# Gaussian fluctuations for the wave equation
under rough random perturbations

Raluca M. Balan111Corresponding author. University of Ottawa, Department of
Mathematics and Statistics, Ottawa, ON, K1N 6N5, Canada. E-mail address:
<EMAIL_ADDRESS>222Research supported by a grant from the Natural Sciences
and Engineering Research Council of Canada. Jingyu Huang333University of
Birmingham, School of Mathematics, Birmingham, B15 2TT, United Kingdom. E-mail
address<EMAIL_ADDRESS>Xiong Wang444Johns Hopkins University,
Department of Mathematics, Baltimore, MD, 21218, United States. E-mail
address<EMAIL_ADDRESS>Panqiu Xia555Auburn University, Department of
Mathematics and Statistics, Auburn, AL, 36849, United States. E-mail address:
<EMAIL_ADDRESS>Wangjun Yuan666University of Luxembourg, Department of
Mathematics, L-4364 Esch-sur-Alzette, Luxembourg. E-mail address:
<EMAIL_ADDRESS>
###### Abstract

In this article, we consider the stochastic wave equation in spatial dimension
$d=1$, with linear term $\sigma(u)=u$ multiplying the noise. This equation is
driven by a Gaussian noise which is white in time and fractional in space with
Hurst index $H\in(\frac{1}{4},\frac{1}{2})$. First, we prove that the solution
is strictly stationary and ergodic in the spatial variable. Then, we show that
with proper normalization and centering, the spatial average of the solution
converges to the standard normal distribution, and we estimate the rate of
this convergence in the total variation distance. We also prove the
corresponding functional convergence result.

Mathematics Subject Classifications (2020): Primary 60H15; Secondary 60H07,
60G15, 60F05

Keywords: stochastic wave equation, rough noise, Malliavin calculus, Stein’s
method

###### Contents

  1. 1 Introduction
  2. 2 Preliminary estimates
    1. 2.1 Properties of the solution
    2. 2.2 (HAM) with delta initial velocity
    3. 2.3 Malliavin derivatives and (HAM) with the delta initial velocity
    4. 2.4 Moment estimates for the Malliavin derivatives
  3. 3 Proofs of the main results
    1. 3.1 Spatial Ergodicity—Proof of Theorem 1.1
    2. 3.2 Limiting covariance—Proof of Theorem 1.2
    3. 3.3 Quantitative CLT—Proof of Theorem 1.3
    4. 3.4 Functional CLT—Proof of Theorem 1.4

##  1 Introduction

The study of stochastic partial differential equations (SPDEs) is an active
research area in stochastic analysis which has been growing steadily in the
last 40 years, using two approaches: the semigroup approach developed by Da
Prato and Zabczyk in [24], and the random field approach initiated in Walsh’s
lecture notes [61]. The two approaches rely on different infinite-dimensional
extensions of Itô’s martingale theory of stochastic integration, and lead to
different concepts of solutions. A comparison between the two approaches can
be found in [22]. Classical equations which have been studied using the random
field approach are: the stochastic heat equation (SHE), and the stochastic
wave equation (SWE). When these equations are perturbed by a space-time
Gaussian white noise, random field solutions exist only in spatial dimension
$d=1$. A systematic study of SPDEs in higher dimensions was initiated by
Dalang in the seminal article [20], by considering a spatially-homogeneous
Gaussian noise with spatial covariance given by a non-negative-definite
function $\gamma:\mathbb{R}^{d}\to[0,\infty]$. A typical example is the Riesz
kernel $\gamma(x)=|x|^{-\beta}$ with $\beta\in(0,d)$. The case
$\gamma=\delta_{0}$, where $\delta_{0}$ is the Dirac distribution at $0$,
corresponds formally to the white noise in space. Subsequent investigations
revealed that the solutions to these equations have many interesting
properties, such as: intermittency [29], Hölder continuity [23], strict
positivity [17], dense blow-up [13], to mention just a few. Recent
investigations focus on relaxing the conditions on the coefficients of the
equation, as for instance in [21, 30, 58].

In the 1990’s, fractional Brownian motion (fBm) became a popular model for the
noise in various problems in stochastic analysis. Recall that a fBm is a zero-
mean Gaussian process $(B_{x}^{(H)})_{x\in\mathbb{R}}$ with covariance

$\mathbb{E}[B_{x}^{(H)}B_{y}^{(H)}]=\frac{1}{2}(|x|^{2H}+|y|^{2H}-|x-y|^{2H})\eqqcolon
R_{H}(x,y),$

where $H\in(0,1)$ is the Hurst index. If $H=1/2$, the fBm is a Brownian
motion. The paths of fBm are Hölder continuous of order less than $H$, and
hence, are smoother or rougher than the Brownian paths, depending on whether
$H>1/2$ or $H<1/2$. In the “regular” case $H>1/2$, the covariance of fBm can
be written as:

$R_{H}(x,y)=H(2H-1)\int_{0}^{x}\int_{0}^{y}|u-v|^{2H-2}dudv,$ (1.1)

using the Riesz kernel $\gamma(x)=H(2H-1)|x|^{2H-2}$, which is the second
derivative $(|x|^{2H})^{\prime\prime}$ in the sense of distributions. In the
“rough” case $H<1/2$, (1.1) does not hold since $(|x|^{2H})^{\prime\prime}$ is
not a function. In this case, it is useful to work with the spectral
representation:

$R_{H}(x,y)=c_{H}\int_{\mathbb{R}}\mathcal{F}1_{[0,x]}(\xi)\overline{\mathcal{F}1_{[0,y]}(\xi)}|\xi|^{1-2H}d\xi,$
(1.2)

where $c_{H}=\frac{\Gamma(2H+1)\sin(\pi H)}{2\pi}$ and
$\mathcal{F}\varphi(\xi)=\int_{\mathbb{R}}e^{-i\xi x}\varphi(x)dx$ is the
Fourier transform.

The fBm is not a semi-martingale, and Itô calculus cannot be used. Two methods
were proposed for developing stochastic analysis with respect to fBm, using
either (i) Malliavin calculus, or (ii) pathwise integration (which exploits
the Hölder continuity of the paths). Pioneer works in this direction are: [1,
25, 52, 62]. Method (i) is relevant for the present article, and can be
explained briefly as follows: we endow the space $\mathcal{E}$ of linear
combinations of indicator functions of the form $1_{[0,x]}$ with the inner
product $\langle 1_{[0,x]},1_{[0,y]}\rangle_{\mathcal{P}_{0}}=R_{H}(x,y)$, so
that the map $1_{[0,x]}\mapsto B_{x}^{(H)}\in L^{2}(\Omega)$ becomes an
isometry; then, the closure $\mathcal{P}_{0}$ of $\mathcal{E}$ is the domain
of the Wiener integral with respect to $B^{(H)}$. In [42], Jolis proved that
$\mathcal{P}_{0}$ coincides with the fractional Sobolev space
$W^{\frac{1}{2}-H,2}(\mathbb{R})$, and therefore it is a space of
distributions if $H>1/2$, and a space of functions if $H<1/2$. An alternative
representation in [27] for the norm $\|\cdot\|_{\mathcal{P}_{0}}$ was obtained
in the case $H<1/2$, namely:

$\|\varphi\|_{\mathcal{P}_{0}}^{2}=C_{H}\int_{\mathbb{R}^{2}}|\varphi(x)-\varphi(y)|^{2}|x-y|^{2H-2}dxdy,$
(1.3)

where $C_{H}=\frac{H(1-2H)}{2}$. Relation (1.3) is called the Gagliardo
representation and is very useful for problems in stochastic analysis, and in
particular for the present article.

SPDEs with colored noise in time have been considered for the first time in
[51]. Since then, this area has been growing steadily. However, the basic
question of existence of solutions to (SHE) or (SWE) with colored (or
fractional) noise in time is still an open problem in the case when the noise
is multiplied by a Lipschitz function $\sigma(u)$ of the solution. The only
case when it is known that these equations have unique solutions is the linear
case, $\sigma(u)=u$. This case, which is known in the literature as the
parabolic Anderson model (PAM) for the heat equation, respectively the
hyperbolic Anderson model (HAM) for the wave equation, is studied using tools
from Malliavin calculus, since the solution has an explicit Wiener chaos
expansion. This method was initiated by Hu and Nualart in [36]. Various
properties of the solution have been developed in [18, 32, 37] for (PAM),
respectively [4, 5] for (HAM), to name just a few of the recent references.

In the present article, we consider the (HAM) driven by a Gaussian noise
$\dot{W}$ which is white in time and fractional in space with Hurst index
$H\in(\frac{1}{4},\frac{1}{2})$:

$\displaystyle\begin{dcases}\dfrac{\partial^{2}u}{\partial
t^{2}}(t,x)=\dfrac{\partial^{2}u}{\partial
x^{2}}(t,x)+u(t,x)\dot{W}(t,x),&(t,x)\in\mathbb{R}_{+}\times\mathbb{R},\\\
u(0,x)=1,\ \dfrac{\partial u}{\partial t}(0,x)=0,&\forall
x\in\mathbb{R}.\end{dcases}$ (1.4)

Formally, $\dot{W}$ is a zero-mean Gaussian noise with covariance

$\mathbb{E}[\dot{W}(t,x)\dot{W}(s,y)]=\delta_{0}(t-s)\gamma(x-y),\quad\mbox{where}\quad\gamma(x)=(|x|^{2H-2})^{\prime\prime}.$

Rigorously, $W=\\{W(\varphi);\varphi\in\mathcal{D}\\}$ is a zero-mean Gaussian
process defined on a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and indexed by the set $\mathcal{D}$ of
infinitely differentiable functions on $\mathbb{R}_{+}\times\mathbb{R}$, with
compact support. The covariance of $W$ is inspired by (1.2) (in which we
replace $1_{[0,x]}$ and $1_{[0,y]}$ by smooth functions), and is given by:

$\displaystyle\mathbb{E}[W(\varphi)W(\psi)]=c_{H}\int_{0}^{\infty}\int_{\mathbb{R}}\mathcal{F}\varphi(t,\cdot)(\xi)\overline{\mathcal{F}\psi(t,\cdot)(\xi)}|\xi|^{1-2H}d\xi
dt\eqqcolon\langle\varphi,\psi\rangle_{\mathcal{H}}.$ (1.5)

We denote by $\mathcal{H}$ the Hilbert space defined as the completion of
$\mathcal{D}$ with respect to the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$. Then the map
$\mathcal{D}\ni\varphi\mapsto W(\varphi)\in L^{2}(\Omega)$ becomes an isometry
which can be extended to $\mathcal{H}$. The process
$\\{W(\varphi);\varphi\in\mathcal{H}\\}$ is an isonormal Gaussian process, as
defined in Malliavin calculus (see [50]), and $\mathcal{H}$ is isomorphic to
$L^{2}(\mathbb{R}_{+};\mathcal{P}_{0})$. Indicator functions of the form
$1_{[0,t]\times[0,x]}$ lie in $\mathcal{H}$ and the process
$\\{W_{t}(x)=W(1_{[0,t]\times[0,x]})\\}_{x\in\mathbb{R}}$ has the same
distribution as $\sqrt{t}B^{(H)}$, since its covariance matches (1.2).

A predictable process $u=\\{u(t,x);t\geq 0,x\in\mathbb{R}\\}$ is a (mild)
solution to equation (1.4) if it satisfies the following integral equation:

$u(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}}G_{t-s}(x-y)u(s,y)W(ds,dy),$ (1.6)

where $G_{t}$ is the fundamental solution to the deterministic wave equation
on $\mathbb{R}_{+}\times\mathbb{R}$:

$G_{t}(x)\coloneqq\dfrac{1}{2}1_{\\{|x|<t\\}}.$ (1.7)

The integral on the right-hand side of (1.6) is an Itô integral, which
coincides with the Skorohod integral, as observed in [3].

The existence and weak intermittency of the solution to (1.4) was proved in
[2], respectively [3]. In [45], the existence of solution was studied in a
more general scenario, where $u$ is replaced by a Lipschitz function
$\sigma(u)$. The existence of the solution to (PAM) with the same noise
$\dot{W}$ as above was obtained in [34], while the exact asymptotic behaviour
of its moments was established in [39]. The existence and Hölder continuity of
the solution to (PAM) with space-time fractional noise of indices $H_{0}>1/2$
in time and $H<1/2$ in space was obtained in [35], under the condition
$H_{0}+H>3/4$. The same problem for (HAM) was studied in [59] under the
assumption $H\in(\frac{1}{4},\frac{1}{2})$. The (SHE) with the same noise and
a general Lipschitz function $\sigma(u)$ multiplying the noise was studied in
[33], under the restriction $\sigma(0)=0$; this condition was later removed in
[38].

In all these references, the noise is spatially-homogeneous, i.e. it is
invariant under translations. This property is transmitted to the solution $u$
in the form of strict stationarity of the process
$\\{u(t,x)\\}_{x\in\mathbb{R}}$. Without considerable effort, it is possible
to prove that this process is also ergodic. The spatial ergodicity of the
solution to an SPDE was proved for the first time in [14] for the SHE with
spatially homogeneous Gaussian noise (white noise in time), and a Lipshitz
function $\sigma(u)$ multiplying the noise.

In the recent years, there has been a lot of interest in examining the
asymptotic behaviour of the spatial average:

$F_{R}(t)=\int_{-R}^{R}\big{(}u(t,x)-1\big{)}dx.$

Since $\\{u(t,x)\\}_{x\in\mathbb{R}}$ is strictly stationary and ergodic, by
Brirkoff and von Neumann mean ergodic theorem, the following law of large
numbers holds:

$\frac{F_{R}(t)}{R}\to 0\quad\mbox{a.s. and in $L^{2}(\Omega)$},\quad\mbox{as
$R\to\infty$}.$

A natural question is to investigate if $F_{R}(t)$ satisfies also a central
limit theorem. For this, a novel technique was initiated in [40], which
combines Stein’s method for normal approximation with tools from Malliavin
calculus. This method was originally developed for (SHE) with space-time white
noise, and has been rapidly extended to other models. The paramount result is
the Quantitative Central Limit Theorem (QCLT), which gives an estimate for the
total variation distance $d_{TV}$ between $F_{R}(t)/\sigma_{R}(t)$ and a
standard normal random variable $Z$, as a quantifier for the speed of
convergence in distribution, when $R\to\infty$. We recall that the total
variation distance between random variables $X$ and $Y$ is given by:

$d_{\rm TV}(X,Y)=\sup_{B\in\mathcal{B}(\mathbb{R})}|\mathbb{P}(X\in
B)-\mathbb{P}(Y\in B)|,$

where $\mathcal{B}(\mathbb{R})$ stands for the collections of all Borel
subsets of $\mathbb{R}$. The QCLT is closely related to a study of the order
of magnitude of $\sigma_{R}^{2}(t)$, and can be extended to functional
convergence.

The following table summarizes the most important contributions to date,
related to the problem of QCLT for solutions to SPDEs with spatially-
homogeneous Gaussian noise, which can be white in time, fractional in time
with index $H_{0}>1/2$, or time-independent. In some of these references, the
temporal covariance of the noise can be more general, given by a non-negative-
definite function $\gamma_{0}:\mathbb{R}\to[0,\infty]$. For the sake of
conciseness, we present only the fractional noise in time, when
$\gamma_{0}(t)=|t|^{2H_{0}-2}$ with $H_{0}\in(\frac{1}{2},1)$. In this table,
$\gamma(x)$ denotes the spatial covariance of the noise,
$\sigma_{R}^{2}=\sigma_{R}^{2}(1)$, and $d_{TV}=d_{TV}(F_{R}/\sigma_{R},Z)$
where $F_{R}=F_{R}(1)$. The notation $a_{R}\sim b_{R}$ indicates that
$a_{R}/b_{R}\to C$ when $R\to\infty$, and $a_{R}\lesssim b_{R}$ means that
$a_{R}\leq Cb_{R}$, where $C>0$ is a constant.

Noise | (SHE) | (SWE)
---|---|---
White | Regular in space | Regular in space
in time |  $\bullet$ [40]: $d=1,\gamma=\delta_{0}\hskip 48.36967pt$ |  $\bullet$ [26]: $d=1,\gamma(x)=|x|^{2H-2},H\in[\frac{1}{2},1)$
($H_{0}=\frac{1}{2}$) | $\sigma_{R}^{2}\sim R,d_{TV}\lesssim R^{-1/2}$ | $\sigma_{R}^{2}\sim R^{2H},d_{TV}\lesssim R^{H-1}$
$\sigma$ arb. |  $\bullet$ [41]: $\gamma(x)=|x|^{-\beta},\beta\in(0,d)$ |  $\bullet$ [11]: $d=2,\gamma(x)=|x|^{-\beta},\beta\in(0,2)\hskip 14.22636pt$
| $\sigma_{R}^{2}\sim R^{2d-\beta},d_{TV}\lesssim R^{-\beta/2}$ | $\sigma_{R}^{2}\sim R^{4-\beta},d_{TV}\lesssim R^{-\beta/2}$
|  |  $\bullet$ [57]: $d\leq 2,\gamma\in L^{1}(\mathbb{R}^{d})\hskip 71.13188pt$
|  | $\sigma_{R}^{2}\sim R^{d},d_{TV}\lesssim R^{-d/2}$
|  Rough in space [54] | Rough in space
| $d=1,H\in(\frac{1}{4},\frac{1}{2})$ | $d=1,H\in(\frac{1}{4},\frac{1}{2})$
| $\sigma_{R}^{2}\sim R,d_{TV}\lesssim R^{-1/2}$ | Open Problem 1
Fractional | Regular in space [54, 55], | Regular in space [4]
in time |  $\bullet$ $\gamma\in L^{1}(\mathbb{R}^{d})\hskip 85.35826pt$ |  $\bullet$ $d\leq 2,\gamma\in L^{1}(\mathbb{R}^{d})\hskip 99.58464pt$
$H_{0}\in(\frac{1}{2},1)$ | $\sigma_{R}^{2}\sim R^{d},d_{TV}\lesssim R^{-d/2}$ | $\sigma_{R}^{2}\sim R^{d},d_{TV}\lesssim R^{-d/2}$
$\sigma(u)=u$ |  $\bullet$ $\gamma(x)=|x|^{-\beta},\beta\in(0,d)\hskip 25.60747pt$ |  $\bullet$ $d\leq 2,\gamma(x)=|x|^{-\beta},\beta\in(0,d)\hskip 36.98857pt$
| $\sigma_{R}^{2}\sim R^{2d-\beta},d_{TV}\lesssim R^{-\beta/2}$ | $\sigma_{R}^{2}\sim R^{2d-\beta},d_{TV}\lesssim R^{-\beta/2}$
|  Rough in space [53, 54] | Rough in space
| $d=1,H<\frac{1}{2},H_{0}+H>\frac{3}{4}$ | $d=1,H\in(\frac{1}{4},\frac{1}{2})$
| $\sigma_{R}^{2}\sim R,d_{TV}\lesssim R^{-1/2}$ | Open Problem 2
Time- |  Regular in space [7] |  Regular in space [6]
independent |  $\bullet$ $\gamma\in L^{1}(\mathbb{R}^{d})\hskip 85.35826pt$ |  $\bullet$ $d\leq 2,\gamma\in L^{1}(\mathbb{R}^{d})\hskip 93.89409pt$
($H_{0}=1$) | $\sigma_{R}^{2}\sim R^{d},d_{TV}\lesssim R^{-d/2}$ | $\sigma_{R}^{2}\sim R^{d},d_{TV}\lesssim R^{-d/2}$
$\sigma(u)=u$ |  $\bullet$ $\gamma(x)=|x|^{-\beta},\beta\in(0,d)\hskip 22.76219pt$ |  $\bullet$ $d\leq 2,\gamma(x)=|x|^{-\beta},\beta\in(0,d)\hskip 34.1433pt$
|  Rough in space [7] |  Rough in space [8]
| $d=1,H\in(\frac{1}{4},\frac{1}{2})$ | $d=1,H\in(\frac{1}{4},\frac{1}{2})$
| $\sigma_{R}^{2}\sim R,d_{TV}\lesssim R^{-1/2}$ | $\sigma_{R}^{2}\sim R,d_{TV}\lesssim R^{-1/2}$

In the case of the white noise in time (regular in space), reference [16]
gives the CLT for a function $g(u(t,x))$ of the spatial average of the
solution to (SHE), while references [15, 44] studied the QCLT problem for a
normalized version of the solution to (PAM) with delta initial condition. In
the case of the (SWE) with white noise in time, the spatial ergodicity of the
solution was proved in [56] for dimensions $d\leq 3$, while [28] proved the
convergence in the Wasserstein distance for the spatial average of the
solution, in dimension $d=3$.

In this paper, we study the Open Problem 1 mentioned above, in the case
$\sigma(u)=u$. We first show that the solution is strictly stationary and
ergodic in the space variable, then we prove that $\sigma_{R}^{2}\sim R$. The
major effort is dedicated to the proof of QCLT. For this, we use the same
method as in [54] for (PAM), which relies on a second-order Poincaré
inequality (Proposition 2.4 of [54]). Due to the Gagliardo representation
(1.3) of the norm $\|\cdot\|_{\mathcal{P}_{0}}$, we encounter the problem of
estimating the fourth moments of the increments of the Malliavin derivative
$Du(t,x)$, and of the rectangular increments of the second Malliavin
derivative $D^{2}u(t,x)$. In the case of (PAM), these estimates are obtained
in [54] using highly non-trivial methods, that cannot be applied for (HAM).

Our method is simpler than that of [54], and relies on the following key
relation

$D_{r,z}u(t,x)=u(r,z)v^{(r,z)}(t,x),$ (1.8)

between $Du$ and the solution $v=v^{(r,z)}$ of the integral equation (of
Volterra type):

$v(t,x)=G_{t-r}(x-z)+\int_{r}^{t}\int_{\mathbb{R}}G_{t-s}(x-y)v(s,y)W(ds,dy),\quad
t\geq r,x\in\mathbb{R}.$ (1.9)

Relation (1.8) is valid for the solution $u$ of any SPDE with white noise in
time, constant initial condition, and linear function $\sigma(u)=u$
multiplying the noise. But it may not hold for equations with colored noise in
time, or time-independent noise. (Nevertheless, in the case of (HAM) with
time-independent noise, it is still possible to derive an indirect argument
relating $Du$ and $v$ which leads to estimates for the moments of $Du$ and its
increments; see [8, Theorem 4.1].) Key properties of $v^{(r,z)}$, which lie at
the core of our methods, are:
(i) the moments of $v^{(r,z)}(t,x)$ are uniformly bounded in $(r,z,t,x)$ (see
[8, Example B.2]);
(ii) $v^{(r,z)}$ satisfies the following relation (see the proof of Lemma 2.4
below):

$\displaystyle v^{(r,z)}(t,x)=2G_{t-r}(x-z)v^{(r,z)}(t,x);$ (1.10)

(iii) the increments of $v^{(r,z)}$ have translation-invariance properties
(see Lemma 2.5 below).
Properties (i) and (ii) may not hold for other models, such as (PAM); see [8,
Remark B.3].

We are now ready to state the main results of the present article.

######  Theorem 1.1 (Spatial ergodicity).

For any $t>0$, the process $\\{u(t,x)\\}_{x\in\mathbb{R}}$ is strictly
stationary and ergodic.

######  Theorem 1.2 (Limiting covariance).

For any $t>0$ and $s>0$

$\lim_{R\to\infty}\frac{\mathbb{E}[F_{R}(t)F_{R}(s)]}{R}=K(t\wedge s),$ (1.11)

where

$\displaystyle K(t)=4\pi\sum_{n\geq
2}C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n-1}}$
$\displaystyle\prod_{j=1}^{n-1}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}$
$\displaystyle\times\prod_{j=1}^{n-1}|\eta_{j}-\eta_{j-1}|^{1-2H}|\eta_{n-1}|^{1-2H}d\boldsymbol{\eta}_{n-1}d\boldsymbol{t}_{n},$

with $\boldsymbol{\eta}_{n-1}=(\eta_{1},\dots,\eta_{n-1})$,
$\boldsymbol{t}_{n}=(t_{1},\ldots,t_{n})$ and by convention $\eta_{0}=0$. In
particular, $\sigma_{R}^{2}(t)/R\to K(t)>0$ as $R\to\infty$.

######  Theorem 1.3 (QCLT).

For any $t>0$,

$d_{\rm TV}\left(\frac{F_{R}(t)}{\sigma_{R}(t)},Z\right)\leq
C_{t,H}^{(2)}R^{-1/2},$

where $C_{t,H}^{(2)}>0$ is a constant depending on $t$ and $H$.

######  Theorem 1.4 (Functional CLT).

The process $\\{R^{-1/2}F_{R}(t);t\geq 0\\}$ converges in distribution as
$R\to\infty$, to a zero-mean Gaussian process $\\{\mathcal{G}(t);t\geq 0\\}$
with covariance

$\mathbb{E}[\mathcal{G}(t)\mathcal{G}(s)]=K(t\wedge s).$

Theorem 1.1 is derived from an ergodicity criterion credited to Maruyama [46].
By combining this criterion with the Gaussian-Poincaré inequality (see e.g.,
[31, 47]), the spatial ergodicity of the solution to (SHE) was investigated in
[14]. For the solution to (SWE), the spatial ergodicity in dimensions $d\leq
3$ was proved in [56]. However, in the paper [56], the spatial correlation
function has to be a function (or a measure), rendering the result
inapplicable in our case, when the correlation is a distribution.
Consequently, new methods must be employed to establish the validity of
Theorem 1.1.

The proof of Theorem 1.3 is based on the second-order Gaussian-Poincaré
inequality (referenced as in e.g., [12, 49, 60]) following the strategy
initially discovered in [4], and further applied/developed in [6, 7, 8, 54].
It is important to note that several estimates for (PAM) with rough noise in
space obtained in [54] are precise but not essential for achieving the desired
goal. Those estimates unnecessarily increase complexity and cannot be applied
to (HAM). By thoroughly analyzing the implementation of the second-order
Gaussian-Poincaré inequality in the spatial average of SPDE’s, certain
unnecessary computations were successfully eliminated in [8]. This reduction
in complexity significantly streamlined the proof and marked the final stage
of the progress made in this paper.

This article is organized as follows. In Section 2, we include the preliminary
results, leading to the estimates for Malliavin derivatives of the solution,
which are essential for our developments. Section 3 contains the proofs of
Theorems 1.1,1.2,1.3 and 1.4.

Throughout this paper, we denote by $\|\cdot\|_{p}$ the $L^{p}(\Omega)$-norm
of a random variable on the probability space
$(\Omega,\mathcal{F},\mathbb{P})$. For any positive integer $n$, we make use
of the notation $\boldsymbol{x}_{n}\coloneqq(x_{1},\dots,x_{n})$ for an
element in $\mathbb{R}^{n}$, and for any $t\in\mathbb{R}$, $T_{n}(t)$ stands
for a subset of $[0,t]^{n}$:
$T_{n}(t)\coloneqq\\{\boldsymbol{t}_{n}=(t_{1},\ldots,t_{n});0<t_{1}<\ldots<t_{n}<t\\}$.

##  2 Preliminary estimates

In this section, we present an overview of (HAM) driven by noise $W$ as above,
including relevant preliminaries. We reference existing literature for certain
results, while also introducing a few novel estimates supported by detailed
proofs.

The next two identities will be used, whose proof follows from elementary
calculus, and thus omitted. Let $t>0$, then the following two identities hold.
For any $\alpha\in(-1,1)$,

$\int_{\mathbb{R}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|\xi|^{\alpha}d\xi=C_{\alpha}t^{1-\alpha},$
(2.1)

where $C_{\alpha}>0$ is a constant depending on $\alpha$. For any
$\alpha_{1}>-1,\ldots,\alpha_{n}>-1$,

$\int_{T_{n}(t)}\prod_{j=1}^{n}(t_{j+1}-t_{j})^{\alpha_{j}}d\boldsymbol{t}_{n}=\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}+1)}{\Gamma(|\alpha|+n+1)}t^{|\alpha|+n},$
(2.2)

where we use the convention $t_{n+1}=t$, and we denote
$|\alpha|=\sum_{j=1}^{n}\alpha_{j}$.

###  2.1 Properties of the solution

In this section, we include some properties of the solution that will be
needed in the sequel, including the spatial strict stationarity mentioned in
Theorem 1.1.

As discussed in [59], the solution to equation (1.4) has the Wiener chaos
expansion:

$u(t,x)=1+\sum_{n\geq 1}I_{n}(f_{n}(\cdot,t,x)),$

where $I_{n}$ is the multiple integral with respect to $W$ and the kernel
$f_{n}(\cdot,t,x)$ is given by:

$\displaystyle f_{n}(\boldsymbol{t}_{n},\boldsymbol{x}_{n},t,x)=$
$\displaystyle f_{n}(t_{1},\dots,t_{n},x_{1},\dots,x_{n},t,x)$
$\displaystyle\coloneqq$ $\displaystyle G_{t-t_{n}}(x-x_{n})\times\dots\times
G_{t_{2}-t_{1}}(x_{2}-x_{1})\mathbf{1}_{T_{n}(t)}(\boldsymbol{t}_{n})$

In this case,

$\mathbb{E}|u(t,x)|^{2}=\sum_{n\geq
1}n!\|\widetilde{f}_{n}(\cdot,t,x)\|_{\mathcal{H}_{0}^{\otimes n}}^{2},$

where $\widetilde{f}_{n}(\cdot,t,x)$ is the symmetrization of
$f_{n}(\cdot,t,x)$:

$\widetilde{f}_{n}(\boldsymbol{t}_{n},\boldsymbol{x}_{n},t,x)=\frac{1}{n!}\sum_{\rho\in
S_{n}}f_{n}(t_{\rho(1)},\ldots,t_{\rho(n)},x_{\rho(1)},\ldots,x_{\rho(n)},t,x),$

with $S_{n}$ denoting the set of all permutations on $\\{1,\dots,n\\}$.

The following result was proved in [2, 3]; see [2, Theorem 3.9] or [3, Theorem
3.5].

######  Lemma 2.1.

For any $H\in(1/4,1/2)$, equation (1.4) has a unique solution. Moreover, for
any $p\geq 2$ and $t\geq 0$,

$\sup_{(r,x)\in[0,t]\times\mathbb{R}}\|u(r,x)\|_{p}<C,$ (2.3)

where $C>0$ is a constant that depends on $(t,p,H)$.

The following result gives the spatial stationarity of $u$.

######  Lemma 2.2.

For any $t>0$, the process $\\{u(t,x)\\}_{x\in\mathbb{R}}$ is strictly
stationary. In particular, for any $x,h\in\mathbb{R}$,

$u(t,x)-u(t,x+h)\stackrel{{\scriptstyle d}}{{=}}u(t,0)-u(t,h).$

###### Proof.

We show that for any $m\geq 1$ and for any $z_{1},\ldots,z_{m},h\in\mathbb{R}$

$\big{(}u(t,z_{1}),\ldots,u(t,z_{m})\big{)}\stackrel{{\scriptstyle
d}}{{=}}\big{(}u(t,z_{1}+h),\ldots,u(t,z_{m}+h)\big{)}.$

For any $j=1,\ldots,m$, the variable $u(t,z_{j}+h)$ has the chaos expansion:

$\displaystyle u(t,z_{j}+h)=\sum_{n\geq 1}\int_{T_{n}(t)\times\mathbb{R}^{n}}$
$\displaystyle
G_{t-t_{n}}(z_{j}+h-x_{n})\prod_{i=1}^{n-1}G_{t_{i+1}-t_{i}}(x_{i+1}-x_{i})$
$\displaystyle\times W(dt_{1},dx_{1})\ldots W(dt_{n},dx_{n}),$ (2.4)

where $T_{n}(t)=\\{0<t_{1}<\ldots<t_{n}<t\\}$. Let
$W^{(h)}=\\{W^{(h)}(\varphi);\varphi\in\mathcal{D}(\mathbb{R}_{+}\times\mathbb{R})\\}$
be the shifted noise, where $W^{(h)}(\varphi)=W(\varphi(\cdot-h))$. Performing
the formal change of variables $y_{i}=x_{i}-h$ for $i=1,\ldots,n$, we see that
the multiple integral above is equal to

$\displaystyle\int_{T_{n}(t)\times\mathbb{R}^{n}}G_{t-t_{n}}(z_{j}-y_{n})\prod_{i=1}^{n-1}G_{t_{i+1}-t_{i}}(y_{i+1}-y_{i})W^{(h)}(dt_{1},dy_{1})\ldots
W^{(h)}(dt_{n},dy_{n}).$

Since $W$ is spatially homogeneous, $W^{(h)}\stackrel{{\scriptstyle
d}}{{=}}W$. Therefore, the vector of dimension $m$ whose $j$-th component is
given by the series (2.1) has the same distribution as the vector whose $j$-th
component is $u(t,z_{j})$. ∎

######  Lemma 2.3.

For any $t>0$ and $p\geq 2$, we have

$\displaystyle\sup_{0\leq s\leq
t}\int_{\mathbb{R}}\big{\|}u(s,0)-u(s,h)\big{\|}_{p}^{2}\,|h|^{2H-2}dh\leq C,$

where $C>0$ is a constant that depends on $(t,H,p)$.

###### Proof.

Denote by $C$ a constant that depends on $(t,H,p)$ and may be different in
each of its appearances. Using the uniform bound for $\|u(s,x)\|_{p}$ given by
(2.3), we have:

$\int_{|h|>1}\|u(s,0)-u(s,h)\|_{p}|h|^{2H-2}dh\leq
C\int_{|h|>1}|h|^{2H-2}dh<\infty.$

It remains to treat the integral over the set $|h|\leq 1$. By using the chaos
expansion (2.1) for $u(t,0)-u(t,x)$, and hypercontractivity (see e.g., [48,
Corollary 2.8.14]), for any $p\geq 2$,

$\|u(s,0)-u(s,h)\|_{p}\leq\sum_{n\geq
1}(p-1)^{n/2}\|I_{n}({f}_{n,h}(\cdot,0,s))\|_{2}=\sum_{n\geq
1}(p-1)^{n/2}[J_{n,h}(s)]^{1/2}$ (2.5)

where $J_{n,h}(s)\coloneqq n!\|{f}_{n,h}(\cdot,0,s)\|_{\mathcal{H}^{\otimes
n}}^{2}$. It follows from (1.5) that

$\displaystyle J_{n,h}(s)$
$\displaystyle=c_{H}^{n}\int_{T_{n}(s)}\int_{\mathbb{R}^{n}}\big{|}\mathcal{F}f_{n,h}(\boldsymbol{t}_{n},\cdot,s,x)(\xi_{1},\dots,\xi_{n})\big{|}^{2}\prod_{j=1}^{n}|\xi_{j}|^{1-2H}d\boldsymbol{\xi}_{n}d\boldsymbol{t}_{n}$
$\displaystyle=c_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\big{|}1-e^{-i(\xi_{1}+\dots+\xi_{n})h}\big{|}^{2}\prod_{j=1}^{n}\big{|}\mathcal{F}G_{t_{j+1}-t_{j}}(\xi_{1}+\dots+\xi_{j})\big{|}^{2}\prod_{j=1}^{n}|\xi_{j}|^{1-2H}d\boldsymbol{\xi}_{n}d\boldsymbol{t}_{n}$
$\displaystyle=c_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\big{|}1-e^{-i\eta_{n}h}\big{|}^{2}\prod_{j=1}^{n}\big{|}\mathcal{F}G_{t_{j+1}-t_{j}}(\eta_{j})\big{|}^{2}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n},$

where by convention $t_{n+1}=s$ and $\eta_{0}=0$. Taking account of the
(in)equalities $|a+b|^{1-2H}\leq|a|^{1-2H}+|b|^{1-2H}$ for all
$a,b\in\mathbb{R}$, and

$x_{1}\prod_{j=2}^{n}(x_{j}+x_{j-1})=\sum_{\boldsymbol{\alpha}_{n}\in
A_{n}}x_{j}^{\alpha_{j}},\quad\text{for all
}x_{1},\dots,x_{n}\in\mathbb{R}_{+},$

where $A_{n}$ is a set of multi-indices
$\boldsymbol{\alpha}_{n}=(\alpha_{1},\ldots,\alpha_{n})$ such that
$\alpha_{1}\in\\{1,2\\}$, $\alpha_{n}\in\\{0,1\\}$,
$\alpha_{2},\ldots,\alpha_{n-1}\in\\{0,1,2\\}$, $\sum_{j=1}^{n}\alpha_{j}=n$
and $\alpha_{j}+\alpha_{j-1}\in\\{1,2,3\\}$ for any $j=1,\ldots,n-1$. Then the
cardinality ${\rm card}(A_{n})=2^{n-1}$ (see [59, Page 7]). Hence

$\displaystyle\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}\leq\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\prod_{j=1}^{n}|\eta_{j}|^{\alpha_{j}},$ (2.6)

where $D_{n}$ is a subset of $A_{n}$ fulfilling $\alpha_{j}=(1-2H)\alpha_{j}$,
$j=1,\dots,n$. It follows that

$\displaystyle J_{n,h}(s)\leq c_{H}^{n}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\int_{T_{n}(t)}$
$\displaystyle\prod_{j=1}^{n-1}\left(\int_{\mathbb{R}}|\mathcal{F}G_{t_{j+1}-t_{j}}(\eta_{j})||\eta_{j}|^{\alpha_{j}}d\eta_{j}\right)$
$\displaystyle\times\left(\int_{\mathbb{R}}|1-e^{-i\eta_{n}h}|^{2}|\mathcal{F}G_{t-t_{n}}(\eta_{n})|^{2}|\eta_{n}|^{\alpha_{n}}d\eta_{n}\right)d\boldsymbol{t}_{n}$

As a result of identity (2.1) and the fact that $|\eta_{n}|^{\alpha_{n}}\leq
1+|\eta_{n}|^{1-2H}$ (since $\alpha_{n}\in\\{0,1-2H\\}$),

$\displaystyle J_{n,h}(s)$ $\displaystyle\leq C^{n}\sum_{\alpha\in
A_{n}}\int_{0}^{t}\left(\int_{\mathbb{R}}|1-e^{-i\eta_{n}h}|^{2}|\mathcal{F}G_{t-t_{n}}(\eta_{n})|^{2}(1+|\eta_{n}|^{1-2H})d\eta_{n}\right)$
$\displaystyle\qquad\qquad\qquad\left(\int_{T_{n-1}(t_{n})}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{1-\alpha_{j}}d\boldsymbol{t}_{n-1}\right)dt_{n}$

Due to inequality (2.2). and the fact that $\Gamma(an+b+1)\geq
c_{a}^{n}(n!)^{a}$ for any $a>0$ and $b\in\mathbb{R}$ with some constant
$c_{a}$ depending only on $a$, it can be proved that

$\int_{T_{n-1}(t_{n})}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{1-a_{j}}d\boldsymbol{t}_{n-1}\leq\frac{C^{n}}{(n!)^{2H+1}}t_{n}^{(2H+1)n-2}.$

Therefore,

$\displaystyle J_{n,h}(s)\leq$
$\displaystyle\frac{C^{n}}{(n!)^{2H+1}}\int_{0}^{t}\int_{\mathbb{R}}|1-e^{-i\eta_{n}h}|^{2}(1+|\eta_{n}|^{1-2H})\frac{\sin^{2}((t-t_{n})|\eta_{n}|)}{|\eta_{n}|^{2}}t_{n}^{(2H+1)n-2}d\eta_{n}dt_{n}$
$\displaystyle\leq$
$\displaystyle\frac{C^{n}}{(n!)^{2H+1}}s^{(2H+1)n-1}\int_{\mathbb{R}}|1-e^{-i\eta_{n}h}|^{2}(1+|\eta_{n}|^{1-2H})\frac{1}{|\eta_{n}|^{2}}d\eta_{n},$

where the last inequality follows from the bound
$\sin^{2}((s-t_{n})|\eta_{n}|)\leq 1$. Using the fact that
$|1-e^{-ix}|^{2}=2(1-\cos x)$ and the identity

$\int_{\mathbb{R}}(1-\cos(\xi
x))|x|^{-\alpha-1}dx=C_{\alpha}|\xi|^{\alpha},\quad\mbox{for any
$\alpha\in(0,2)$},$

we obtain that
$J_{n,h}(s)\leq\frac{C^{n}}{(n!)^{2H+1}}\big{(}|h|+|h|^{2H}\big{)}$. Plugging
this inequality to (2.5), we get

$\|u(s,0)-u(s,h)\|_{p}\leq(|h|^{1/2}+|h|^{H})\sum_{n\geq
1}(p-1)^{n/2}\frac{C^{n}}{(n!)^{H+1/2}}\eqqcolon C(|h|^{1/2}+|h|^{H}).$

This implies

$\displaystyle\int_{|h|\leq 1}\|u(s,x)-u(s,x+h)\|_{p}^{2}|h|^{2H-2}dh\leq
C\int_{|h|\leq 1}\big{(}|h|^{1/2}+|h|^{H}\big{)}^{2}|h|^{2H-2}dh<\infty,$

because $H>1/4$. The proof of this lemma is complete. ∎

###  2.2 (HAM) with delta initial velocity

In this section, we study equation (1.9). We begin by recalling a basic fact
which states that the solution to the deterministic wave equation:

$\frac{\partial^{2}w}{\partial^{2}t}(t,x)=\frac{\partial^{2}w}{\partial^{2}x}(t,x),\quad
t>r,x\in\mathbb{R}$

with initial condition $w(r,x)=u_{0}(x)$ and $\frac{\partial w}{\partial
t}(r,x)=v_{0}(x)$ is given by:

$w(t,x)=(G_{t-r}*v_{0})(x)+\frac{\partial}{\partial t}(G_{t-r}*u_{0})(x).$

When $u_{0}=0$ and $v_{0}=\delta_{z}$ (for fixed $z\in\mathbb{R}$), this
becomes $w(t,x)=G_{t-r}(x-z)$. This observation motivates the definition
below.

Consider the following model:

$\displaystyle\begin{dcases}\dfrac{\partial^{2}v}{\partial
t^{2}}(t,x)=\dfrac{\partial^{2}v}{\partial x^{2}}(t,x)+v(t,x)\dot{W}(t,x),\
t>r,\ x\in\mathbb{R}\\\ v(r,\cdot)=0,\ \dfrac{\partial v}{\partial
t}(r,\cdot)=\delta_{z},\end{dcases}$ (2.7)

We say that a random field
$v^{(r,z)}=\\{v^{(r,z)}(t,x);(t,x)\in[r,\infty)\times\mathbb{R}\\}$ is a
(mild) solution of equation (2.7) if it satisfies the integral equation (1.9).

It can be proved that equation (2.7) has a unique solution (see e.g., [8,
Example B.2]). Moreover, this solution has the chaos expansion:

$v^{(r,z)}(t,x)=G_{t-r}(x-z)+\sum_{n\geq 1}I_{n}(g_{n}(\cdot,r,z,t,x)),$ (2.8)

where $g_{0}(r,z,t,x)=G_{t-r}(x-z)$ and for $n\geq 1$,

$g_{n}(\boldsymbol{t}_{n},\boldsymbol{x}_{n},r,z,t,x)=G_{t-t_{n}}(x-x_{n})\times\dots\times
G_{t_{1}-r}(x_{1}-z).$ (2.9)

In addition, for any $p\geq 2$ and $t>0$,

$\sup_{\begin{subarray}{c}0\leq r\leq s\leq t\\\
x,z\in\mathbb{R}\end{subarray}}\big{\|}v^{(r,z)}(s,x)\big{\|}_{p}<C,$ (2.10)

where $C>0$ is a constant depends on $(t,p,H)$.

######  Lemma 2.4.

For any $0\leq r\leq s\leq t<\infty$, $x,z\in\mathbb{R}$, and $p\geq 2$, we
have:

$\big{\|}v^{(r,z)}(s,x)\big{\|}_{p}=2G_{s-r}(x-z)\big{\|}v^{(r,z)}(s,x)\big{\|}_{p}\leq
CG_{s-r}(x-z).$ (2.11)

where $C>0$ is a constant that depends on $(t,p,H)$.

###### Proof.

Taking account of the special form (1.7) of $G$, we see that
$g_{n}(\cdot,r,z,t,x)=2G_{t-r}(x-z)g_{n}(\cdot,r,z,t,x)$. From this, we deduce
using the chaos expansion (2.8) that (1.10) holds. Thus, (2.11) follows from
(2.10) and (1.10). The proof of this lemma is complete. ∎

The next lemma presents some translation-invariance properties of $v^{(r,z)}$.

######  Lemma 2.5.

For any $0\leq r\leq t<\infty$ and $x,x^{\prime},z,z^{\prime}\in\mathbb{R}$,
the following properties hold:

  1. a)

$\displaystyle v^{(r,z)}(t,x)\stackrel{{\scriptstyle
d}}{{=}}v^{(r,0)}(t,x-z)$.

  2. b)

$\displaystyle
v^{(r,z)}(t,x)-v^{(r,z^{\prime})}(t,x^{\prime})\stackrel{{\scriptstyle
d}}{{=}}v^{(r,0)}(t,x-z)-v^{(r,z^{\prime}-z)}(t,x^{\prime}-z)$.

  3. c)

$\begin{aligned}
&v^{(r,z)}(t,x)-v^{(r,z)}(t,x^{\prime})-v^{(r,z^{\prime})}(t,x)+v^{(r,z^{\prime})}(t,x^{\prime})\stackrel{{\scriptstyle
d}}{{=}}\\\ &\quad\quad\quad
v^{(r,0)}(t,x-z)-v^{(r,0)}(t,x^{\prime}-z)-v^{(r,z^{\prime}-z)}(t,x-z)+v^{(r,z^{\prime}-z)}(t,x^{\prime}-z).\end{aligned}$

###### Proof.

a) This is similar to the proof of Lemma 2.2.

b) We have the following chaos expansion:

$\displaystyle
v^{(r,z)}(t,x)-v^{(r,z^{\prime})}(t,x^{\prime})=G_{t-r}(x-z)-G_{t-r}(x^{\prime}-z^{\prime})+$
$\displaystyle\quad\quad\quad\sum_{n=1}^{\infty}\int_{r<t_{1}<\ldots<t_{n}<t}\int_{\mathbb{R}^{n}}\Big{(}G_{t-t_{n}}(x-x_{n})G_{t_{1}-r}(x_{1}-z)-G_{t-t_{n}}(x^{\prime}-x_{n})G_{t_{1}-r}(x_{1}-z^{\prime})\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\prod_{i=1}^{n-1}G_{t_{i+1}-t_{i}}(x_{i+1}-x_{i})W(dt_{1},dx_{1})\ldots
W(dt_{n},dx_{n}).$ (2.12)

Performing the formal change of variables $y_{i}=x_{i}-z$ for $i=1,\ldots,n$,
we see that the multiple integral above is equal to

$\displaystyle\int_{r<t_{1}<\ldots<t_{n}}\int_{\mathbb{R}^{n}}\Big{(}G_{t-t_{n}}(x-z-
y_{n})G_{t_{1}-r}(y_{1})-G_{t-t_{n}}(x^{\prime}-z-y_{n})G_{t_{1}-r}(y_{1}-z^{\prime}+z)\Big{)}$
$\displaystyle\qquad\qquad\qquad\prod_{i=1}^{n-1}G_{t_{i+1}-t_{i}}(y_{i+1}-y_{i})W^{(z)}(dt_{1},dy_{1})\ldots
W^{(z)}(dt_{n},dy_{n}).$

Since $W\stackrel{{\scriptstyle d}}{{=}}W^{(z)}$, the series (2.12) has the
same distribution as the series which gives the chaos expansion of
$v^{(r,0)}(t,x-z)-v^{(r,z^{\prime}-z)}(t,x^{\prime}-z)$.

c) The chaos expansion of the term on the left-hand side is:

$\displaystyle
G_{t-r}(x-z)-G_{t-r}(x^{\prime}-z)-G_{t-r}(x-z^{\prime})+G_{t-r}(x^{\prime}-z^{\prime})+$
$\displaystyle\quad\sum_{n\geq
1}\int_{r<t_{1}<\ldots<t_{n}<t}\int_{\mathbb{R}^{n}}\big{(}G_{t-t_{n}}(x-x_{n})-G_{t-t_{n}}(x^{\prime}-x_{n})\big{)}\prod_{i=1}^{n-1}G_{t_{i+1}-t_{i}}(x_{i+1}-x_{i})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\big{(}G_{t_{1}-r}(x_{1}-z)-G_{t_{1}-r}(x_{1}-z^{\prime})\big{)}W(dt_{1},dx_{1})\ldots
W(dt_{n},dx_{n})$

Then we perform the formal change of variables $y_{i}=x_{i}-z$ for
$i=1,\ldots,n$ in the multiple integral of order $n$. The conclusion follows
using the fact that $W\stackrel{{\scriptstyle d}}{{=}}W^{(z)}$. ∎

######  Lemma 2.6.

For any $p\geq 2$, $q>0$ and $t\in\mathbb{R}_{+}$, we have

$\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq s\leq t\\\
z\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\big{\|}v^{(r,z)}(s,x^{\prime})\big{\|}_{p}^{q}dx^{\prime}+\sup_{\begin{subarray}{c}0\leq
r\leq s\leq t\\\
x\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\big{\|}v^{(r,z^{\prime})}(s,x)\big{\|}_{p}^{q}dz^{\prime}\leq
C,$

where $C>0$ is a constant that depends on $(t,p,q,H)$.

###### Proof.

This is a direct consequence of Lemma 2.4. ∎

The properties listed in the following lemma have been proved in [8]: parts a)
and c) correspond to relations (98)–(99), ibid. and part b) was shown in the
proof of Lemma 5.4c), ibid. Note that $v^{(r,z)}$ is denoted by
$V_{1}^{(r,z)}$ in [8].

######  Lemma 2.7.

For any $p\geq 2$ and $t\in\mathbb{R}_{+}$, we have:

  1. a)

$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{2}}\left\|v^{(r,0)}(s,x)-v^{(r,h)}(s,x)\right\|_{p}^{2}|h|^{2H-2}dxdh\leq
C,$

  2. b)

$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{2}}\left\|v^{(r,0)}(s,x)-v^{(r,0)}(s,x+h)\right\|_{p}^{2}|h|^{2H-2}dxdh\leq
C,$

  3. c)

$\begin{aligned} \sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{3}}&\left\|v^{(r,0)}(s,x)-v^{(r,h)}(s,x)-v^{(r,0)}(s,x+\hbar)+v^{(r,h)}(s,x+\hbar)\right\|_{p}^{2}\\\
&\times|h|^{2H-2}|\hbar|^{2H-2}dxdhd\hbar\leq C,\end{aligned}$

where $C>0$ is a constant that depends on $(t,p,H)$.

######  Lemma 2.8.

For any $p\geq 2$ and $t\in\mathbb{R}_{+}$, we have:

  1. a)

$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\big{\|}v^{(r,0)}(s,x)-v^{(r,h)}(s,x)\big{\|}_{p}dx\right)^{2}|h|^{2H-2}dh\leq
C,$

  2. b)

$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\big{\|}v^{(r,0)}(s,x)-v^{(r,0)}(s,x+h)\big{\|}_{p}dx\right)^{2}|h|^{2H-2}dh\leq
C,$

  3. c)

$\begin{aligned} \sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{2}}&\left(\int_{\mathbb{R}}\big{\|}v^{(r,0)}(s,x)-v^{(r,h)}(s,x)-v^{(r,0)}(s,x+\hbar)+v^{(r,h)}(s,x+\hbar)\big{\|}_{p}dx\right)^{2}\\\
&\times|h|^{2H-2}|\hbar|^{2H-2}dhd\hbar\leq C,\end{aligned}$

where $C>0$ is a constant that depends on $(t,p,H)$.

###### Proof.

a) The proof follows from the same argument as in that of [8, Lemma 5.5], with
minor modifications. Denote by $I_{1}$ and $I_{2}$ the integral in a) on the
region $\\{h;|h|>1\\}$ and $\\{h;|h|\leq 1\\}$, respectively. Then, by Lemma
2.6,

$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}}\left\|v^{(r,0)}(s,x)-v^{(r,h)}(s,x)\right\|_{p}dx\leq$
$\displaystyle\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}}\Big{(}\left\|v^{(r,0)}(s,x)\right\|_{p}+\left\|v^{(r,h)}(s,x)\right\|_{p}dx\Big{)}$
$\displaystyle\leq$ $\displaystyle C,$

and hence

$I_{1}\leq C\int_{|h|>1}|h|^{2H-2}dh=C.$

Using (1.10), we have $I_{2}\leq C\big{(}I_{2,1}+I_{2,2}\big{)}$, where

$\displaystyle I_{2,1}$ $\displaystyle\coloneqq\sup_{0\leq r\leq s\leq
t}\int_{|h|\leq
1}\left(\int_{\mathbb{R}}G_{s-r}(x)\left\|v^{(r,0)}(s,x)-v^{(r,h)}(s,x)\right\|_{p}dx\right)^{2}|h|^{2H-2}dh$
$\displaystyle I_{2,2}$ $\displaystyle\coloneqq\sup_{0\leq r\leq s\leq
t}\int_{|h|\leq
1}\left(\int_{\mathbb{R}}\big{|}G_{s-r}(x)-G_{s-r}(x-h)\big{|}\left\|v^{(r,h)}(s,x)\right\|_{p}dx\right)^{2}|h|^{2H-2}dh.$

As a result of the Cauchy-Schwarz inequality and the fact that
$\|G_{s-r}\|_{L^{2}(\mathbb{R})}^{2}=(s-r)/2$,

$I_{2,1}\leq C\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{2}}\left\|v^{(r,0)}(s,x)-v^{(r,h)}(s,x)\right\|_{p}^{2}|h|^{2H-2}dxdh.$

The last integral is uniformly bounded, by Lemma 2.7a). Using (2.10) and the
fact that $\|G_{s-r}(\cdot)-G_{s-r}(\cdot-h)\|_{L^{1}(\mathbb{R})}\leq|h|$, we
have:

$\displaystyle I_{2,2}$ $\displaystyle\leq C\sup_{0\leq r\leq s\leq
t}\int_{|h|\leq
1}\left(\int_{\mathbb{R}}\big{|}G_{s-r}(x)-G_{s-r}(x-h)\big{|}dx\right)^{2}|h|^{2H-2}dh\leq
C\int_{|h|\leq 1}|h|^{2H}dh=C.$

b) The proof is similar to that of a), and thus skipped.

c) Denote by $J$ the integral in c). Then,

$J\leq C\big{(}J_{1}+J_{2}+J_{3}+J_{4}\big{)},$

where

$J_{k}\coloneqq\sup_{0\leq r\leq s\leq
t}\int_{\mathbb{R}^{2}}\left(\int_{\mathbb{R}}F_{k}(x,h,\hbar)dx\right)^{2}|h|^{2H-2}|\hbar|^{2H-2}dhd\hbar,\quad
k=1,\dots,4,$

with

$\displaystyle F_{1}\coloneqq
2G_{s-r}(x)\big{\|}v^{(r,0)}(s,x)-v^{(r,h)}(s,x)-v^{(r,0)}(s,x+\hbar)+v^{(r,h)}(s,x+\hbar)\big{\|}_{p},$
$\displaystyle F_{2}\coloneqq
2|G_{s-r}(x)-G_{s-r}(x+\hbar)|\big{\|}v^{(r,0)}(s,x+\hbar)-v^{(r,h)}(s,x+\hbar)\big{\|}_{p},$
$\displaystyle F_{3}\coloneqq
2|G_{s-r}(x)-G_{s-r}(x-h)-G_{s-r}(x+\hbar)+G_{s-r}(x+\hbar-h)|\big{\|}v^{(r,h)}(s,x)\big{\|}_{p},$
and $\displaystyle F_{4}\coloneqq
2|G_{s-r}(x+\hbar)-G_{s-r}(x+\hbar-h)|\big{\|}v^{(r,h)}(s,x)-v^{(r,h)}(s,x+\hbar)\big{\|}_{p}.$

The conclusion follows from the same argument as in the proof of [8, Lemma
5.5e)], by using Lemmas 2.7 and 2.6, as well as relations (58) and (59), ibid.
The proof of this lemma is complete. ∎

###  2.3 Malliavin derivatives and (HAM) with the delta initial velocity

In this section, we establish the connection between the Malliavin derivatives
$Du$ and $D^{2}u$ and the solution $v^{(r,z)}$ of (2.7).

######  Lemma 2.9.

Let $u$ be the solution to (1.4), and let $v^{(r,z)}$ be the solution to (2.7)
with arbitrary $(r,z)\in\mathbb{R}_{+}\times\mathbb{R}$. Then,

  1. a)

$\displaystyle D_{r,z}u(t,x)=u(r,z)v^{(r,z)}(t,x)$, for all $0\leq r\leq
t<\infty$ and $x,z\in\mathbb{R}$.

  2. b)

$\displaystyle
D_{(\theta,w),(r,z)}^{2}u(t,x)=u(\theta,w)v^{(\theta,w)}(r,z)v^{(r,z)}(t,x)$
for all $0\leq\theta\leq r\leq t<\infty$ and $x,z,w\in\mathbb{R}$.

###### Proof.

We only provide the proof of a). One can show b) similarly. Note that the
Malliavin derivative $D_{r,z}u(t,x)$ has the chaos expansion: for any
$(r,z)\in[0,t]\times\mathbb{R}$,

$D_{r,z}u(t,x)=\sum_{n\geq 1}nI_{n-1}(\bar{f}_{n}(\cdot,r,z,t,x))=\sum_{n\geq
1}\sum_{j=1}^{n}I_{n-1}(f_{j}^{(n)}(\cdot,r,z,t,x)),$ (2.13)

where

$\displaystyle
f_{j}^{(n)}(\boldsymbol{t}_{n-1},\boldsymbol{x}_{n-1},r,z,t,x)=$
$\displaystyle
f_{n}(t_{1},\ldots,t_{j-1},r,t_{j},\dots,t_{n-1},x_{1},\dots,x_{j-1},z,x_{j},\ldots,x_{n-1},t,x)$
$\displaystyle=$ $\displaystyle G_{t-t_{n-1}}(x-x_{n-1})\times\dots\times
G_{t_{j}-r}(x_{j}-z)G_{r-t_{j-1}}(z-x_{j-1})$ $\displaystyle\times\dots\times
G_{t_{2}-t_{1}}(x_{2}-x_{1})\mathbf{1}_{T_{n-1}^{j}(t,r)}(\boldsymbol{t}_{n-1}),$

with

$\displaystyle
T_{n-1}^{j}(t,r)\coloneqq\\{\boldsymbol{t}_{n-1}\in[0,t]^{n-1};0<t_{1}<\ldots<t_{j-1}<r<t_{j}<\ldots<t_{n-1}<t\\}.$

Note that the function $f_{j}^{(n)}(\cdot,r,z,t,x)$ can be written as follows:

$\displaystyle
f_{j}^{(n)}(\boldsymbol{t}_{n-1},\boldsymbol{x}_{n-1},r,z,t,x)=$
$\displaystyle
f_{j}(\boldsymbol{t}_{j-1},\boldsymbol{x}_{j-1},r,z)g_{n-j}(\boldsymbol{t}_{j:n-1},\boldsymbol{x}_{j;n-1},r,z,t,x),$
(2.14)

where $g_{n-j}$ is given by (2.9) with $g_{0}(r,z,t,x)\coloneqq G_{t-r}(x-z)$,
$\boldsymbol{t}_{j:n-1}\coloneqq(t_{j},\dots,t_{n-1})$, and
$\boldsymbol{x}_{j:n-1}\coloneqq(x_{j},\dots,x_{n-1})$. Because the two
functions appearing in this decomposition have “disjoint temporal supports”
(see the footnote on page 16 of [4]) and the noise is white in time, it
follows

$f_{j-1}(\cdot,r,z)\otimes_{k}g_{n-j}(\cdot,r,z,t,x)\equiv 0,\quad\text{for
all }k=1,\dots,(j-1)\wedge(n-j).$

As a result, using the product formula e.g., [50, Proposition 1.1.3], we have

$I_{n-1}\big{(}f_{j}^{(n)}(\cdot,r,z,t,x)\big{)}=I_{j-1}\big{(}f_{j-1}(\cdot,r,z)\big{)}I_{n-j}\big{(}g_{n-j}(\cdot,r,z,t,x)\big{)},$

Interchanging the order of summation in (2.13), we obtain:

$\displaystyle D_{r,z}u(t,x)=$ $\displaystyle\sum_{j\geq 1}\sum_{n\geq
j}I_{n-1}(f_{j}^{(n)}\big{(}\cdot,r,z,t,x)\big{)}$ $\displaystyle=$
$\displaystyle\sum_{j\geq 1}I_{j-1}\big{(}f_{j-1}(\cdot,r,z)\big{)}\sum_{n\geq
j}I_{n-j}\big{(}g_{n-j}(\cdot,r,z,t,x)\big{)}.$

This justifies a). The proof of this lemma is complete. ∎

###  2.4 Moment estimates for the Malliavin derivatives

In this section, we derive some estimates for the moments of the first and
second Malliavin derivatives of $u(t,x)$.

Notice that $u$ is adapted and the noise has independent increments in time.
Therefore, $u(r,z)$ is independent of $v^{(r,z)}$. Hence, using (2.3), for any
$p\geq 2$, $r\in[0,t]$ and $x,z\in\mathbb{R}$,

$\|D_{r,z}u(t,x)\|_{p}=\|u(r,z)\|_{p}\|v^{(r,z)}(t,x)\|_{p}\leq
C_{t}\|v^{(r,z)}(t,x)\|_{p}.$

A similar deduction can be done for $D^{2}u(t,x)$ and thus taking account of
Lemma 2.4, we can summarize the next proposition.

######  Proposition 2.10.

Let $u$ be the solution to (1.4), then for any $0\leq r\leq s\leq t<\infty$
and $x,y,z\in\mathbb{R}$, the next inequalities hold:

$\|D_{r,z}u(t,x)\|_{p}\leq C_{t}G_{t-r}(x-z),$ (2.15)

and

$\|D_{(r,z)(s,y)}^{2}u(t,x)\|_{p}\leq C_{t}G_{t-s}(x-y)G_{s-y}(y-z),$ (2.16)

where $C_{t}>0$ depending only on $t$.

The next proposition about the estimates for the increments of the Malliavin
derivatives of $u(t,x)$ will be used in the proof of the main results. To this
end, we should first introduce the following notation. For any
$r,s,t\in\mathbb{R}_{+}$ such that $0\leq r\vee s\leq t$, and
$x,y,y^{\prime},z,z^{\prime}\in\mathbb{R}$, we denote

$\displaystyle\Delta_{h}(r,z,t,x)\coloneqq D_{r,z+h}u(t,x)-D_{r,z}u(t,x),$
(2.17)

and

$\displaystyle\square_{h,\hbar}(r,z,s,y,t,x)\coloneqq$ $\displaystyle
D_{(r,z+h),(s,y+\hbar)}^{2}u(t,x)-D_{(r,z),(s,y+\hbar)}^{2}u(t,x)$
$\displaystyle-D_{(r,z+h),(s,y)}^{2}u(t,x)+D_{(r,z),(s,y)}^{2}u(t,x).$ (2.18)

######  Proposition 2.11.

For any $t\in\mathbb{R}_{+}$, and $p\geq 2$, we have:

  1. a)

$\displaystyle\sup_{0\leq r\leq
t}\int_{\mathbb{R}}\sup_{z\in\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}\Delta_{h}(r,z,t,x)\big{\|}_{p}dx\Big{)}^{2}|h|^{2H-2}dh\leq
C$,

  2. b)

$\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}\big{\|}\Delta_{h}(r,z,t,x)\big{\|}_{p}\big{\|}\Delta_{h}(r,z,t,x^{\prime})\big{\|}_{p}|h|^{2H-2}dhdzdx\leq
C$,

  3. c)

$\displaystyle\sup_{0\leq r\vee s\leq
t}\int_{\mathbb{R}^{2}}\sup_{y\in\mathbb{R}}\Big{(}\int_{\mathbb{R}^{2}}\big{\|}\square_{h,\hbar}(r,z,s,y,t,x)\big{\|}_{p}dxdz\Big{)}^{2}|h|^{2H-2}|\hbar|^{2H-2}dhd\hbar\leq
C$,

  4. d)

$\displaystyle\sup_{\begin{subarray}{c}0\leq r\vee s\leq t\\\
x\in\mathbb{R}\end{subarray}}\Big{(}\int_{\mathbb{R}^{2}}\big{\|}\square_{h,\hbar}(r,z,s,y,t,x)\big{\|}_{p}dydz\Big{)}^{2}|h|^{2H-2}|\hbar|^{2H-2}dhd\hbar<C$,

where $C>0$ is a constant that depends on $(t,p,H)$.

###### Proof.

a) By using Lemma 2.9a) , we can write

$\displaystyle\Delta_{h}(r,z,t,x)=$ $\displaystyle
u(r,z+h)v^{(r,z+h)}(t,x)-u(r,z)v^{(r,z)}(t,x)$ $\displaystyle=$
$\displaystyle\big{(}u(r,z+h)-u(r,z))v^{(r,z)}(t,x)+u(r,z+h)\big{(}v^{(r,z+h)}(t,x)-v^{(r,z)}(t,x)\big{)}.$

Then, it follows from the Cauchy-Schwartz inequality and Lemmas 2.2 and 2.5b)
that

$\displaystyle\big{\|}\Delta_{h}(r,z,t,x)\big{\|}_{p}\leq$
$\displaystyle\big{\|}u(r,h)-u(r,0)\|_{2p}\|v^{(r,z)}(t,x)\|_{2p}$
$\displaystyle+\|u(r,z+h)\|_{2p}\big{\|}v^{(r,h)}(t,x-z)-v^{(r,0)}(t,x-z)\big{\|}_{2p}.$
(2.19)

Hence, Proposition 2.11a) is a consequence of Lemmas 2.1, 2.3, 2.6, and 2.8b).

b) Using (2.4), we can write

$\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}\big{\|}\Delta_{h}(r,z,t,x)\big{\|}_{p}\big{\|}\Delta_{h}(r,z,t,x^{\prime})\big{\|}_{p}|h|^{2H-2}dhdzdx=K_{1}+K_{2}+K_{3}+K_{4},$

where

$\displaystyle K_{1}\coloneqq\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}$
$\displaystyle\big{\|}u(r,h)-u(r,0)\|_{2p}^{2}\|v^{(r,z)}(t,x)\|_{2p}\|v^{(r,z)}(t,x^{\prime})\|_{2p}|h|^{2H-2}dhdzdx,$
$\displaystyle K_{2}\coloneqq\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}$
$\displaystyle\big{\|}u(r,h)-u(r,0)\|_{2p}\|v^{(r,z)}(t,x)\|_{2p}$
$\displaystyle\times\|u(r,z+h)\|_{2p}\big{\|}v^{(r,h)}(t,x^{\prime}-z)-v^{(r,0)}(t,x^{\prime}-z)\big{\|}_{2p}|h|^{2H-2}dhdzdx,$
$\displaystyle K_{3}\coloneqq\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}$
$\displaystyle\|u(r,z+h)\|_{2p}\big{\|}v^{(r,h)}(t,x-z)-v^{(r,0)}(t,x-z)\big{\|}_{2p}$
$\displaystyle\times\big{\|}u(r,h)-u(r,0)\|_{2p}\|v^{(r,z)}(t,x^{\prime})\|_{2p}|h|^{2H-2}dhdzdx,$
and $\displaystyle K_{4}\coloneqq\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}^{3}}$
$\displaystyle\|u(r,z+h)\|_{2p}^{2}\big{\|}v^{(r,h)}(t,x-z)-v^{(r,0)}(t,x-z)\big{\|}_{2p}$
$\displaystyle\times\big{\|}v^{(r,h)}(t,x^{\prime}-z)-v^{(r,0)}(t,x^{\prime}-z)\big{\|}_{2p}|h|^{2H-2}dhdzdx.$

It follows from Lemmas 2.3 and 2.6 that

$\displaystyle K_{1}\leq$ $\displaystyle\sup_{0\leq r\leq
t}\int_{\mathbb{R}}\big{\|}u(r,h)-u(r,0)\|_{2p}^{2}|h|^{2H-2}dh\times\sup_{\begin{subarray}{c}0\leq
r\leq t\\\
z\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\|v^{(r,z)}(t,x)\|_{2p}dx$
$\displaystyle\times\sup_{\begin{subarray}{c}0\leq r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\|v^{(r,z)}(t,x^{\prime})\|_{2p}dz\leq
C.$

Due to Cauchy-Schwarz’s inequality and Lemmas 2.1, 2.3, 2.6, and 2.8a), we
have

$\displaystyle K_{2}\leq$ $\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq
t\\\
z^{\prime}\in\mathbb{R}\end{subarray}}\|u(r,z^{\prime})\|_{2p}\times\sup_{\begin{subarray}{c}0\leq
r\leq t\\\
z\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\|v^{(r,z)}(t,x)\|_{2p}dx$
$\displaystyle\times\sup_{0\leq r\leq
t}\bigg{(}\int_{\mathbb{R}}\big{\|}u(r,h)-u(r,0)\|_{2p}^{2}|h|^{2H-2}dh\bigg{)}^{1/2}$
$\displaystyle\times\sup_{0\leq r\leq
t}\bigg{(}\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}v^{(r,h)}(t,z^{\prime\prime})-v^{(r,0)}(t,z^{\prime\prime})dz^{\prime\prime}\Big{)}^{2}\big{\|}_{2p}|h|^{2H-2}dh\bigg{)}^{1/2}\leq
C,$

and, with a change of variable $x-z\to x^{\prime\prime}$,

$\displaystyle K_{3}\leq$ $\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq
t\\\
z^{\prime}\in\mathbb{R}\end{subarray}}\|u(r,z^{\prime})\|_{2p}\times\sup_{\begin{subarray}{c}0\leq
r\leq t\\\
x^{\prime}\in\mathbb{R}\end{subarray}}\int_{\mathbb{R}}\|v^{(r,z)}(t,x^{\prime})\|_{2p}dz$
$\displaystyle\times\sup_{0\leq r\leq
t}\bigg{(}\int_{\mathbb{R}}\big{\|}u(r,h)-u(r,0)\|_{2p}^{2}|h|^{2H-2}dh\bigg{)}^{1/2}$
$\displaystyle\times\sup_{0\leq r\leq
t}\bigg{(}\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}v^{(r,h)}(t,x^{\prime\prime})-v^{(r,0)}(t,x^{\prime\prime})dx^{\prime\prime}\Big{)}^{2}\big{\|}_{2p}|h|^{2H-2}dh\bigg{)}^{1/2}\leq
C.$

Finally, preforming a change of variables
$(x-z,x^{\prime}-z)\to(x^{\prime\prime},z^{\prime\prime})$, and using Cauchy-
Schwarz’s inequality and Lemmas 2.1 and 2.8a),

$\displaystyle K_{4}\leq$ $\displaystyle\sup_{\begin{subarray}{c}0\leq r\leq
t\\\
z^{\prime}\in\mathbb{R}\end{subarray}}\|u(r,z^{\prime})\|_{2p}^{2}\times\sup_{0\leq
r\leq
t}\bigg{(}\int_{\mathbb{R}}\Big{(}\big{\|}v^{(r,h)}(t,z^{\prime\prime})-v^{(r,0)}(t,z^{\prime\prime})\big{\|}_{2p}dz^{\prime\prime}\Big{)}^{2}|h|^{2H-2}dh\bigg{)}^{1/2}$
$\displaystyle\times\sup_{0\leq r\leq
t}\bigg{(}\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}v^{(r,h)}(t,x^{\prime\prime})-v^{(r,0)}(t,x^{\prime\prime})\big{\|}_{2p}dx^{\prime\prime}\Big{)}^{2}|h|^{2H-2}dh\bigg{)}^{1/2}\leq
C.$

The proof of Proposition 2.11b) is complete.

c) Without loss of generality, assume that $r\leq s$. Then, as a result of
Lemmas 2.9b), 2.2, 2.5b) and 2.5c), and Hölder’s inequality, we deduce that

$\displaystyle\big{\|}\square_{h,\hbar}(r,z,s,y,t,x)\big{\|}_{p}\leq
I_{1}+I_{2}+I_{3}+I_{4}$ (2.20)

where

$\displaystyle
I_{1}\coloneqq\|u(r,h)-u(r,0)\|_{3p}\big{\|}v^{(r,0)}(s,y+\hbar-z)-v^{(r,0)}(s,y-z)\big{\|}_{3p}\|v^{(s,y)}(t,x)\|_{3p},$
$\displaystyle\begin{aligned}
I_{2}\coloneqq\|u(r,z+h)\|_{3p}\big{\|}&v^{(r,h)}(s,y+\hbar-z)-v^{(r,h)}(s,y-z)\\\
&-v^{(r,0)}(s,y+\hbar-z)+v^{(r,0)}(s,y-z)\big{\|}_{3p}\|v^{(s,y)}(t,x)\|_{3p},\end{aligned}$
$\displaystyle
I_{3}\coloneqq\|u(r,h)-u(r,0)\|_{3p}\big{\|}v^{(r,z)}(s,y+\hbar)\|_{3p}\big{\|}v^{(s,\hbar)}(t,x-y)-v^{(s,0)}(t,x-y)\big{\|}_{3p},$
and $\displaystyle\begin{aligned}
I_{4}\coloneqq&\|u(r,z+h)\|_{3p}\big{\|}v^{(r,z)}(s,y+\hbar)-v^{(r,z+h)}(s,y+\hbar)\big{\|}_{3p}\\\
&\times\big{\|}v^{(s,\hbar)}(t,x-y)-v^{(s,0)}(t,x-y)\big{\|}_{3p}.\end{aligned}$

Then, preforming a change of variable $(z-y,y)\to(\widetilde{z},y)$, and
applying Lemmas 2.3, 2.6 and 2.8b), we get

$\displaystyle\int_{\mathbb{R}^{2}}\sup_{y\in\mathbb{R}}\Big{(}\int_{\mathbb{R}^{2}}I_{1}dxdz\Big{)}^{2}|\hbar|^{2H-2}|h|^{2H-2}dhd\hbar$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}v^{(r,0)}(s,\hbar-\widetilde{z})-v^{(r,0)}(s,\widetilde{z})\big{\|}_{3p}d\widetilde{z}\Big{)}^{2}\sup_{y\in\mathbb{R}}\Big{(}\int_{\mathbb{R}}\|v^{(s,y)}(t,x)\|_{3p}dx\Big{)}^{2}|\hbar|^{2H-2}d\hbar$
$\displaystyle\times\int_{\mathbb{R}}\|u(r,h)-u(r,0)\|_{3p}^{2}|h|^{2H-2}dh\leq
C,$

with $C>0$ depending on $(t,p,H)$. Similarly, one can show that

$\displaystyle\int_{\mathbb{R}^{2}}\sup_{y\in\mathbb{R}}\Big{(}\int_{\mathbb{R}^{2}}I_{k}dxdz\Big{)}^{2}|\hbar|^{2H-2}|h|^{2H-2}dhd\hbar<C\quad\text{for
all }k=2,3,4.$

This completes that proof of Proposition 2.11c).

d) Similarly as in the proof of Proposition 2.11c), we decompose
$\|\square_{h,\hbar}(r,z,s,y,t,x)\|_{p}$ by (2.20). Then, one can deduce that

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}$
$\displaystyle\int_{\mathbb{R}^{2}}I_{1}dydz\Big{)}^{2}|\hbar|^{2H-2}|h|^{2H-2}dhd\hbar=\int_{\mathbb{R}}\|u(r,h)-u(r,0)\|_{3p}^{2}|h|^{2H-2}dh$
$\displaystyle\times\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}\big{\|}v^{(r,0)}(s,\widetilde{z}+\hbar)-v^{(r,0)}(s,\widetilde{z})\big{\|}_{3p}d\widetilde{z}\Big{)}^{2}\Big{(}\int_{\mathbb{R}}\|v^{(s,y)}(t,x)\|_{3p}dy\Big{)}^{2}|\hbar|^{2H-2}d\hbar<C.$

Again, similar arguments can be applied to the integrations of $I_{2}$,
$I_{3}$ and $I_{4}$, and we skip them for conciseness. The proof of this
proposition is complete. ∎

##  3 Proofs of the main results

In this section, we present the proof of Theorems 1.1, 1.2, 1.3 and 1.4.

###  3.1 Spatial Ergodicity—Proof of Theorem 1.1

In this section, we include the proof of Theorem 1.1. Recall that the
stationarity of $\\{u(t,x)\\}_{x\in\mathbb{R}}$ was proved in Lemma 2.2. In
this section, we prove that this process is also ergodic. For this, we use a
version of the ergodicity criterion given by [14, Lemma 7.2], as stated in [9,
Lemma 4.2]. More precisely, we prove that:

$\lim_{R\to\infty}\frac{1}{R^{2}}{\rm Var}(U_{R})=0,\quad\mbox{where}\quad
U_{R}=\int_{-R}^{R}g\Big{(}\sum_{j=1}^{k}b_{j}u(t,x+\zeta_{j})\Big{)}dx,$

where $k$ is an arbitrary positive integer, $b_{1},\ldots,b_{k}\in\mathbb{R}$
and $\zeta_{1},\ldots,\zeta_{k}\in\mathbb{R}$ are arbitrary and
$g\in\\{\cos,\sin\\}$.

Without loss of generality, we assume that $g(x)=\cos x$, the case when
$g(x)=\sin x$ being similar. By the Gaussian-Poincaré inequality (see e.g,
[48, Exerice 2.11.1]),

${\rm
Var}(U_{R})\leq\mathbb{E}\|DU_{R}\|_{\mathcal{H}}^{2}=\int_{0}^{t}\int_{\mathbb{R}^{2}}\mathbb{E}|D_{r,y}U_{R}-D_{r,z}U_{R}|^{2}|y-z|^{2H-2}dydzdr.$

Using the chain rule $D\varphi(F)=\varphi^{\prime}(F)DF$, we see that

$D_{r,y}U_{R}=\int_{-R}^{R}\sin\Big{(}\sum_{j=1}^{k}b_{j}u(s,x+\zeta_{j})\Big{)}\sum_{j=1}^{k}b_{j}D_{r,y}u(t,x+\zeta_{j})dx.$

Using Minkowski inequality, Cauchy-Schwarz inequality and the fact that
$|\sin(x)|\leq 1$,

$\displaystyle\mathbb{E}|D_{r,y}U_{R}-D_{r,z}U_{R}|^{2}=\|D_{r,y}U_{R}-D_{r,z}U_{R}\|_{2}^{2}$
$\displaystyle\quad\leq\left(\int_{-R}^{R}\left\|\sin\left(\sum_{j=1}^{k}b_{j}u(t,x+\zeta_{j})\right)\sum_{j=1}^{k}b_{j}\left(D_{r,y}u(t,x+\zeta_{j})-D_{r,z}u(t,x+\zeta_{j})\right)\right\|_{2}dx\right)^{2}$
$\displaystyle\quad\leq\left(\int_{-R}^{R}\sum_{j=1}^{k}|b_{j}|\big{\|}D_{r,y}u(t,x+\zeta_{j})-D_{r,z}u(t,x+\zeta_{j})\big{\|}_{4}dx\right)^{2},$

and therefore,

${\rm Var}(U_{R})\leq
C\sum_{j=1}^{k}b_{j}^{2}\times\sup_{\zeta\in\mathbb{R}}\int_{0}^{t}\int_{\mathbb{R}^{2}}\left(\int_{-R}^{R}\big{\|}\Delta_{h}(r,z,t,x+\zeta)\big{\|}_{4}dx\right)^{2}|h|^{2H-2}dhdzdr,$

where $\Delta$ is defined as in (2.17). Notice that Proposition 2.11b) yields
that

$\displaystyle\sup_{\zeta\in\mathbb{R}}\int_{\mathbb{R}^{2}}\left(\int_{-R}^{R}\big{\|}\Delta_{h}(r,z,t,x+\zeta)\big{\|}_{4}dx\right)^{2}|h|^{2H-2}dhdz$
$\displaystyle=\sup_{\zeta\in\mathbb{R}}\int_{\mathbb{R}^{2}}\left(\int_{-R}^{R}\int_{-R}^{R}\big{\|}\Delta_{h}(r,z,t,x_{1}+\zeta)\big{\|}_{4}\big{\|}\Delta_{h}(r,z,t,x_{2}+\zeta)\big{\|}_{4}dx_{1}dx_{2}\right)|h|^{2H-2}dhdz$
$\displaystyle\leq\sup_{\zeta\in\mathbb{R}}\int_{-R}^{R}\left(\int_{\mathbb{R}^{3}}\big{\|}\Delta_{h}(r,z,t,x_{1})\big{\|}_{4}\big{\|}\Delta_{h}(r,z,t,x_{2}+\zeta)\big{\|}_{4}|h|^{2H-2}dhdzdx_{1}\right)dx_{2}$
$\displaystyle\leq\int_{-R}^{R}\left(\sup_{y\in\mathbb{R}}\int_{\mathbb{R}^{3}}\big{\|}\Delta_{h}(r,z,t,x_{1})\big{\|}_{4}\big{\|}\Delta_{h}(r,z,t,y)\big{\|}_{4}|h|^{2H-2}dhdzdx_{1}\right)dx_{2}\leq
CR,$

and thus

$\displaystyle\mathrm{Var}(U_{R})\leq CR.$

It follows that

$\displaystyle\frac{1}{R^{2}}\mathrm{Var}(U_{R})\leq CR^{-1}\to
0,\quad\text{as }R\to\infty.$

The proof of Theorem 1.1 is complete.

###  3.2 Limiting covariance—Proof of Theorem 1.2

In this section, we give the proof of Theorem 1.2. Note that

$\sigma_{R}^{2}(t)={\rm
Var}(F_{R}(t))=\int_{-R}^{R}\int_{-R}^{R}\rho_{t}(x-y)dxdy$

where

$\rho_{t}(x-y)\coloneqq\mathbb{E}\left[\big{(}u(t,x)-1\big{)}\big{(}u(t,y)-1\big{)}\right]=\sum_{n\geq
1}\frac{1}{n!}\gamma_{n}(t,x-y)$ (3.1)

and

$\displaystyle\gamma_{n}(t,x-y)\coloneqq(n!)^{2}\langle\widetilde{f}_{n}(\cdot,t,x),\widetilde{f}_{n}(\cdot,t,y)\rangle_{\mathcal{H}^{\otimes
n}}$
$\displaystyle\quad=n!C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\mathcal{F}f_{n}(\cdot,t,x)(\xi_{1},\ldots,\xi_{n})\overline{\mathcal{F}f_{n}(\cdot,t,y)(\xi_{1},\ldots,\xi_{n})}\prod_{j=1}^{n}|\xi_{j}|^{1-2H}d\boldsymbol{\xi}_{n}d\boldsymbol{t}_{n}$
$\displaystyle\quad=n!C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}e^{-i(\xi_{1}+\ldots+\xi_{n})(x-y)}\prod_{j=1}^{n}|\mathcal{F}G_{t_{j+1}-t_{j}}(\xi_{1}+\ldots+\xi_{j})|^{2}\prod_{j=1}^{n}|\xi_{j}|^{1-2H}d\boldsymbol{\xi}_{n}d\boldsymbol{t}_{n}$
$\displaystyle\quad=n!C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}e^{-i\eta_{n}(x-y)}\prod_{j=1}^{n}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n},$
(3.2)

with convention $t_{n+1}=t$ and $\eta_{0}=0$. This shows that
$\alpha_{n}(t,x-y)$ and $\rho_{t}(x-y)$ depend on $x$ and $y$ only through the
difference $x-y$. In particular, $\\{u(t,x);x\in\mathbb{R}\\}$ is a (wide-
sense) stationary process with covariance function $\rho_{t}$.

###### Proof of Theorem 1.2.

The proof is divided in three steps.

Step 1. In this step, we prove (1.11) in the case $t=s$. We write

$\sigma_{R}^{2}(t)=\sum_{n\geq
1}\frac{1}{n!}\int_{-R}^{R}\int_{-R}^{R}\gamma_{n}(t,x-y)dxdy.$ (3.3)

Note that

$\int_{-R}^{R}\int_{-R}^{R}e^{-i\xi(x-y)}dxdy=\left|\int_{-R}^{R}e^{-i\xi
x}dx\right|^{2}=\frac{4\sin^{2}(R|\xi|)}{|\xi|^{2}}=4\pi R\ell_{R}(\xi)$ (3.4)

where $\ell_{R}(x)\coloneqq(\pi R|x|^{2})^{-1}\sin^{2}(|x|R)$ is an
approximation of the identity as $R\to\infty$; see [55, Lemma 2.1]. On the
other hand, [59, Theorem 3.2] shows that for every $n\geq 1$,

$\displaystyle\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\bigg{|}e^{-i\eta_{n}(x-y)}\prod_{j=1}^{n}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}\bigg{|}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n}<\infty.$

Thus, by Fubini theorem,

$\displaystyle\frac{1}{n!}\int_{-R}^{R}\int_{-R}^{R}\gamma_{n}(t,x-y)dxdy$
$\displaystyle=$ $\displaystyle
C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\left(\int_{-R}^{R}\int_{-R}^{R}e^{-i\eta_{n}(x-y)}dxdy\right)\prod_{j=1}^{n}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n}$
$\displaystyle=$ $\displaystyle
4C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\frac{\sin^{2}(R|\eta_{n}|)}{|\eta_{n}|^{2}}\prod_{j=1}^{n}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n}.$
(3.5)

We treat separately the case $n=1$. From [8, Page 30], we know that for any
$t_{1}\in[0,t]$, $\theta\in(H,\frac{1}{2})$ and
$\varepsilon\in(0,\frac{\pi}{4t})$,

$\int_{\mathbb{R}}\frac{\sin^{2}(R|\eta_{1}|)}{|\eta_{1}|^{2}}\frac{\sin^{2}((t-t_{1})|\eta_{1}|)}{|\eta_{1}|^{2}}|\eta_{1}|^{1-2H}d\eta_{1}\leq
C_{\varepsilon,\theta,H}(t-t_{1})^{2}R^{2\theta}+C_{\varepsilon,H}.$

Therefore,

$\lim_{R\to\infty}\frac{1}{R}\int_{-R}^{R}\int_{-R}^{R}\gamma_{1}(t,x-y)dxdy\leq
4C_{H}\lim_{R\to\infty}R^{2\theta-1}\int_{0}^{t}\big{(}C_{\varepsilon,\theta,H}(t-t_{1})^{2}+C_{\varepsilon,H}\big{)}dt_{1}=0.$

Next, we examine the terms corresponding to $n\geq 2$. For any $n\geq 2$,
denote

$g_{\boldsymbol{t}_{n}}^{(n)}(\eta)\coloneqq\frac{\sin^{2}((t-t_{n})|\eta|)}{|\eta|^{2}}\int_{\mathbb{R}^{n-1}}\prod_{j=1}^{n-1}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n-1},$

with

$\displaystyle g_{\boldsymbol{t}_{n}}^{(n)}(0)\coloneqq$
$\displaystyle\lim_{\eta\to 0}g_{\boldsymbol{t}_{n}}^{(n)}(\eta)$
$\displaystyle=$
$\displaystyle(t-t_{n})^{2}\int_{\mathbb{R}^{n-1}}\prod_{j=1}^{n-1}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n-1}|\eta_{j}-\eta_{j-1}|^{1-2H}|\eta_{n-1}|^{1-2H}d\boldsymbol{\eta}_{n-1},$

where the last equality is due to the fact that $\lim_{x\to 0}\sin x/x=1$.
Then, $g_{\cdot}^{(n)}(*)$ is a non-negative function on
$T_{n}(t)\times\mathbb{R}$, and thus

$\displaystyle\frac{1}{n!}\int_{-R}^{R}$
$\displaystyle\int_{-R}^{R}\gamma_{n}(t,x-y)dxdy$ $\displaystyle=$
$\displaystyle 4\pi
C_{H}^{n}R\int_{T_{n}(t)}\int_{\mathbb{R}^{n}}\ell_{R}(\eta_{n})\prod_{j=1}^{n}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n}$
$\displaystyle=$ $\displaystyle 4\pi
C_{H}^{n}R\int_{T_{n}(t)}\int_{\mathbb{R}}\ell_{R}(\eta_{n})g_{\boldsymbol{t}_{n}}^{(n)}(\eta_{n})d\eta_{n}d\boldsymbol{t}_{n}=4\pi
C_{H}^{n}R\int_{T_{n}(t)}\big{(}\ell_{R}*g_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)d\boldsymbol{t}_{n}.$
(3.6)

Using (2.6), we obtain that

$\displaystyle g_{\boldsymbol{t}_{n}}^{(n)}(\eta_{n})$
$\displaystyle\leq\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\frac{\sin^{2}((t-t_{n})|\eta_{n}|)}{|\eta_{n}|^{2}}|\eta_{n}|^{\alpha_{n}}\prod_{j=1}^{n-1}\left(\int_{\mathbb{R}}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}|\eta_{j}|^{\alpha_{j}}d\eta_{j}\right)$
$\displaystyle\leq C^{n-1}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\frac{\sin^{2}((t-t_{n})|\eta_{n}|)}{|\eta_{n}|^{2}}|\eta_{n}|^{\alpha_{n}}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{1-\alpha_{j}},$
(3.7)

where for the last inequality we used (2.1). It is not difficult to see that
$g_{\boldsymbol{t}_{n}}^{(n)}$ is continuous, locally bounded and integrable
on $\mathbb{R}$. Recall that $\ell_{R}$ is an approximation of the identity as
$R\to\infty$. Hence, for any $n\geq 2$,

$\lim_{R\to\infty}\big{(}\ell_{R}*g_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)=g_{\boldsymbol{t}_{n}}^{(n)}(0).$

Combining (3.3) and (3.2), by using the dominated convergence theorem, we see
that

$\displaystyle\lim_{R\to\infty}\frac{\sigma_{R}^{2}(t)}{R}=\lim_{R\to\infty}4\pi\sum_{n\geq
2}C_{H}^{n}\int_{T_{n}(t)}\big{(}\ell_{R}*g_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)d\boldsymbol{t}_{n}=4\pi\sum_{n\geq
2}C_{H}^{n}\int_{T_{n}(t)}g_{\boldsymbol{t}_{n}}^{(n)}(0)d\boldsymbol{t}_{n}.$
(3.8)

As the dominated convergence theorem is applied in (3.8), one needs to justify
the applicability. In other words, we have to find a sequence of functions
$\\{h_{n}\\}_{n\geq 2}$ on $T_{n}(t)$ such that
$\big{(}\ell_{R}*g_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)\leq
h_{n}(\boldsymbol{t}_{n})$ for all $\boldsymbol{t}_{n}\in T_{n}(t)$ and $n\geq
1$, and

$\sum_{n\geq
2}C_{H}^{n}\int_{T_{n}(t)}h_{n}(\boldsymbol{t}_{n})d\boldsymbol{t}_{n}<\infty.$
(3.9)

In particular, this shows that

$K(t)=4\pi\sum_{n\geq
2}C_{H}^{n}\int_{T_{n}(t)}g_{\boldsymbol{t}_{n}}^{(n)}(0)d\boldsymbol{t}_{n}<\infty.$

Thanks to (3.2) and the inequality
$\frac{\sin^{2}((t-t_{n})|\eta|)}{|\eta|^{2}}\leq(t-t_{n})^{2}$, we have:

$\displaystyle\big{(}\ell_{R}*g_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)$
$\displaystyle\leq C^{n-1}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{\alpha_{j}}\int_{\mathbb{R}}\frac{\sin^{2}(R|\eta_{n}|)}{\pi
R|\eta|^{2}}\frac{\sin^{2}((t-t_{n})|\eta_{n}|)}{|\eta_{n}|^{2}}|\eta_{n}|^{\alpha_{n}}d\eta_{n}$
$\displaystyle\leq C^{n-1}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{\alpha_{j}}(t-t_{n})^{2}\int_{\mathbb{R}}\frac{\sin^{2}(R|\eta_{n}|)}{\pi
R|\eta_{n}|^{2}}|\eta_{n}|^{\alpha_{n}}d\eta_{n}$ $\displaystyle\leq
C^{n}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{1-\alpha_{j}}(t-t_{n})^{2}R^{-\alpha_{n}}$
$\displaystyle\leq C^{n}\sum_{\boldsymbol{\alpha}_{n}\in
D_{n}}\prod_{j=1}^{n-1}(t_{j+1}-t_{j})^{1-\alpha_{j}}(t-t_{n})^{2}\eqqcolon
h_{n}(\boldsymbol{t}_{n}),$

for any $R\geq 1$, where identity (2.1) is used for the second last line. As a
consequence of identity (2.2) and the fact that $\Gamma(an+b+1)\geq
C^{n}(n!)^{a}$ for any $a>0,b\in\mathbb{R}$, we get:

$\displaystyle\int_{T_{n}(t)}h_{n}(\boldsymbol{t}_{n})d\boldsymbol{t}_{n}\leq\frac{C^{n}}{(n!)^{2H+1}}(t^{(1+2H)n+1}+t^{(1+2H)n+2-2H}),$

which yields (3.9).

Step 2. In this step, we show that

$\lim_{R\to\infty}\frac{\sigma_{R}^{2}(t)}{R}>0.$

Recall (3.2), we have

$\int_{-R}^{R}\int_{-R}^{R}\gamma_{n}(t,x-y)dxdy>0,$

for all $n\geq 2$, $R>0$ and $t>0$. Then, taking account of (3.2), we can
write

$\displaystyle\lim_{R\to\infty}\frac{\sigma_{R}^{2}(t)}{R}>$
$\displaystyle\lim_{R\to\infty}\frac{1}{2R}\int_{-R}^{R}\int_{-R}^{R}\gamma_{2}(t,x-y)dxdy=4\pi
C_{H}^{2}\int_{0<r<s<t}g^{(2)}_{r,s}(0)drds$ $\displaystyle=$ $\displaystyle
4\pi
C_{H}^{2}\int_{0<r<s<t}\int_{\mathbb{R}}\frac{\sin^{2}((s-r)\eta_{1})}{|\eta_{1}|^{2}}|\eta_{1}|^{2(1-2H)}d\eta_{1}drds$
$\displaystyle=$ $\displaystyle C\int_{0<r<s<t}(s-r)^{4H-1}drds>0,$

where $C>0$ is a constant depending on $H$, and the last line is due to (2.1)
and the fact that $H>1/4$.

Step 3. In this step, we complete the proof of (1.11). Without loss of
generality, assume that $s\leq t$. Similarly to (3.2),

$\mathbb{E}[F_{R}(t)F_{R}(s)]=\sum_{n\geq
1}\frac{1}{n!}\int_{-R}^{R}\int_{-R}^{R}\widehat{\gamma}_{n}(t,s,x-y)dxdy,$

where

$\displaystyle\widehat{\gamma}_{n}(t,s,x-y)$
$\displaystyle=(n!)^{2}\langle\widetilde{f}_{n}(\cdot,t,x),\widetilde{f}_{n}(\cdot,s,y)\rangle_{\mathcal{H}^{\otimes
n}}$
$\displaystyle=n!C_{H}^{n}\int_{T_{n}(s)}\int_{\mathbb{R}^{n}}e^{-i\eta_{n}(x-y)}\prod_{j=1}^{n-1}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}$
$\displaystyle\quad\quad\quad\quad\quad\frac{\sin((t-t_{n})|\eta_{n}|)\sin((s-t_{n})|\eta_{n}|)}{|\eta_{n}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n}d\boldsymbol{t}_{n}.$

The same argument as for (3.2) shows that

$\displaystyle\frac{1}{n!}\int_{-R}^{R}\int_{-R}^{R}\widehat{\gamma}_{n}(t,s,x-y)dxdy=4\pi
C_{H}^{n}R\int_{T_{n}(s)}\big{(}\ell_{R}*\widehat{g}_{\boldsymbol{t}_{n}}^{(n)}\big{)}(0)d\boldsymbol{t}_{n},$

where

$\displaystyle\widehat{g}_{\boldsymbol{t}_{n}}^{(n)}(\eta)\coloneqq$
$\displaystyle\frac{\sin((t-t_{n})|\eta|)\sin((s-t_{n})|\eta|)}{|\eta|^{2}}$
$\displaystyle\times\int_{\mathbb{R}^{n-1}}\prod_{j=1}^{n-1}\frac{\sin^{2}((t_{j+1}-t_{j})|\eta_{j}|)}{|\eta_{j}|^{2}}\prod_{j=1}^{n}|\eta_{j}-\eta_{j-1}|^{1-2H}d\boldsymbol{\eta}_{n-1}.$

Then, relation (1.11) follows by the dominated convergence theorem, as in Step
1. The proof of this theorem is complete. ∎

######  Remark 3.1.

There might be an alternative method for proving Theorem 1.2, which would give
a different representation of the limiting covariance. We explain this method
here. If

$\int_{\mathbb{R}}|\rho_{t}(x)|dx<\infty,$ (3.10)

then by the dominated convergence theorem

$\frac{\sigma_{R}^{2}(t)}{R}=\int_{\mathbb{R}}\rho_{t}(x)\frac{|B_{R}\cap
B_{R}(-x)|}{R}dx\to 2\int_{\mathbb{R}}\rho_{t}(x)dx\quad\mbox{as}\quad
R\to\infty$

where $B_{R}=[-R,R]$ (see [55, Page 27]). Recalling definition (3.1) of
$\rho_{t}(x)$, (3.10) follows if one can show that

$\sum_{n\geq 1}\frac{1}{n!}\int_{\mathbb{R}}|\gamma_{n}(t,x)|dx<\infty.$

Unfortunately, we could not prove that $\gamma_{n}(t,\cdot)$ is integrable on
$\mathbb{R}$.

######  Remark 3.2.

Suppose we can exchange the order of integrals arbitrarily and treat the
Diract distribution $\delta$ as a regular function. Then, we can write

$\displaystyle 4\pi
C_{H}^{n}\int_{T_{n}(t)}g_{\boldsymbol{t}_{n}}^{(n)}(0)d\boldsymbol{t}_{n}=$
$\displaystyle 4\pi
C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}}\delta_{0}(\eta_{n})g_{\boldsymbol{t}_{n}}^{(n)}(\eta_{n})d\eta_{n}d\boldsymbol{t}_{n}$
$\displaystyle=$ $\displaystyle
2C_{H}^{n}\int_{T_{n}(t)}\int_{\mathbb{R}}\Big{(}\int_{\mathbb{R}}e^{-i\eta_{n}x}\mathbf{1}_{\mathbb{R}}(x)dx\Big{)}g_{\boldsymbol{t}_{n}}^{(n)}(\eta_{n})d\eta_{n}d\boldsymbol{t}_{n}$
$\displaystyle=$ $\displaystyle\frac{2}{n!}\int_{\mathbb{R}}\langle
f_{n}(\cdot,t,x),f_{n}(\cdot,t,0)\rangle_{\mathcal{H}}dx.$

Thus it is natural to conjecture that

$\displaystyle\lim_{R\to\infty}\frac{1}{R}\sigma_{R}^{2}=2\int_{\mathbb{R}}\mathrm{Cov}(u(t,x),u(t,0))dx.$
(3.11)

Actually, this has been confirmed in parabolic cases (see [53, 55]) and also
in hyperbolic cases assuming the non-negative correlation in space of driven
noises (see [4]). The aforementioned results rely on the nonnegativity of
$\gamma_{n}(t,x)$, which does not hold in our setting. This prevents us to
provide a proof for (3.11). We expect it can be verified in the future.

###  3.3 Quantitative CLT—Proof of Theorem 1.3

In this section, we prove Theorem 1.3. Applying [54, Proposition 2.4], we get:

$d_{TV}\left(\frac{F_{R}(t)}{\sigma_{R}(t)},Z\right)\leq\frac{2\sqrt{3}}{\sigma_{R}^{2}(t)}\sqrt{\mathcal{A}},$

where

$\mathcal{A}=C_{H}^{3}\int_{[0,t]^{3}}\mathcal{A}_{0}^{*}(r,s,\theta)drdsd\theta,$

and

$\displaystyle\mathcal{A}_{0}^{*}(r,s,$
$\displaystyle\theta)=\int_{\mathbb{R}^{6}}\|D_{r,z}F_{R}(t)-D_{r,z^{\prime}}F_{R}(t)\|_{4}\|D_{\theta,w}F_{R}(t)-D_{\theta,w^{\prime}}F_{R}(t)\|_{4}$
$\displaystyle\times\|D_{(r,z),(s,y)}^{2}F_{R}(t)-D_{(r,z),(s,y^{\prime})}^{2}F_{R}(t)-D_{(r,z^{\prime}),(s,y)}^{2}F_{R}(t)+D_{(r,z^{\prime}),(s,y^{\prime})}^{2}F_{R}(t)\|_{4}$
$\displaystyle\times\|D_{(\theta,w),(s,y)}^{2}F_{R}(t)-D_{(\theta,w),(s,y^{\prime})}^{2}F_{R}(t)-D_{(\theta,w^{\prime}),(s,y)}^{2}F_{R}(t)+D_{(\theta,w^{\prime}),(s,y^{\prime})}^{2}F_{R}(t)\|_{4}$
$\displaystyle\times|y-y^{\prime}|^{2H-2}|z-z^{\prime}|^{2H-2}|w-w^{\prime}|^{2H-2}dydy^{\prime}dzdz^{\prime}dwdw^{\prime}.$

Since Theorem 1.2 concludes that $\sigma_{R}^{2}(t)\sim CR$ as $R\to\infty$,
it is enough to show that $\mathcal{A}\leq CR$.

By Minkowski’s inequality, we get
$\mathcal{A}_{0}^{*}(r,s,\theta)\leq\mathcal{A}_{0}(r,s,\theta)$, where
$\mathcal{A}_{0}$ can be written, after a change of variables, as follows:

$\displaystyle\mathcal{A}_{0}$
$\displaystyle(r,s,\theta)=\int_{[-R,R]^{4}}dx_{1}dx_{2}dx_{3}dx_{4}\int_{\mathbb{R}^{6}}dydy^{\prime}dzdz^{\prime}dwdw^{\prime}dw^{\prime}|y^{\prime}|^{2H-2}|z^{\prime}|^{2H-2}|w^{\prime}|^{2H-2}$
$\displaystyle\times\big{\|}\Delta_{z^{\prime}}(r,z,t,x_{1})\big{\|}_{4}\big{\|}\Delta_{w^{\prime}}(\theta,w,t,x_{2})\big{\|}_{4}\big{\|}\square_{z^{\prime},y^{\prime}}(r,z,s,y,t,x_{3})\big{\|}_{4}\big{\|}\square_{w^{\prime},y^{\prime}}(\theta,w,s,y,t,x_{4})\big{\|}_{4}.$

where $\Delta$ and $\square$ are defined as in (2.17) and (2.4), respectively.

Hence,

$\displaystyle\mathcal{A}\leq\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3}+\mathcal{A}_{4},$

where

$\displaystyle\mathcal{A}_{1}\coloneqq\int_{0<r\vee\theta<s<t}\mathcal{A}_{0}(r,s,\theta)drdsd\theta,\qquad\mathcal{A}_{2}\coloneqq\int_{0<r<s<\theta<t}\mathcal{A}_{0}(r,s,\theta)drdsd\theta,$
$\displaystyle\mathcal{A}_{3}\coloneqq\int_{0<\theta<s<r<t}\mathcal{A}_{0}(r,s,\theta)drdsd\theta,\quad\text{and}\quad\mathcal{A}_{4}\coloneqq\int_{0<s<r\wedge\theta<t}\mathcal{A}_{0}(r,s,\theta)drdsd\theta.$

The estimates for $\mathcal{A}_{1},\dots,\mathcal{A}_{4}$ are quite similar.
Here we only provide a detailed deduction of the estimate for
$\mathcal{A}_{1}$.

$\displaystyle\mathcal{A}_{0}(r,s,\theta)\leq
T_{1}(r,\theta)^{\frac{1}{2}}T_{2}(s,r)^{\frac{1}{2}}\int_{-R}^{R}T_{3}(s,\theta,x_{4})^{\frac{1}{2}}dx_{4}$
(3.12)

where

$\displaystyle\begin{aligned}
T_{1}(r,\theta)\coloneqq\int_{\mathbb{R}^{2}}&\Big{(}\sup_{z\in\mathbb{R}}\int_{\mathbb{R}}\big{\|}\Delta_{z^{\prime}}(r,z,t,x_{1})\big{\|}_{4}dx_{1}\Big{)}^{2}\Big{(}\sup_{w\in\mathbb{R}}\int_{\mathbb{R}}\big{\|}\Delta_{w^{\prime}}(\theta,w,t,x_{2})\big{\|}dx_{2}\Big{)}^{2}\\\
&\times|z^{\prime}|^{2H-2}|w^{\prime}|^{2H-2}dz^{\prime}dw^{\prime}<C,\end{aligned}$
$\displaystyle
T_{2}(s,r)\coloneqq\int_{\mathbb{R}^{2}}\left(\sup_{y\in\mathbb{R}}\int_{\mathbb{R}^{2}}\big{\|}\square_{z^{\prime},y^{\prime}}(r,z,s,y,t,x_{3})\big{\|}_{4}dx_{3}dz\right)^{2}|y^{\prime}|^{2H-2}|z^{\prime}|^{2H-2}dy^{\prime}dz^{\prime}<C,$
and $\displaystyle
T_{3}(s,\theta,x_{4})\coloneqq\int_{\mathbb{R}^{2}}\left(\int_{\mathbb{R}^{2}}\big{\|}\square_{w^{\prime},y^{\prime}}(\theta,w,s,y,y,x_{4})\big{\|}_{4}dydw\right)^{2}|y^{\prime}|^{2H-2}|w^{\prime}|^{2H-2}dy^{\prime}dw^{\prime}<C,$

by using Proposition 2.11. This yields immediately that for any
$0<r\vee\theta<s<t$,

$\displaystyle\mathcal{A}_{0}(r,s,\theta)\leq C\int_{-R}^{R}dx_{4}=CR,$

and thus

$\displaystyle\mathcal{A}_{1}=\int_{\\{0<r<s\vee\theta<s\\}}\mathcal{A}_{0}(r,s,\theta)drdsd\theta\leq
CR.$

Similar arguments ensure that $\mathcal{A}_{k}\leq CR$ for $k=2,3,4$.
Therefore, $\mathcal{A}\leq CR$. This completes the proof of Theorem 1.3.

###  3.4 Functional CLT—Proof of Theorem 1.4

In this section, we include the proof of Theorem 1.4. We fix $T>0$. It
suffices to show the following properties:

  1. (1)

The tightness of the collection of $C([0,T])$-valued random variables
$\\{\sigma_{R}^{-1}F_{R}(r);r\in[0,T]\\}_{R\in\mathbb{R}_{+}}$.

  2. (2)

The convergence in distribution of
$\\{(\sigma_{R}^{-1}F_{R}(t_{1}),\dots,\sigma_{R}^{-1}F_{R}(t_{m})\\}_{R\in\mathbb{R}_{+}}$
to $(\mathcal{G}(t_{1}),\ldots,\mathcal{G}(t_{m}))$ as $R\to\infty$, for all
positive integer $m$ and for all $0\leq t_{1}\leq\dots\leq t_{m}\leq T$.

###### Proof of tightness.

By Kolmogorov-Chentsov criterion ([43, Theorem 23.7]), it is enough to prove
that

$\displaystyle\|F_{R}(t)-F_{R}(s)\|_{p}\leq CR^{1/2}(t-s)^{1/2},$

for all $0\leq s\leq t\leq T$, where $C>0$ is a constant that depends on
$(T,p,H)$. Using (1.6) and the convention $G_{t}(x)=0$ for $t<0$, we write:

$u(t,x)-u(s,x)=\int_{0}^{t}\int_{\mathbb{R}}\big{(}G_{t-r}(x-y)-G_{s-r}(x-y)\big{)}u(r,y)W(dr,dy).$

By stochastic Fubini theorem,

$F_{R}(t)-F_{R}(s)=\int_{0}^{t}\int_{\mathbb{R}}\left(\int_{-R}^{R}\big{(}G_{t-r}(x-y)-G_{s-r}(x-y)\big{)}dx\right)u(r,y)W(dr,dy).$

We use the following inequality: for any predictable process $S$,

$\left\|\int_{0}^{T}\int_{\mathbb{R}}S(t,x)W(dt,dx)\right\|_{p}^{2}\leq
C_{p,H}\int_{0}^{T}\int_{\mathbb{R}^{2}}\|S(t,x)-S(t,y)\|_{p}^{2}|x-y|^{2H-2}dxdydt.$

This inequality follows from the Burhholder-Davis-Gundy inequality for the
stochastic integral with respect to $W$ (given by [2, Theorem 2.9]) followed
by Minkowski inequality for the $\|\cdot\|_{p/2}$-norm. It follows that

$\|F_{R}(t)-F_{R}(s)\|_{p}^{2}\leq
C_{p,H}\int_{0}^{t}\int_{\mathbb{R}^{2}}\|S(r,y)-S(r,z)\|_{p}^{2}|y-z|^{2H-2}dydzdr,$

with
$S(r,y)=\left(\int_{-R}^{R}\big{(}G_{t-r}(x-y)-G_{s-r}(x-y)\big{)}dx\right)u(r,y)$.
It follows that:

$\displaystyle\|F_{R}(t)-F_{R}(s)\|_{p}^{2}\leq
C_{p,H}\big{(}I_{1}+I_{2}\big{)},$

where

$\displaystyle I_{1}$
$\displaystyle\coloneqq\int_{0}^{t}\int_{\mathbb{R}^{2}}\left(\int_{-R}^{R}\big{(}G_{t-r}(x-y)-G_{s-r}(x-y)-G_{t-r}(x-z)+G_{s-r}(x-z)\big{)}dx\right)^{2}$
$\displaystyle\quad\quad\quad\times\|u(r,y)\|_{p}^{2}|y-z|^{2H-2}dydzdr$
$\displaystyle I_{2}$
$\displaystyle\coloneqq\int_{0}^{t}\int_{\mathbb{R}^{2}}\left(\int_{-R}^{R}\big{(}G_{t-r}(x-z)-G_{s-r}(x-z)\big{)}dx\right)^{2}\|u(r,y)-u(r,z)\|_{p}^{2}|y-z|^{2H-2}dydzdr.$

Note that

$\displaystyle\int_{-R}^{R}\big{(}G_{t-r}(x-z)-G_{s-r}(x-z)\big{)}dx=\frac{1}{2}\int_{-R}^{R}1_{\\{s-r<|x-z|<t-r\\}}dx\in[0,t-s].$
(3.13)

Thus, due to Lemmas 2.2 and 2.3, and the fact that $G_{s-r}(x)\leq
G_{t-r}(x)$, we deduce that

$\displaystyle I_{2}\leq$ $\displaystyle
C(t-s)\int_{-R}^{R}\int_{0}^{t}\left(\int_{\mathbb{R}}G_{t-r}(x-z)dz\right)drdx=C(t-s)R\int_{0}^{t}(t-r)dr\leq
C(t-s)R.$

Next, we treat $I_{1}$. By (3.13),

$\displaystyle\left|\int_{-R}^{R}\big{(}G_{t-r}(x-y)-G_{s-r}(x-y)-G_{t-r}(x-z)+G_{s-r}(x-z)\big{)}dx\right|\leq
2(t-s).$

Using this bound and Lemma 2.1, we find that

$\displaystyle I_{1}$ $\displaystyle\leq
2(t-s)\int_{-R}^{R}\int_{0}^{t}\int_{\mathbb{R}^{2}}\big{(}|G_{t-r}(x-y)-G_{t-r}(x-z)|+|G_{s-r}(x-y)-G_{s-r}(x-z)|\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times|y-z|^{2H-2}dydzdrdx$

Note that $|G_{t}(x)-G_{t}(y)|\in\\{1/2,0\\}$ for all $t\geq 0$ and
$x,y\in\mathbb{R}$, thus $|G_{t}(x)-G_{t}(y)|=2|G_{t}(x)-G_{t}(y)|^{2}$. As a
result,

$\displaystyle I_{1}$ $\displaystyle\leq
4(t-s)\bigg{(}\int_{-R}^{R}\int_{0}^{t}\int_{\mathbb{R}^{2}}|G_{t-r}(x-y)-G_{t-r}(x-z)|^{2}|y-z|^{2H-2}dydzdrdx$
$\displaystyle+\int_{-R}^{R}\int_{0}^{t}\int_{\mathbb{R}^{2}}|G_{s-r}(x-y)-G_{s-r}(x-z)|^{2}|y-z|^{2H-2}dydzdrdx\bigg{)}\leq
C(t-s)R,$

where the last inequality is due to [8, Inequality (58)]. ∎

###### Proof of finite dimensional convergence.

The proof follows the same idea as in [4, Section 4.2]. More precisely, it
suffices to show that:

$\displaystyle\mathrm{Var}\big{(}\big{\langle}DF_{R}(t_{i}),-DL^{-1}F_{R}(t_{j})\big{\rangle}_{\mathcal{H}}\big{)}\leq
CR,\quad\mbox{for any $i,j=1,\ldots,m$}.$

This inequality is proved using the same argument as for $\mathcal{A}$ in
Section 3.3, based on an estimate for ${\rm Var}(\langle
DF,-DL^{-1}G\rangle_{\mathcal{H}})$ for two random variables $F$ and $G$,
which can be deduced similarly to the estimate derived for $F=G$ in the proof
of [54, Proposition 2.4]. ∎

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