Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Abstract

Suppose that $\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}$ is the solution to a d-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x$, as $N\rightarrow \infty$, where g is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf–Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

Appendix: Association for parabolic SPDEs

We conclude the paper by showing how we can adapt the ideas of this paper to also document a result about the association of the solution to (1.1). First, let us recall the following.

Definition A.1

(Esary et al. [20]) A random vector $X:=(X_1,\ldots ,X_n)$ is said to be associated if

$$\begin{aligned} {{\,\mathrm{\text {Cov}}\,}}[h_1(X),h_2(X)]\ge 0, \end{aligned}$$

(A.1)

for every pair of functions $h_1,h_2:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ that are nondecreasing in every coordinate and satisfy $h_1(X),h_2(X)\in L^2(\Omega )$.

It is easy to see that $h_1$ and $h_2$ could instead be assumed to be nonincreasing in every coordinate: One simply replaces $(h_1,h_2)$ by $(-h_1,-h_2)$. Other simple variations abound. Esary, Proschan, and Walkup [20] prove the more interesting result that X is associated iff (A.1) holds for all $h_1,h_2\in C_b({\mathbb {R}}^n)$ that are nondecreasing coordinatewise.⁷ This and the dominated convergence theorem together imply the following simple criterion for association; see also Pitt [35].

Lemma A.2

$X=(X_1,\ldots ,X_n)$ is associated iff (A.1) holds for all $h_1,h_2\in C^\infty _c({\mathbb {R}}^n)$ such that $\partial _i h_j\ge 0$ for every $i=1,2$ and $j=1,\ldots ,n$.⁸

The notion of association has a natural extension to random processes. Here is one.

Definition A.3

A random field $\Phi =\{\Phi (t,x)\}_{t\ge 0,x\in {\mathbb {R}}^d}$ is associated if $(\Phi (t_1,x_1),\ldots , \Phi (t_n,x_n))$ is associated for every $t_1,\ldots ,t_n>0$ and $x_1,\ldots ,x_n\in {\mathbb {R}}^d$.

The following is the principal result of this appendix.

Theorem A.4

Let u be the solution to (1.1) with $u(0,\cdot )$ being a nonnegative measure such that $\left( u(0,\cdot )*{\varvec{p}}_t\right) <\infty$ for all $t>0$. Under Dalang’s condition (1.5), if $\sigma \circ u$ a.s. does not change signs; that is,

$$\begin{aligned} \mathrm {P}\left\{ \sigma (u(t,x))\sigma (u(s,y))\ge 0\right\} =1 \quad \text {for every }s,t>0 \text { and }x,y\in {\mathbb {R}}^d, \end{aligned}$$

(A.2)

then, u is associated.

Theorem A.4 reduces to simpler results in two important special cases that we present in the following two examples.

Example A.5

Condition (A.2) holds tautologically when $\sigma$ is a constant, say $\sigma \equiv \sigma _0\in {\mathbb {R}}$. In that case, it is not hard to see that the solution to (1.1) is a Gaussian random field with covariance function,

$$\begin{aligned} {{\,\mathrm{\text {Cov}}\,}}[u(t,x),u(t',x')] = \sigma _0^2\int _0^t\left( {\varvec{p}}_{2s}*f\right) (x-x')\,\mathrm {d}s. \end{aligned}$$

Because this quantity is $\ge 0$ for every $t,t'>0$ and $x,x'\in {\mathbb {R}}^d$, Pitt’s characterization of Gaussian random vectors [35] immediately implies the association of u when $\sigma$ is a constant.

Example A.6

If $\sigma (u)=u$, $d=1$, and $f=\delta _0$, then the sign condition (A.2) holds simply because $u(t,x)\ge 0$ a.s. In this case, Corwin and Quastel [14, Proposition 1] have observed that u is associated; see also Corwin and Ghosal [13, Proposition 1.9]. Strictly speaking, the statements of the latter two results are weaker that the association of u. But the proof of Proposition 1.9 of Corwin and Quastel (ibid.) goes through unhindered to imply association. Indeed, Corwin and Quastel note the said association from the well-known fact that all exclusion processes are associated (the FKG inequality for i.i.d. random variables).

The preceding examples are proved using two different “direct hands-on” methods. The general case is quite a bit more involved. In order to avoid writing a lengthy proof, we merely outline the proof, with detailed pointers to the requisite literature and preceding results of the present paper.

Outline of the proof of Theorem A.4

By considering $-\sigma$ in place of $\sigma$ if need be, we may—and will—assume without loss in generality that

$$\begin{aligned} \sigma (u(t,x))\ge 0\qquad \hbox { a.s. for every}\ (t,x)\in (0,\infty )\times {\mathbb {R}}^d. \end{aligned}$$

We begin by rehashing through the proof of Theorem 3.2, and recall that the Malliavin derivative of u(t, x) a.s. solves

$$\begin{aligned} D_{s,z}u(t,x)= & {} {\varvec{p}}_{t-s}(x-z) \sigma (u(s,z))\nonumber \\&+ \int _{\left( s,t\right) \times {\mathbb {R}}^d} {\varvec{p}}_{t-r}(x-y) \sigma '(u(r,y)) D_{s,z} u(r,y)\,\eta (\mathrm {d}r\,\mathrm {d}y), \end{aligned}$$

(A.3)

for almost all $(s,z)\in (0,t)\times {\mathbb {R}}^d$. And, of course, $D_{s,z}u(t,x)=0$ a.s. for a.e. $(s,z)\in (t,\infty )\times {\mathbb {R}}^d$. Note that the right-hand side of (A.3) is in fact a modification of $D_{s,z}u(t,x)$. Therefore, we can—and will without loss of generality—redefine $D_{s,z}u(t,x)$ so that (A.3) is an almost-sure identity for every $(s,z)\in (0,t)\times {\mathbb {R}}^d$, and that $D_{s,z}u(t,x):=0$ whenever $s\ge t$. It is not hard to deduce from (A.3), and arguments involving continuity in probability, that the use of this particular modification does not change the law of the Malliavin derivative. However, it makes the following discussion simpler as we can avoid worrying about null sets.

We aim to prove that, under Dalang’s condition (1.5) alone,

$$\begin{aligned} D_{s,z}u(t,x)\ge 0\qquad \hbox { a.s. for every} (t,x)\in (0,\infty )\times {\mathbb {R}}^d. \end{aligned}$$

(A.4)

Once this is proved, one can appeal to the Clark–Ocone formula (3.3) to conclude the theorem; see the proof of Theorem 4.6 for a similar argument.

Choose and fix $(s,z)\in (0,\infty )\times {\mathbb {R}}^d$, and consider a space-time random field $\Phi _{s,z}$ defined via

$$\begin{aligned} \Phi _{s,z}(t,x) := D_{s,z}u(s+t,z+x)\qquad \hbox {for all} t\ge 0 \hbox {and} x\in {\mathbb {R}}^d. \end{aligned}$$

Our claim (A.4) is equivalent to the a.s.-nonnegativity of $\Phi _{s,z}(t,x)$.

Observe (similarly to what we did in the proof of Theorem 3.2), that $\Phi _{s,z}$ solves

$$\begin{aligned} \Phi _{s,z}(t,x)= & {} {\varvec{p}}_t(z+x)\sigma (u(s,z))\\&+ \int _{(0,t)\times {\mathbb {R}}^d} {\varvec{p}}_{t-r}(x-y) H_{s,z}(t,y)\Phi _{s,z} (r,y)\,\eta _{s,z}(\mathrm {d}r\,\mathrm {d}y), \end{aligned}$$

for $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$, where $\eta _{s,z}$ is the same shifted Gaussian noise that arose in the course of the proof of Theorem 3.2, and

$$\begin{aligned} H_{s,z}(t,y) = \sigma '\left( u(s+t,z+y)\right) , \end{aligned}$$

also as in the proof of Theorem 3.2. In other words, conditional on the sigma-algebra ${\mathcal {F}}_s$, the random field $\Phi _{s,z}$ solves the SPDE,

$$\begin{aligned} \partial _t\Phi _{s,z}= & {} \tfrac{1}{2}\Delta \Phi _{s,z} + \Phi _{s,z} H_{s,z} \eta _{s,z} \quad \text {subject to }\nonumber \\ \Phi _{s,z}(0,x)= & {} \sigma (u(s,z))\delta _0(x). \end{aligned}$$

(A.5)

This is a standard Walsh-type SPDE. In order to understand why this is the case, note first that u(s, z) is ${\mathcal {F}}_s$-measurable and $\eta _{s,z}$ is independent of ${\mathcal {F}}_s$; see also the proof of Lemma 3.2. In order to complete the assertion about (A.5) being a standard Walsh-type SPDE we note that $M_t(A) := \int _{(0,t)\times A}H_{s,z}(r,y)\,\eta _{s,z}(\mathrm {d}r\,\mathrm {d}y)$ defines a martingale measure in the sense of Walsh. Hence, we view (A.5) conditionally on ${\mathcal {F}}_s$ to see that it is a reformulation of the Walsh SPDE,

$$\begin{aligned} \partial _t \Phi _{s,z} = \tfrac{1}{2}\Delta \Phi _{s,z} + \Phi _{s,z}{\dot{M}}_t(x) \quad \text {subject to}\quad \Phi _{s,z}(0,x)=\sigma (u(s,z))\delta _0(x), \end{aligned}$$

also viewed conditionally on ${\mathcal {F}}_s$. In fact, we can interpret (A.5) as a parabolic Anderson model with respect to the martingale measure M. Because of (1.11), and since $\sigma \in \text {Lip}$, M is in fact a worthy martingale measure because $H_{s,z}$ is bounded; see Walsh [38].

For every $\varepsilon \in (0,1)$ let $\eta _{s,z}^\varepsilon$ denote the Gaussian noise

$$\begin{aligned} \eta _{s,z}^\varepsilon (t,x)\,\mathrm {d}t := \int _{{\mathbb {R}}^d}{\varvec{p}}_\varepsilon (x-y)\,\eta _{s,z}(\mathrm {d}t\,\mathrm {d}y). \end{aligned}$$

The spatial correlation measure $f_\varepsilon$ of $\eta _{s,z}^\varepsilon$ is in fact a function [not just a measure] and is given by $f_\varepsilon ={\varvec{p}}_{2\varepsilon }*f.$ Because f is a finite measure, it follows that $f_\varepsilon$ satisfies the extended Dalang condition (1.6) with $\alpha =1$. Let $\Phi _{s,z}^\varepsilon$ denote the unique solution to the SPDE,

$$\begin{aligned} \partial _t \Phi _{s,z}^\varepsilon = \tfrac{1}{2} \Delta \Phi _{s,z}^\varepsilon + (\sigma '\circ u)\Phi _{s,z}^\varepsilon \eta ^{\varepsilon }_{s,z} \quad \text {subject to}\quad \Phi _{s,z}(0,x)=\sigma (u(s,z))\delta _0(x). \end{aligned}$$

This is the same SPDE as (A.5), but with respect to the mollified version $\eta _{s,z}^\varepsilon$ of the Gaussian noise $\eta _{s,z}$ in place of $\eta _{s,z}$. Because $f_\varepsilon$ satisfies (1.6) for some $\alpha >0$, Theorem 1.6 of Chen and Huang [5] [applied conditionally on ${\mathcal {F}}_s$] implies that, if $H_{s,z}$ were a constant [that is, $\sigma (u)\propto u$], then $\Phi _{s,z}^\varepsilon (t,x)>0$ a.s. for every $t>0$ and $x\in {\mathbb {R}}^d$. Careful scrutiny of the proof of Theorem 1.6 of [5], using the boundedness of $H_{s,z}$, yields that $\Phi _{s,z}^\varepsilon >0$ a.s. in all cases; see also Theorem 1.8 of Chen and Huang [6]. Finally, we let $\varepsilon \downarrow 0$ and appeal to Theorem 1.9 of Chen and Huang [5] to see that $\lim _{\varepsilon \rightarrow 0}\Phi _{s,z}^\varepsilon (t,x)= \Phi _{s,z}(t,x)$ in $L^2(\Omega )$ for all $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$. These facts, together imply $\Phi _{s,z}(t,x)\ge 0$ a.s. for all $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$, which is another way to write (A.4). This concludes the proof. $\square$