$. In other words, $W_{\epsilon}$ is white in time and correlated in space with spatial covariance function $\gamma_{\epsilon}$, which satisfies 1. Under condition 2c, $\gamma_{\epsilon}$ satisfies the scaling relation

[TABLE]

Let $u_{\epsilon}$ be the solution to equation (1.1) with $\dot{W}$ replaced by $\dot{W}_{\epsilon}$. It is expected that as $\epsilon\downarrow 0$, $u_{\epsilon}(t,x)$ converges to $u(t,x)$ in $L^{2}(\Omega)$ for each $(t,x)$, see [ChenHuang] for a proof when the initial data is a bounded function. The following result describes spatial asymptotic of the family of random fields $\{u_{\epsilon}\}_{\epsilon\in(0,1)}$.

Theorem 1.5**.**

Assume that $u_{0}$ is a non-negative measure with compact support and either 1 or 2 holds. Then

[TABLE]

If, in particular, $u_{0}=\delta(\cdot-x_{0})$ for some $x_{0}\in\mathbb{R}^{\ell}$, then

[TABLE]

Neither one of (1.16) and (1.26) is stronger than the other. While the result of Theorem 1.5 relates to the solution of (1.1) indirectly, it is certainly interesting. In Hairer’s theory of regularity structures (cf. [MR3274562]), one first regularizes the noise to obtain a sequence of approximated solutions. The solution of the corresponding stochastic partial differential equation is then constructed as the limiting object of this sequence. From this point of view, (1.26) provides a unified characteristic of the sequence of approximating solutions $\{u_{\epsilon}\}_{\epsilon\in(0,1)}$, which approaches the solution $u$ as $\epsilon\downarrow 0$. The proof of (1.26) does not rely on localization, rather, on the Gaussian nature of the noise. This leads to a possibility of extending (1.26) to temporal colored noises, which will be a topic for future research.

The remainder of the article is structured as follows: In Section 2 we briefly summarize the theory of stochastic integrations and well-posedness results for (1.1). In Section 3 we introduce some variational quantities which are related to the spatial asymptotics. In Section 4 we derive some Feynman-Kac formulas of the solution and its moments, these formulas play a crucial role in our consideration. In Section 5 we investigate the high moment asymptotics and Hölder regularity of the solutions of (1.1) with respect to various parameters. The results in Section 5 are used to obtain upper bounds in (1.15) and (1.16). This is presented in Section 6, where we also give a proof of the lower bounds in Theorems 1.3, 1.4 and 1.5.

2. Preliminaries

We introduce some notation and concepts which are used throughout the article. The space of Schwartz functions is denoted by ${\mathcal{S}}(\mathbb{R}^{\ell})$. The Fourier transform of a function $g\in{\mathcal{S}}(\mathbb{R}^{\ell})$ is defined with the normalization

[TABLE]

so that the inverse Fourier transform is given by ${\mathcal{F}}^{-1}g(\xi)=(2\pi)^{-\ell}{\mathcal{F}}g(-\xi)$. The Plancherel identity with this normalization reads

[TABLE]

Let us now describe stochastic integrations with respect to $W$. We can interpret $W$ as a Brownian motion with values in an infinite dimensional Hilbert space. In this context, the stochastic integration theory with respect to $W$ can be handled by classical theories (see for example, [DQ]). We briefly recall the main features of this theory.

We denote by $\mathfrak{H}_{0}$ the Hilbert space defined as the closure of $\mathcal{S}(\mathbb{R}^{\ell})$ under the inner product

[TABLE]

which can also be written as

[TABLE]

If $\gamma$ satisfies 1, then $\mathfrak{H}_{0}$ contains distributions such as Dirac delta masses. The Gaussian family $W$ can be extended to an isonormal Gaussian process $\{W(\phi),\phi\in L^{2}(\mathbb{R}_{+},\mathfrak{H}_{0})\}$ parametrized by the Hilbert space $\mathfrak{H}:=L^{2}(\mathbb{R}_{+},\mathfrak{H}_{0})$. For any $t\geq 0$, let $\mathcal{F}_{t}$ be the $\sigma$-algebra generated by $W$ up to time $t$. Let $\Lambda$ be the space of $\mathfrak{H}_{0}$-valued predictable processes $g$ such that $\mathbb{E}\|g\|_{\mathfrak{H}}^{2}<\infty$. Then, one can construct (cf. [HLN15]) the stochastic integral $\int_{0}^{\infty}\int_{\mathbb{R}^{\ell}}g(s,x)\,W(ds,dx)$ such that

[TABLE]

Stochastic integration over finite time interval can be defined easily

[TABLE]

Finally, the Burkholder’s inequality in this context reads

[TABLE]

which holds for all $p\geq 2$ and $g\in\Lambda$. A useful application of (2.30) is the following result

Lemma 2.1**.**

Let $m\geq 2$ be an integer, $f$ be a deterministic function on $[0,\infty)\times\mathbb{R}^{\ell}$ and $u=\{u(s,x):s\geq 0,x\in\mathbb{R}^{\ell}\}$ be a predictable random field such that

[TABLE]

Under hypothesis 2, we have

[TABLE]

and under hypothesis 1, we have

[TABLE]

Proof.

We consider only the hypothesis 2, the other case is obtained similarly. In view of Burkholder inequality (2.30) and Minkowski inequality, it suffices to show

[TABLE]

In fact, using (2.28) and Minkowski inequality, the left-hand side in the above is at most

[TABLE]

Note in addition that by Cauchy-Schwarz inequality,

[TABLE]

From here, (2.31) is transparent and the proof is complete. ∎

We now state the definition of the solution to equation (1.1) using the stochastic integral introduced previously.

Definition 2.2*.*

Let $u=\{u(t,x),t\geq 0,x\in\mathbb{R}^{\ell}\}$ be a real-valued predictable stochastic process such that for all $t\geq 0$ and $x\in\mathbb{R}^{\ell}$ the process $\{p_{t-s}(x-y)u(s,y)\mathbf{1}_{[0,t]}(s),0\leq s\leq t,y\in\mathbb{R}^{\ell}\}$ is an element of $\Lambda$. We say that $u$ is a mild solution of (1.1) if for all $t\in[0,T]$ and $x\in\mathbb{R}^{\ell}$ we have

[TABLE]

The following existence and uniqueness result has been proved in [HLN15] under hypothesis 2. Under hypothesis 1, one can proceed as in [Huang], using a simple Picard iteration argument to obtain the existence and uniqueness of the solution.

Theorem 2.3**.**

Suppose that $u_{0}$ satisfies (1.13) and the spectral measure $\mu$ satisfies hypotheses 1 or 2. Then there exists a unique solution to equation (1.1).

When $u_{0}=\delta(\cdot-z)$, we denote the corresponding unique solution by $\mathcal{Z}(z;t,x)$. In particular $\mathcal{Z}(z;\cdot,\cdot)$ is predictable and satisfies

[TABLE]

for all $t\geq 0$ and $x\in\mathbb{R}^{\ell}$.

Next, we record a Gronwall-type lemma which will be useful later.

Lemma 2.4**.**

Suppose $\alpha\in[0,2)$ and $f$ is a locally bounded function on $[0,\infty)$ such that

[TABLE]

where $A,B$ are positive constants and $g$ is non-decreasing function. Then there exists a constant $C_{\alpha}$ such that

[TABLE]

Proof.

Fix $T>0$. For each $\rho>0$, denote $D_{\rho}=\sup_{t\in[0,T]}f_{t}e^{-\rho t}$. It follows that

[TABLE]

It is easy to see

[TABLE]

for some suitable constant $C$ depending only on $\alpha$. We then choose $\rho=(2AC)^{\frac{2}{2-\alpha}}$ so that $AC\rho^{-\frac{2-\alpha}{2}}=\frac{1}{2}$. This leads to $D_{\rho}\leq 2Bg_{T}$, which implies the result. ∎

Let us conclude this section by introducing a few key notation which we will use throughout the article. Let $B=(B(t),t\geq 0)$ denote a standard Brownian motion in $\mathbb{R}^{\ell}$ starting at the origin. For each $t>0$, we denote

[TABLE]

The process $B_{0,t}=(B_{0,t}(s),0\leq s\leq t)$ is independent from $B(t)$ and is a Brownian bridge which starts and ends at the origin. An important connection between $B$ and $B_{0,t}$ is the following identity. For every $\lambda\in(0,1)$ and every bounded measurable function $F$ on $C([0,\lambda t];\mathbb{R}^{d})$ we have

[TABLE]

This is in fact an application of Girsanov’s theorem, see [HLN15]*Eq. (2.8) for more details. Let $B^{1},B^{2},\dots$ be independent copies of $B$ and $B^{1}_{0,t},B^{2}_{0,t},\dots$ be the corresponding Brownian bridges. An important quantity which appears frequently in our consideration is

[TABLE]

From the proof of Proposition 4.2 in [HLN15], it is easy to see that under one of the hypotheses 1 and 2, $\Theta_{t}(m)<\infty$ for any $t>0$. Finally, $A\lesssim E$ means $A\leq CE$ for some positive constant $C$, independent from all the terms appearing in $E$.

3. Variations

We introduce two variational quantities and give their basic properties and relations. The high moment asymptotic is governed by a variational quantity which is known as the Hartree energy (cf. [ChPh15]). If there exists a locally integrable function $\gamma$ whose Fourier transform is $\mu$, then the Hartree energy can be expressed as

[TABLE]

where ${\mathcal{G}}$ is the set

[TABLE]

The subscript $H$ stands for “Hartree”. We can also write this variation in Fourier mode. Indeed, the presentation (1.3) leads to

[TABLE]

Setting $h=(2\pi)^{-\frac{\ell}{2}}{\mathcal{F}}g$ so that $\|h\|_{L^{2}}=1$, we arrive at

[TABLE]

where

[TABLE]

Under 1, from (3.37), we upper bound $\gamma(x-y)$ by $\gamma(0)$, it follows that ${\mathcal{E}}_{H}(\gamma)\leq\gamma(0)$, which is finite. The fact that this variation (either in the form (3.37) or (3.39)) is finite under the condition 2 is not immediate. In some special cases, this is verified in [MR3414457] and [Chetal16].

Proposition 3.1**.**

Suppose (1.5) holds. Then ${\mathcal{E}}_{H}(\gamma)$ is finite.

Proof.

Our proof is based on the argument in [Chetal16]*Proposition 3.1. Here, however, we work on the frequency space and use the presentation (3.39). Let $h$ be in ${\mathcal{A}}$. Applying Cauchy-Schwarz inequality yields

[TABLE]

On the other hand, using the elementary inequality

[TABLE]

and Cauchy-Schwarz inequality, we also get

[TABLE]

Then, for every $R>0$ we have

[TABLE]

We now choose $R$ sufficiently large so that ${4}(2\pi)^{-\ell}\int_{|\xi|>R}\frac{\mu(\xi)}{|\xi|^{2}}d\xi<1$. This implies

[TABLE]

for all $g$ in ${\mathcal{A}}$, which finishes the proof. ∎

In establishing the lower bound of spatial asymptotic, another variation arises, which is given by

[TABLE]

or alternatively in frequency mode

[TABLE]

Under the scaling condition 2c, ${\mathcal{E}}_{H}$ and ${\mathcal{M}}$ are linked together by the following result.

Proposition 3.2**.**

Assuming condition 2c, ${\mathcal{E}}_{H}(\gamma)$ is finite if and only if ${\mathcal{M}}(\gamma)$ is finite. In addition,

[TABLE]

Before giving the proof, let us see how (3.37) and (3.40) are connected to a certain interpolation inequality. Under scaling condition 2c, it is a routine procedure in analysis to connect the finiteness of ${\mathcal{E}}_{H}(\gamma)$ with a certain interpolation inequality. For instance, when $\gamma=\delta$ and $\ell=1$, the fact that

[TABLE]

is equivalent to the following Gagliardo–Nirenberg inequality

[TABLE]

for all $g$ in $W^{1,2}(\mathbb{R})$. For readers convenience, we provide a brief explanation below.

Proposition 3.3**.**

Assume that the scaling relation 2c holds.

(i)* If ${\mathcal{E}}_{H}(\gamma)$ is finite then there exists $\kappa>0$ such that for all $g$ in $W^{1,2}(\mathbb{R}^{\ell})$*

[TABLE]

In addition the constant $\kappa$ can be chosen to be

[TABLE]

(ii)* If (3.42) holds for some finite constant $\kappa>0$, then ${\mathcal{E}}_{H}(\gamma)$ is finite and the best constant in (3.42) is $\kappa(\gamma)$.*

Proof.

Recall that ${\mathcal{G}}$ is defined in (3.38).

(i) Let $g$ be in ${\mathcal{G}}$. For each $\theta>0$, the function $x\mapsto g_{\theta}(x):=\theta^{\frac{\ell}{2}}{g(\theta x)}$ also belongs to ${\mathcal{G}}$. Hence,

[TABLE]

Writing these integrals back to $g$ and using 2c yields

[TABLE]

for all $\theta>0$. Optimizing the left-hand side (with respect to $\theta$) leads to

[TABLE]

Removing the normalization $\|g\|_{L^{2}}=1$ and some algebraic manipulation yields the result.

(ii) Let $\kappa_{0}$ be the best constant in (3.42). Then for every $g\in{\mathcal{G}}$,

[TABLE]

This shows ${\mathcal{E}}_{H}(\gamma)$ is finite and at most $\frac{2-\alpha}{\alpha}(\frac{\alpha}{2}\kappa_{0})^{\frac{2}{2-\alpha}}$, which also means $\kappa(\gamma)\leq\kappa_{0}$. On the other hand, (i) already implies $\kappa_{0}\leq\kappa(\gamma)$, hence completes the proof. ∎

Proof of Proposition 3.2.

Reasoning as in Proposition 3.3, we see that ${\mathcal{M}}(\gamma)$ is finite if and only if (3.42) holds for some constant $\kappa>0$. In addition, the best constant $\kappa(\gamma)$ in (3.42) satisfies the relation

[TABLE]

Together with (3.43), this yields the result. ∎

The following result preludes the connection between ${\mathcal{E}}_{H},{\mathcal{M}}$ with exponential functional of Brownian motions.

Lemma 3.4**.**

Let $\{B(s),s\geq 0\}$ be a Brownian motion in $\mathbb{R}^{n}$ and $D$ be a bounded open domain in $\mathbb{R}^{n}$ containing 0. Let $h(s,x)$ be a bounded function defined on $[0,1]\times\mathbb{R}^{n}$ which is continuous in $x$ and equicontinuous (over $x\in\mathbb{R}^{n}$) in $s$. Then

[TABLE]

where ${\mathcal{G}}_{D}$ is the class of functions $g$ in $W^{1,2}(\mathbb{R}^{n})$ such that $\int_{D}|g(x)|^{2}dx=1$ and $\tau_{D}$ is the exit time $\tau_{D}:=\inf\{t\geq 0:B_{t}\notin D\}$.

Proof.

The process $\{B_{0,t}(s)=B(s)-\frac{s}{t}B(t)\}_{s\in[0,t]}$ is a Brownian bridge. We fix $\theta\in(0,1)$ and consider first the limit

[TABLE]

Let $M$ be such that $|x|\leq M$ for all $x\in D$. Using Girsanov theorem (see [HLN15]*Eq. (2.38)), we can write

[TABLE]

The result of [MR3414457, Proposition 3.1] asserts that

[TABLE]

This leads to

[TABLE]

Observing that

[TABLE]

we can send $\theta\uparrow 1$ in (3.45) to obtain the lower bound for (3.44). The upper bound for (3.44) is proved analogously, we omit the details. ∎

We conclude this section with an observation: 2c induces the following scaling relation on ${\mathcal{E}}_{H}(\gamma)$

[TABLE]

4. Feynman-Kac formulas and functionals of Brownian Bridges

We derive Feynman-Kac formulas for the moments $\mathbb{E}u^{m}(t,x)$ for integers $m\geq 2$. These formulas play important roles in proving upper and lower bounds of (1.15) and (1.26).

To discuss our contributions in the current section, let us assume for the moment that $\dot{W}$ is a space-time white noise and $\ell=1$. The most well-known Feynman-Kac formula for second moment is

[TABLE]

where $B^{1},B^{2}$ are two independent Brownian motions starting at 0. If $u_{0}$ is merely a measure, some efforts are needed to make sense of $u_{0}(B(t)+x)$, which appears on the right-hand side above. An attempt is carried out in [ChenNualart2016] using Meyer-Watanabe’s theory of Wiener distributions.

The Feynman-Kac formulas presented here (see (4.4) below) have appeared in [HLN15]. However, there seems to have a minor gap in that article. Namely, Eq. (4.52) there has not been proven if $u_{0}$ is a measure. In the current article, we take the chance to fill this gap. Our approach is in the same spirit as [HLN15] and is different from [ChenNualart2016]. In particular, we do not make use of Wiener distributions.

Since $W_{\epsilon}$ has bounded covariance, it is easy to see that the stochastic heat equation

[TABLE]

has a unique random field solution $u_{\epsilon}$. In addition, for each $t>0$ and $x\in\mathbb{R}^{\ell}$, $u_{\epsilon}(t,x)$ admits a chaos expansion (see, for instance [HuNu09])

[TABLE]

where $f_{0}[u_{0}](t,x)=p_{t}*u_{0}(x)$ and for each $n\geq 1$

[TABLE]

Here, $\sigma$ denotes the permutation of $\{1,2\dots,n\}$ such that $0<s_{\sigma(1)}<\cdots<s_{\sigma(n)}<t$ and $I_{\epsilon,n}$ is the $n$-th multiple Itô-Wiener integral with respect to the Gaussian field $W_{\epsilon}$.

Proposition 4.1**.**

Let $u_{0}$ be a measure satisfying (1.13). Then

[TABLE]

In addition, if 1 holds, then

[TABLE]

Proof.

Let $v(t,x)$ be the integral on the right-hand side of (4.50). From (2.33), integrating $z$ with respect to $u_{0}(dz)$ and applying the stochastic Fubini theorem (cf. [DPZ, Theorem 4.33]), we have

[TABLE]

Hence, $v$ is a solution of (1.1) with initial datum $u_{0}$. By unicity, Theorem 2.3, we see that $u=v$ and (4.50) follows.

Next, we show (4.51) assuming 1. Fix $t>0$ and $x\in\mathbb{R}^{\ell}$. For every $u_{0}\in C_{c}^{\infty}(\mathbb{R}^{\ell})$, the following Feynman-Kac formula holds

[TABLE]

Using the decomposition (2.34) and the fact that $B_{0,t}$ and $B(t)$ are independent, we see that

[TABLE]

where

[TABLE]

Together with (4.50) we obtain

[TABLE]

for all $u_{0}\in C^{\infty}_{c}(\mathbb{R}^{\ell})$.

Next we show that $z\mapsto Y(z;t,x)$ is continuous. Fix $p>2$. From the elementary relation $|e^{x}-e^{y}|\leq(e^{x}+e^{y})|x-y|$ and the Cauchy-Schwarz inequality, it follows

[TABLE]

Since $\gamma$ is bounded, it is easy to see that

[TABLE]

for some constant $C_{p,t}$. We now resort to Minkowski inequality, our exponential bound for $V_{t,x}(z)$ and the relation between $L^{p}$ and $L^{2}$ moments for Gaussian random variables in order to obtain

[TABLE]

In addition, under 1, $\gamma$ is Hölder continuous with order $\kappa>0$ at 0, it follows that

[TABLE]

We have shown

[TABLE]

Thus, the process $z\rightarrow Y(z;t,x)$ has a continuous version. On the other hand, $z\mapsto\mathcal{Z}(z;t,x)$ is also continuous (see Proposition 5.5 below). It follows that $\mathcal{Z}(z;t,x)=Y(z-x;t,x)p_{t}(z-x)$, which is exactly (4.51). ∎

Proposition 4.2**.**

Assuming 1, we have

[TABLE]

and

[TABLE]

Proof.

We observe that conditioned on $B$,

[TABLE]

is a normal random variable with mean zero. In addition, for every $x,x^{\prime},z,z^{\prime}\in\mathbb{R}^{\ell}$, applying (1.23), we have

[TABLE]

For every $(x_{1},\dots,x_{m})\in(\mathbb{R}^{\ell})^{m}$, using (4.51) and (4.55), we have

[TABLE]

Note that in the exponent above, the diagonal terms (with $j=k$) are removed because there are cancellations with the normalization factor $-\frac{t}{2}\gamma(0)$ in (4.51), which occur after taking expectation with respect to $W$. Finally, apply [HLN15, Lemma 4.1], we obtain (4.54) from (4.56). ∎

To extend the previous result to nosies satisfying 2, we need the following result.

Proposition 4.3**.**

Assuming 2. There exists a constant $c$ depending only on $\alpha$ such that for any $\beta\in(0,4\wedge(\ell-\alpha))$,

[TABLE]

where $\Theta_{t}(m)$ is defined in (2.36)

Proof.

Let us put

[TABLE]

From (2.33), we have

[TABLE]

Then we obtain

[TABLE]

To estimate $I_{1}$, we use Lemma 2.1 to obtain

[TABLE]

To estimate $I_{2}$, we first note that the noise $W-W_{\epsilon}$ has spectral density $(1-e^{-\epsilon|\xi|^{2}})^{2}\mu(\xi)$. Applying Lemma 2.1, we obtain

[TABLE]

Let us fix $\beta\in(0,4\wedge(\ell-\alpha))$. Applying the elementary inequality $1-e^{-\epsilon|\xi|^{2}}\leq\epsilon^{\beta/4}|\xi|^{\beta/2}$ together with the estimate

[TABLE]

we get

[TABLE]

Reasoning as in [HLN15, Lemma 4.1], we see that

[TABLE]

Two key observations here are $\gamma_{\epsilon},\gamma$ have spectral measures $\mu(\xi),e^{-\epsilon|\xi|^{2}}\mu(\xi)$ respectively and $e^{-\epsilon|\xi|^{2}}\mu(\xi)\leq\mu(\xi)$. Hence, it follows from (4.54) and the previous estimate that

[TABLE]

In summary, we have shown

[TABLE]

Applying Lemma 2.4, this yields

[TABLE]

for some constant $c$ depending only on $\alpha$. ∎

We are now ready to derive Feynman-Kac formulas for positive moments.

Proposition 4.4**.**

Let $u_{0}$ be a measure satisfying (1.13). Under 1 or 2, for every $x_{1},\dots,x_{m}\in\mathbb{R}^{\ell}$, we have

[TABLE]

and

[TABLE]

Proof.

We prove the result under the hypothesis 2. The proof under hypothesis 1 is easier and omitted.

Step 1: we first consider (4.4) and (4.60) when the initial data are Dirac masses. More precisely, we will show that

[TABLE]

and

[TABLE]

Fix $\epsilon>0$, (4.61) with $\mathcal{Z},\gamma$ replaced by $\mathcal{Z}_{\epsilon},\gamma_{\epsilon}$ has been obtained in (4.53). Namely, we have

[TABLE]

Using analogous arguments with [HLN15, Proposition 4.2], we can show that for every $\kappa\in\mathbb{R}$, as $\epsilon\downarrow 0$, the functions

[TABLE]

converges uniformly on $\mathbb{R}^{2m\ell}$ to the function

[TABLE]

In addition, in view Proposition 4.3,

[TABLE]

Sending $\epsilon\downarrow 0$ in (4.63), we obtain (4.61). (4.62) is obtained analogously using (4.54). We omit the details.

Step 2: For general initial data satisfying (1.13), we note that from (4.50),

[TABLE]

From here, it is evident that (4.4), (4.60) are consequences of (4.61), (4.62) and Fubini’s theorem. ∎

We conclude this section with the following observation.

Remark 4.5*.*

Under 1, it is evident from (4.51) that $\mathcal{Z}(z;t,x)$ is non-negative for every $z,t,x$. Under 2, thanks to Proposition 4.3, $\mathcal{Z}(z;t,x)$ is the limit of non-negative random variables, hence $\mathcal{Z}(z;t,x)$ is also non-negative for every $z,t,x$. Furthermore, in view of (4.50), if $u_{0}$ is non-negative then $u(t,x)$ is non-negative for every $t,x$.

5. Moment asymptotic and regularity

Moment asymptotic

We begin with a study on high moments. Under hypothesis 1, the high moment asymptotic is governed by the value of $\gamma$ at the origin.

Proposition 5.1**.**

Under 1, for every $T>0$, we have

[TABLE]

Proof.

Since $\gamma$ is positive definite, $\gamma(x)\leq\gamma(0)$ for all $x\in\mathbb{R}^{\ell}$. It follows from (4.60) that

[TABLE]

This immediately yields (5.64). ∎

The following result is needed to obtain moment asymptotic under 2.

Lemma 5.2**.**

Suppose that $\mu(\mathbb{R}^{\ell})<\infty$. For each $t,T,m$ we put $t_{m}=m^{\frac{2}{2-\alpha}}t$ and $T_{m}=m^{\frac{2}{2-\alpha}}T$. Then

[TABLE]

Proof.

For each $\lambda\in(0,1)$, we note that

[TABLE]

Using (2.35), we see that the expectation above is at most

[TABLE]

In addition, reasoning as in [HLN15, Lemma 4.1], we see that

[TABLE]

It follows that

[TABLE]

Applying [ChPh15]*Theorem 1.1, we get

[TABLE]

Thus we have shown

[TABLE]

Finally, we send $\lambda\to 1^{-}$ to finish the proof. ∎

Proposition 5.3**.**

Assuming 2, for every fixed $T>0$,

[TABLE]

where ${\mathcal{E}}_{H}(\gamma)$ is the Hartree energy defined in (3.37).

Proof.

Applying inequality (4.60), we have

[TABLE]

In addition, by the change of variable $s\rightarrow sm^{-\frac{2}{2-\alpha}}$ and the scaling property of Brownian bridge, $\{B_{0,\lambda t}(\lambda s),s\in[0,t]\}\stackrel{{\scriptstyle\mbox{law}}}{{=}}\{\sqrt{\lambda}B_{0,t}(s),s\in[0,t]\}$, the right hand side in the above expression is the same as

[TABLE]

Hence, denoting $t_{m}=m^{\frac{2}{2-\alpha}}t$ and $T_{m}=m^{\frac{2}{2-\alpha}}T$, we see that (5.66) is equivalent to the statement

[TABLE]

Let $p,q>1$ such that $p^{-1}+q^{-1}=1$. By Hölder inequality

[TABLE]

where

[TABLE]

From Lemma 5.2 and the fact that ${\mathcal{E}}_{H}(\gamma_{\epsilon})\leq{\mathcal{E}}_{H}(\gamma)$ (see (3.39)), we have

[TABLE]

Hence, it suffices to show for every fixed $q>1$,

[TABLE]

By Cauchy-Schwarz inequality and the fact that $B_{0,t}{\stackrel{{\scriptstyle\text{law}}}{{=}}}B_{0,t}(t-\cdot)$, we have

[TABLE]

Together with (2.35), we arrive at

[TABLE]

note that the right hand side of the above inequality is the $m$-th moment of the solution to the equation (1.1) driven by the noise with spatial covariance $\frac{2q}{m}\left(\gamma-\gamma_{\epsilon}\right)$, i.e., $\mathbb{E}u(\frac{t_{m}}{2},x)^{m}$, the initial condition is $u_{0}(x)\equiv 2^{\ell}$. Using the hyper-contractivity as in [HLN15, MR3531492], we have

[TABLE]

where in the last line we have used the estimate (3.7) in [MR3354615] and $\mu(\eta)d\eta$ is abbreviation for $\prod_{j=1}^{k}\mu(\eta_{j})d\eta_{j}$. Since $\alpha<2$, we can find a $\beta>0$ such that $\beta<1-\frac{\alpha}{2}$. Then using the elementary inequality

[TABLE]

we obtain

[TABLE]

Hence, we have shown

[TABLE]

from which (5.68) follows. The proof for (5.66) is complete. ∎

Hölder continuity

We investigate the regularity of the process $\frac{\mathcal{Z}(x;t,y)}{p_{t}(y-x)}$ in the variables $x$ and $y$. These properties will be used in the proof of upper bound. For each integer $m\geq 2$ and $t>0$, we recall that $\Theta_{t}(m)$ is defined in (2.36).

Note that from Proposition 4.4, we have

[TABLE]

Lemma 5.4**.**

For every $r>0$ and $y_{1},y_{2}\in\mathbb{R}^{\ell}$

[TABLE]

under 2; and

[TABLE]

under 1. In the above, the constant $C$ does not depend on $y_{1},y_{2}$ nor $r$.

Proof.

We denote $f(\cdot)=|p_{r}(\cdot-y_{1})-p_{r}(\cdot-y_{2})|$. Assuming first 2, we observe the following simple estimate

[TABLE]

Noting that

[TABLE]

and

[TABLE]

the result easily follows. Under 1, we used the following inequality

[TABLE]

together with (5.70) to obtain the result. ∎

Proposition 5.5**.**

Assuming 1 or 2. There exists a constant $\eta\in(0,1)$ such that for every compact set $K$ and every integer $m\geq 2$,

[TABLE]

and

[TABLE]

where $B(w,1)$ is the closed unit ball in $\mathbb{R}^{\ell}$ centered at $w$. In the above, the constant $c$ depends only on $\bar{\alpha}$ and $\eta$ and $c_{K}(t)$ depends only on $K,t,\eta$.

Proof.

We present the proof under hypothesis 2 in detail. The proof for the other case is similar and is omitted. We first show that for every $\eta\in(0,2-\alpha)$,

[TABLE]

Fix $t>0$ and $x_{1},x_{2},x\in\mathbb{R}^{\ell}$. From (2.33), we have

[TABLE]

where

[TABLE]

Obviously $f$ also depends on $t,x_{1},x_{2}$ and $x$, however these parameters will be omitted. For each integer $m\geq 2$, applying Lemma 2.1 we see that

[TABLE]

Applying Lemma 5.4, for every $\eta\in(0,2-\alpha)$, there exists $c_{\eta}>0$ such that

[TABLE]

Hence,

[TABLE]

For each $s>0$, we set

[TABLE]

It follows from Lemma 2.1 that

[TABLE]

where $c$ is some constant. Applying these estimates in (5.74) yields

[TABLE]

We now apply Lemma 2.4 to get

[TABLE]

which is exactly (5.73).

To complete the proof of the estimate (5.71). Fix $t>0$ and $x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}^{\ell}$. Observe that

[TABLE]

where

[TABLE]

Similar to (5.75), we have

[TABLE]

$I_{2}$ can be estimated using Lemma 2.1 and (5.73)

[TABLE]

Hence, we have shown

[TABLE]

At this point, the estimate (5.71) follows from the Garsia-Rodemich-Rumsey inequality (cf. [garsiarodemich]).

The proof of (5.72) is simpler. Actually, by writing

[TABLE]

we get an estimate for $\|\frac{\mathcal{Z}(x;t,y_{1})}{p_{t}(y_{1}-x)}-\frac{\mathcal{Z}(x;t,y_{2})}{p_{t}(y_{2}-x)}\|_{L^{m}(\Omega)}$ as in (5.76). The estimate (5.72) again follows from the Garsia-Rodemich-Rumsey inequality (cf. [garsiarodemich]). We omit the details. ∎

In proving (1.26), we need to handle the asymptotic of $\sup_{\epsilon<1}\sup_{x\in K,|y|\leq R}\frac{\mathcal{Z}_{\epsilon}(x;t,y)}{p_{t}(y-x)}$, thus we write down the Hölder continuity result for $\frac{\mathcal{Z}_{\epsilon}(x;t,y)}{p_{t}(y-x)}$ with respect to $\epsilon,x,y$. The proof is similar with Proposition 5.5 and is left to the reader.

Proposition 5.6**.**

Assuming 1 or 2. There exists a constant $\eta\in(0,1)$ such that for every compact set $K$ and every integer $m\geq 2$,

[TABLE]

and

[TABLE]

In the above, the constant $c$ depends only on $\bar{\alpha}$ and $\eta$ and $c_{K}(t)$ depends only on $K,t,\eta$.

6. Spatial asymptotic

In this section we study the asymptotic of

[TABLE]

as described in Theorems 1.3, 1.4 and 1.5. In what follows, we denote

[TABLE]

where we recall that $\bar{\alpha}$ is defined in (1.19). Since $0\leq\bar{\alpha}<2$, $a$ ranges inside the interval $[1/2,1)$. Because $R\mapsto\sup_{|y|\leq R}\frac{u(t,y)}{p_{t}*u_{0}(y)}$ is monotone, it suffices to show these results along lattice sequence $R\in\{e^{n}\}_{n\geq 1}$.

6.1. The upper bound

This subsection is devoted to the proof of upper bounds in Theorems 1.3 and 1.4 by combining the moment asymptotic bounds and the regularity estimates obtained in Section 5. We also recall that $\Theta_{t}(m)$ is defined in (2.36). Propositions 5.1, 5.3 together with (5.69) imply

[TABLE]

where ${\mathcal{E}}$ is defined in (1.19). The following result gives an upper bound for spatial asymptotic of $\mathcal{Z}(x;\cdot,\cdot)$.

Theorem 6.1**.**

For every compact set $K$, we have

[TABLE]

Proof.

We begin by noting that according Remark 4.5, $\mathcal{Z}(x;t,y)$ is non negative a.s. for each $x,y,t$. Let $t$ be fixed and put

[TABLE]

where we have omitted the dependence on $t$. For every $n>1$ and every $\lambda>0$, we consider the probability

[TABLE]

Let $b$ be a fixed number such that $a<b<1$. We can find the points $x_{i}$, $i=1,\dots,M_{n}$, such that $K\subset\cup_{i=1}^{M_{n}}B(x_{i},e^{-n^{b}})$ and $M_{n}\lesssim e^{\ell n^{b}}$. In addition, by partitioning the ball $B(0,e^{n})$ into unit balls, we see that $P_{n}$ is at most

[TABLE]

Applying Chebychev inequality, we see that

[TABLE]

The above $m$-th moment is estimated by triangle inequality

[TABLE]

Using Proposition 5.5 and (5.69), we see that

[TABLE]

Altogether, we have

[TABLE]

For each $\beta>0$, we choose $m=\lfloor\beta n^{1-a}\rfloor$. In addition, for every fixed $\epsilon>0$, (6.80) yields

[TABLE]

for all $n$ sufficiently large. It follows that

[TABLE]

where

[TABLE]

Since $1-a+b>1$, the term $e^{-\eta\beta n^{1-a+b}}$ is dominant, and hence, $S_{1}$ is finite for every $\lambda,\beta>0$. To ensure the convergence of $S_{2}$, we choose $\lambda$ such that

[TABLE]

It follows that the series on the right hand side of (6.83) is finite. By Borel-Cantelli lemma, we have almost surely

[TABLE]

Evidently, the best choice for $\lambda$ is

[TABLE]

which yields (6.81). ∎

Remark 6.2*.*

Using Proposition 5.6 and analogous arguments in Theorem 6.1, we can show that

[TABLE]

We omit the details.

6.2. The lower bound

We now focus on the lower bound of (1.15) and (1.26). To start with, we explain an issue of using the localization procedure as in [MR3474477, MR3098071]. In these papers, a localized version of the equation (1.1) is introduced, i.e.

[TABLE]

for some $\beta>0$. For fixed $t$ and $\beta$ sufficiently large, $\sup_{|x|\leq R}U^{\beta}(t,x)$ gives a good approximation for $\sup_{|x|\leq R}u(t,x)$ as $R\to\infty$. In our situation, suppose for instance that $u_{0}=\delta(\cdot-x_{0})$, the random field $\frac{\mathcal{Z}(x_{0};t,x)}{p_{t}(x-x_{0})}$ satisfies the equation

[TABLE]

Since the kernel $p_{\frac{s(t-s)}{t}}\left(y-x_{0}-\frac{s}{t}(x-x_{0})\right)$ now involves $s$ and $t$ with $s$ moving from [math] to $t$, the mass concentration of the stochastic integration on the right-hand side of (6.88) varies and depends on $s$. We are not able to find a fixed localized integration domain similar as $\{y:|y-x|\leq\beta\sqrt{t}\}$. To get around this difficulty, we propose an alternative result (Theorem 1.5) which is about the regularized version of $\mathcal{Z}$, i.e., $\mathcal{Z}_{\epsilon}$. To handle the spatial asymptotic of $\mathcal{Z}_{\epsilon}$, we rely on the Feynman-Kac representation (4.51) and adopt an argument developed by Xia Chen in [MR3474477] with an additional scaling procedure.

Hereafter, $t$ and $\epsilon$ are fixed positive constants, $n$ is the driving parameter which tends to infinity,

[TABLE]

Let $y_{1},\dots,y_{N}$ be $N$ points in $B(0,e^{n})$ and $d$ be a positive number such that

[TABLE]

Under 1, $d$ is chosen to be sufficiently large, depending on the shape of $\gamma$, while under 2, we can simply choose $d=1$. See Lemma 6.4 below for more details.

Theorem 6.3**.**

For every $x_{0}\in\mathbb{R}^{\ell}$

[TABLE]

Proof.

Step 1: Let $m=m_{n}$ be a natural number such that

[TABLE]

Under hypothesis 1, for each $j$, we define the stopping time

[TABLE]

where $r_{0}>0$ is chosen so that

[TABLE]

Such a constant always exists since $\gamma$ is continuous and $\gamma(0)>0$. Under hypothesis 2, the stopping times depends on $n$ and an arbitrary domain. More precisely, let $D$ be an open bounded ball in $\mathbb{R}^{\ell}$ which contains 0. For each $j$, $\tau^{j}=\tau^{j}_{n}(D)$ denotes the stopping time

[TABLE]

As previously, we denote

[TABLE]

omitting the dependence on $t$. We note that from (4.51)

[TABLE]

where

[TABLE]

Conditioning on $B$, the variance of $\xi_{m}(x_{0},y)$ is given by

[TABLE]

For every $\lambda\in(0,\sqrt{2\ell})$, it is evident that

[TABLE]

where we have put

[TABLE]

and

[TABLE]

Combining all previous estimates, we arrive at an important inequality

[TABLE]

It follows that

[TABLE]

We put

[TABLE]

Applying the estimate

[TABLE]

we obtain

[TABLE]

Noting that $N^{-\frac{1}{m}}\lesssim e^{\ell\frac{n}{m}}$ and by (1.24), $\gamma_{\epsilon_{n}}(0)=\epsilon_{n}^{-\frac{\alpha}{2}}\gamma_{1}(0)\lesssim n^{\frac{\alpha}{2}a}$, we see that

[TABLE]

In other words, the factor $N^{-\frac{1}{m}}e^{-\frac{t}{2}\gamma_{\epsilon_{n}(0)}}$ in (6.101) is negligible. In addition, we claim that for every $\lambda\in(0,\sqrt{2\ell})$ and every $x\in\mathbb{R}^{\ell}$

[TABLE]

We postpone the proof of this claim till Lemmas 6.4 below. It follows that

[TABLE]

Step 2: We will show that

[TABLE]

We consider first the hypothesis 1. Since $\gamma$ is continuous, for any $\epsilon>0$, there is $\delta$ such that whenever $|z|\leq\delta\wedge r_{0}$, $\gamma(z)\geq\gamma(0)-\epsilon$. Hence,

[TABLE]

Since as $n\to\infty$, $m\to\infty$ too, we have

[TABLE]

which proves (6.105) under 1.

Assume now that 2 holds. We put $t_{n}=t^{1-a}n^{a}$ so that $\epsilon_{n}=\epsilon\frac{t}{t_{n}}$. The Brownian motion scaling and the relation (1.24) yield

[TABLE]

It follows that

[TABLE]

where

[TABLE]

Let $K_{\epsilon}$ be the function defined by

[TABLE]

so that

[TABLE]

Hence, we can write

[TABLE]

Let $\mathcal{D}$ be the set of compactly supported continuous functions on $\mathbb{R}^{\ell}$ with unit $L^{2}(\mathbb{R}^{\ell})$-norm. For every $f\in\mathcal{D}$, applying Cauchy-Schwarz inequality, we see that the right-hand side in the equation above is at least

[TABLE]

where we have set

[TABLE]

Using independency of Brownian motions, we obtain

[TABLE]

where $\tau_{D}:=\inf\{s\geq 0:B(s)\notin D\}$. Applying Lemma 3.4 we obtain

[TABLE]

We now let $D\uparrow\mathbb{R}^{\ell}$ to get

[TABLE]

We now link the variation on the right-hand side with ${\mathcal{M}}(\gamma)$ by observing that

[TABLE]

Indeed, for each fixed $g\in{\mathcal{G}}$, applying Fubini’s theorem, Hahn-Banach theorem and (6.106), we have

[TABLE]

This leads us the identity (6.107). We can send $\epsilon\downarrow 0$ and apply Proposition 3.2 to obtain (6.105) under hypothesis 2.

Step 3: Combining the inequalities (6.104) and (6.105) together, we have for every $\lambda\in(0,\sqrt{2\ell})$

[TABLE]

Finally we let $\lambda\to\sqrt{2\ell}^{-}$ to conclude the proof. ∎

We now provide the proof of (6.103).

Lemma 6.4**.**

For every $\lambda\in(0,\sqrt{2\ell})$, we have

[TABLE]

where we recall $\eta_{n}^{c}$ is defined in (6.100).

Proof.

Assuming first that 1 holds. We recall that $\epsilon_{n}=0$ in this case so that $\gamma_{\epsilon_{n}}=\gamma$. Let $\mathcal{B}$ be the $\sigma$-field generated by the Brownian motions $\{B^{j}\}_{1\leq j\leq m}$. First we will show that for any $0<\rho<\frac{1}{2}$, we can find $d>0$ sufficiently large so that on the event $\{\min_{1\leq j\leq m}\tau^{j}\geq t\}$, for every $z,z^{\prime}\in B(0,e^{n})$ with $|z-z^{\prime}|\geq d$.

[TABLE]

We recall that $d$ and $\tau^{j}$ are defined in (6.90) and (6.93) respectively. We choose and fix $\varkappa\in(0,1)$ such that

[TABLE]

Note that on the event $\{\min_{1\leq j\leq m}\tau^{j}\geq t\}$, we have $\sup_{s\leq t,j\leq m}|B^{j}_{0,t}(s)|\leq r_{0}$. Then for every $j,k\leq m$,

[TABLE]

In addition, from (1.3) and Riemann-Lebesgue lemma, $\lim_{x\to\infty}\gamma(x)=0$. Hence, when $s\in[\varkappa t,t]$, we can choose $d$ large enough such that whenever $|y|\leq 2r_{0}$ and $|z-z^{\prime}|\geq d$

[TABLE]

In particular, for every $|z-z^{\prime}|\geq d$ we have

[TABLE]

It follows that

[TABLE]

which verifies (6.109).

Since $\lambda<\sqrt{2\ell}$, we can choose $\kappa,\rho\in(0,\frac{1}{2})$ sufficiently small so

[TABLE]

Let us now recall Lemma 4.2 in [MR3178468]. For a mean zero n-dimensional Gaussian vector $(\xi_{1},\cdots,\xi_{n})$ with identically distributed components,

[TABLE]

and for any $A,B>0$, we have

[TABLE]

where $U$ is a standard normal random variable. Applying this inequality conditionally with $A=\lambda S_{m}(t)\sqrt{n}$ and $B=\kappa S_{m}\sqrt{n}$, we have for sufficiently large $n$,

[TABLE]

where $v>0$ is independent of $n$. Now for any $\theta>0$, this yields

[TABLE]

An application of Borel-Cantelli lemma yields (6.108) under hypothesis 1.

We now consider the hypothesis 2. The argument is similar to the previous case. There is, however, an additional scaling procedure. Recall that $\mathcal{B}$ is the $\sigma$-field generated by the Brownian motions $\{B^{j}\}_{1\leq j\leq m}$. We choose $d=1$. It suffices to prove (6.109) on the event $\{\min_{0\leq j\leq m}\tau^{j}\geq t\}$, for any $|z-z^{\prime}|\geq 1$. Indeed, we have

[TABLE]

For every $j,k\leq m$, using the scaling relation (1.24), we can write

[TABLE]

We now choose and fix $\theta>0$ such that

[TABLE]

this is always possible since $\gamma_{1}=p_{2}*\gamma$ is a strictly positive function. It follows that

[TABLE]

In addition, on the event $\{\min_{0\leq j\leq m}\tau^{j}\geq t\}$, $\epsilon_{n}^{-\frac{1}{2}}(B_{0,t}^{j}(t-s)-B_{0,t}^{k}(t-s))$ belongs to $2\epsilon^{-\frac{1}{2}}D$ for all $s\in[0,t]$. Hence, for every $s\in[\theta t,t]$ and $|z-z^{\prime}|\geq 1$, we have

[TABLE]

We note that from Riemann-Lebesgue lemma, $\lim_{x\to\infty}\gamma_{1}(x)=0$. Hence, whenever $n$ is sufficiently large,

[TABLE]

for all $|y|\geq\theta\epsilon_{n}^{-\frac{1}{2}}-2\epsilon^{-\frac{1}{2}}\mathrm{diag}(D)$. It follows that for every $z,z^{\prime}$ with $|z-z^{\prime}|\geq 1$,

[TABLE]

Upon combining these estimates, we arrive at (6.109), which in turn, implies (6.108). ∎

6.3. Proofs

The Theorems 1.3, 1.4 and 1.5 follow from the asymptotic results from the previous two subsections. Indeed, Theorem 1.3 follows by combining the upper bound in Theorem 6.1 and the lower bound Theorem 6.3. To obtain Theorem 1.4, we first observe that from (4.50),

[TABLE]

Then, an application of Theorem 6.1 yields the result. For Theorem 1.5, the upper bound of (1.25) follows from Remark 6.2 and the bound (6.116) with $u,\mathcal{Z}$ replaced respectively by $u_{\epsilon},\mathcal{Z}_{\epsilon}$, together with the obvious fact that $\mathcal{E}_{H}(\gamma_{\epsilon})\leq\mathcal{E}_{H}(\gamma)$, see (3.39). The lower bound of (1.26) is immediate from Theorem 6.3.

Acknowledgment

The project was initiated while both authors are visiting the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester. The first named author thanks Professor Davar Khoshnevisan for stimulating discussions and encouragement. The second named author thanks PIMS for its support through the Postdoctoral Training Centre in Stochastics. Both authors thank Professor Yaozhong Hu and Professor David Nualart for support and encouragement.