# Scaling limit of a long-range random walk in time-correlated random
environment

Guanglin Rang  School of Mathematics and Statistics, Wuhan University, Wuhan
430072,China; Computational Science Hubei Key Laboratory, Wuhan University,
Wuhan, 430072, China<EMAIL_ADDRESS>,  Jian Song  Research Center for
Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao
266237, China<EMAIL_ADDRESS>and  Meng Wang  School of Mathematics,
Shandong University, Jinan, Shandong, 250100, China
<EMAIL_ADDRESS>
###### Abstract.

This paper concerns a long-range random walk in random environment in
dimension $1+1$, where the environmental disorder is independent in space but
has long-range correlations in time. We prove that two types of rescaled
partition functions converge weakly to the Stratonovich solution and the Itô-
Skorohod solution respectively of a fractional stochastic heat equation with
multiplicative Gaussian noise which is white in space and colored in time.

###### Key words and phrases:

Directed polymer, scaling limit, stochastic heat equation, fractional noise

###### 2010 Mathematics Subject Classification:

60F05; 60H15; 82C05

###### Contents

  1. 1 Introduction
    1. 1.1 Notations and known results
    2. 1.2 Model description
    3. 1.3 Main result, strategy and discussions
  2. 2 On the SPDE
    1. 2.1 Preliminaries on Gaussian spaces
      1. 2.1.1 Banach spaces associated with $\dot{W}$
      2. 2.1.2 Chaos expansion, Wick product, multiple integrals, etc
    2. 2.2 Mild Stratonovich solution
    3. 2.3 Mild Itô-Skorohod solution
  3. 3 On weak convergences
    1. 3.1 Weak convergence of U-statistics
    2. 3.2 Weak convergence of rescaled partition functions
    3. 3.3 $L^{p}$-bound of rescaled partition functions
  4. A Physical Wick product
  5. B Some preliminaries on convergence of probability measures
  6. C Miscellaneous results

##  1\. Introduction

The model of directed polymer in random environment was first introduced by
Huse and Henley [24] in the study of the Ising model and became a canonical
model for disordered systems (see e.g. the lecture notes by Comets [11]). In
recent years, it has attracted much attention in particular due to its
intimate connections to the stochastic heat equation (parabolic Anderson
model), the stochastic Burgers equation, and the Kardar-Parisi-Zhang (KPZ)
equation and its universality class (see [11] and [12] for a review).

For a simple symmetric random walk in i.i.d. random environment on
$\mathbb{N}\times{\mathbb{Z}}$, Alberts et al. [1] proved that the rescaled
partition function converges weakly to the Itô-Skorohod solution of the
stochastic heat equation (SHE) with multiplicative space-time Gaussian white
noise. This result was extended by Caravenna et al. [8] to a long-range random
walk in i.i.d. random environment and to other disordered models (disordered
pinning model and random field Ising model). Later on, Rang [37] proved that,
for a simple and a long-range random walk in random environment which is white
in time but correlated in space, the rescaled partition function converges
weakly to the Itô-Skorohod solution of SHE with Gaussian noise white in time
and colored in space; this result was extended to long-range random walk by
Chen and Gao [9]; if the random environment is given by the occupation field
of a Poisson system of independent random walks on ${\mathbb{Z}}$ which is now
correlated in both time and space, Shen et al. [41] showed that the scaling
limit is the Stratonovich solution of SHE with space-time colored Gaussian
noise whose covariance coincides with the heat kernel. It is suggested by the
results and methodologies in the above-mentioned papers that the temporal
independence of the random environment plays a critical rule when identifying
the scaling limits for partition functions, which will be further discussed in
Section 1.3.

In this paper, we aim to study the scaling limit of the partition function for
a long-range random walk in random environment on
$\mathbb{N}\times{\mathbb{Z}}$ where the disorder is independent in space but
has long-range correlations in time.

We remark that the model of directed polymer in space-time correlated random
environment whose covariance has a power-law decay was first considered in
physics literature by Medina et al. [32] where the Burgers equation with
colored noise was studied and then applied to analyse directed polymer and
interface growth. We also would like to mention that Rovira and Tindel [39]
introduced Brownian polymer in a centered Gaussian field that is white in time
and correlated in space on $\mathbb{R}_{+}\times\mathbb{R}$ and studied the
asymptotic behavior of the partition function; in the two subsequent papers,
Bezerra et al. [6] obtained the superdiffusivity and Lacoin [30] investigated
the effect of strong spatial correlation. Finally, we remind the reader that
the model of long-range directed polymer has been studied by Comets [10], and
more recently by Wei [44].

###  1.1. Notations and known results

For the reader’s convenience, we collect the mathematical notations that will
be used throughout this article.

######  Notations 1.1.

Let $\mathbb{N}$ denote the set of natural numbers without 0, i.e.,
$\mathbb{N}\overset{\rm{def}}{=}\\{1,2,\dots\\}$, and
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$; for $N\in\mathbb{N}$, $\llbracket
N\rrbracket\overset{\rm{def}}{=}\\{1,2,\dots,N\\}$; for $a\in\mathbb{R}$,
$[a]$ means the greatest integer that is not greater than $a$; $\|\cdot\|$ is
used for the Euclidean norm; let
${\bm{k}}\overset{\rm{def}}{=}(k_{1},\dots,k_{d}),{\bm{l}}\overset{\rm{def}}{=}(l_{1},\dots,l_{d}),{\bm{x}}\overset{\rm{def}}{=}(x_{1},\dots,x_{d}),{\bm{y}}\overset{\rm{def}}{=}(y_{1},\dots,y_{d})$
etc stand for vectors in ${\mathbb{Z}}^{d}$ or $\mathbb{R}^{d}$ depending on
the context; we use $C$ to denote a generic positive constant that may change
from line to line; we say $f(x)\lesssim g(x)$ if $f(x)\leq Cg(x)$ for all $x$;
we write $f(x)\sim g(x)$ (as $x\to\infty$), if
$\lim\limits_{x\to\infty}f(x)/g(x)=1$. We use $\overset{d}{\longrightarrow}$
to denote the convergence in distribution (also called weak convergence) for
random variables/vectors. For a random variable $X$,
$\|X\|_{L^{p}}=\left({\mathbb{E}}[|X|^{p}]\right)^{1/p}$ for $p\geq 1$.

Let $S=\\{S_{i},i\in\mathbb{N}_{0}\\}$ be a random walk in ${\mathbb{Z}}$ and
${\omega}=\\{{\omega}(i,k),(i,k)\in\mathbb{N}\times{\mathbb{Z}}\\}$ be a
family of random variables independent of $S$ serving as the random
environment (disorder). We shall use ${\mathbb{P}}_{S}$ and ${\mathbb{E}}_{S}$
(resp. ${\mathbb{P}}_{{\omega}}$ and ${\mathbb{E}}_{\omega}$) to denote the
probability and expectation in the probability space of $S$ (resp.
${\omega}$), respectively. The probability and expectation in the product
probability space of $(S,{\omega})$ is denoted by ${\mathbb{P}}$ and
${\mathbb{E}}$, respectively.

The partition function is defined by

(1.1) $\displaystyle Z_{\omega}^{(N)}({\beta})=\sum_{S}{\rm
e}^{{\beta}\sum_{i=1}^{N}{\omega}(i,S_{i})}{\mathbb{P}}(S)={\mathbb{E}}_{S}\left[{\rm
e}^{{\beta}\sum_{i=1}^{N}{\omega}(i,S_{i})}\right],$

where ${\beta}>0$ is the inverse temperature, and the point-to-line partition
function (or partition function for simplicity) is given by

(1.2) $\displaystyle
Z_{\omega}^{(N)}(\beta,k)\overset{\rm{def}}{=}{\mathbb{E}}_{S}\left[{\rm
e}^{{\beta}\sum_{i=1}^{N}{\omega}(i,S_{i})}\Big{|}S_{N}=k\right],~{}\text{ for
}k\in{\mathbb{Z}}.$

We stress that the partition function (1.2) is point-to-line in the sense that
there is no restriction on the value $S_{0}$ of the random walk at time 0.
Heuristically, we may view $S$ in (1.2) as a backward random walk on the time
interval $[0,N]$ with $S_{N}=k$.

When $S$ is a simple symmetric random walk with i.i.d. increments and
${\omega}(i,k)$ are i.i.d. random variables with

$\lambda(\beta)\overset{\rm{def}}{=}\log{\mathbb{E}}[{\rm
e}^{\beta{\omega}(i,k)}]<\infty$

for some $\beta$ sufficiently small, Alberts et al. [1] introduced the so-
called intermediate disorder regime. More precisely, if $\beta$ is scaled in
the way

$\beta\to\hat{\beta}_{N}\overset{\rm{def}}{=}\beta N^{-\frac{1}{4}},$

one has the following weak convergence

${\rm
e}^{-N\lambda(\hat{\beta}_{N})}Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/2}x)\overset{d}{\longrightarrow}u(1,x)\text{
as }N\to\infty,$

where $u(t,x)$ is the Itô-Skorohod solution of the equation

(1.3) $\begin{cases}\dfrac{\partial u(t,x)}{\partial t}=\frac{1}{2}\Delta
u(t,x)+\sqrt{2}\beta u(t,x)\dot{W}(t,x),~{}t>0,x\in\mathbb{R},\\\
u(0,x)=1.\end{cases}$

Here, $\dot{W}$ is space-time Gaussian white noise and Itô-Skorohod solution
means that the product in $u(t,x)\dot{W}(t,x)$ is a Wick product, or
equivalently the associated stochastic integral is an Itô-Skorohod integral.

Under the same setting except that the random walk $S$ may be long-range
(i.e., the increments of $S$ now lie in the domain of attraction of a
$\rho$-stable law with $\rho\in(1,2]$), the result of [1] was extended in
Caravenna et al. [8] where a unified framework based on the Lindeberg
principle for polynomial chaos expansion was developed to study the scaling
limits for pinning model, directed polymer model, and Ising model. Still using
the setting of [1] but assuming that the disorder
$\\{{\omega}(i,k),(i,k)\in\mathbb{N}\times\mathbb{Z}\\}$ is correlated in
space (but still independent in time), Rang [37] and Chen-Gao [9] obtained the
weak convergence of the rescaled partition function to the Itô-Skorohod
solution of (1.3) with the Gaussian noise $\dot{W}(t,x)$ being colored in
space (still white in time).

###  1.2. Model description

Motivated by the above-mentioned works, we aim to study the weak convergence
of the rescaled partition function for a long-range random walk
$\\{S_{n},n\in\mathbb{N}\\}$ in random environment on
$\mathbb{N}\times{\mathbb{Z}}$, where the disorder
$\\{{\omega}(i,k),i\in\mathbb{N},k\in{\mathbb{Z}}\\}$ is correlated in time
and independent in space.

In our model, the directed polymer (random walk) is given by
$S_{n}=\sum_{i=0}^{n}Y_{i}$, where $\\{Y_{i}\\}_{i\in\mathbb{N}_{0}}$ are
independent and identically distributed random variables which have a
1-lattice distribution belonging to the domain of attraction of a
$\rho$-stable distribution with density function denoted by
$\mathfrak{g}_{\rho}(\cdot)$. We assume $1<\rho\leq 2$ and

(1.4) $\begin{cases}{\mathbb{P}}(Y_{i}=k)\lesssim|k|^{-1-\rho}\text{ for
}k\in{\mathbb{Z}}\backslash\\{0\\},&\text{ if }\rho\in(1,2),\vspace{0.2cm}\\\
{\mathbb{E}}[Y_{i}]=0\text{ and }{\mathbb{E}}[Y_{i}^{2}]=1,&\text{ if
}\rho=2.\end{cases}$

Let $\psi(u)\overset{\rm{def}}{=}{\mathbb{E}}[{\rm e}^{{\imath}uY_{i}}]$ be
the characteristic function of $Y_{i}$. Then the 1-lattice distribution of
$Y_{i}$ implies that $\psi(u)$ is periodic with period $2\pi$, and
furthermore, $|\psi(u)|<1$ for all $u\in[-\pi,\pi]\backslash\\{0\\}$ (see,
e.g., [25, Theorem 1.4.2]); this property of $\psi(u)$ usually makes analysis
easier. The assumption of $1$-lattice distribution on $Y_{i}$ also yields that
the random walk $S_{n}=\sum_{i=1}^{n}Y_{i}$ has period 1 and hence is
aperiodic . We remind the reader that a simple symmetric random walk has
period 2.

Denote by $P_{n}(k)={\mathbb{P}}(S_{n}=k)$ for
$n\in\mathbb{N},k\in{\mathbb{Z}}$ the probability of $S$ being at $k$ at time
$n$. Then by the local limit theorem for $\rho$-stable distribution (see [40,
Theorem 6.1]), the convergence

(1.5) $n^{1/\rho}P_{n}(k)-\mathfrak{g}_{\rho}\big{(}k/n^{1/\rho}\big{)}\to
0,\text{ as }n\to\infty,$

holds uniformly in $k$. Noting that $\mathfrak{g}_{\rho}$ is a bounded
function, (1.5) yields

(1.6) $\displaystyle P_{n}(k)\lesssim n^{-1/\rho},\text{ for
}n\in\mathbb{N},k\in{\mathbb{Z}}.$

Also under the condition of (1.4), we have a local deviation estimation (see,
e.g., [5, Theorem 2.6]):

(1.7) $\displaystyle P_{n}(k)\lesssim n|k|^{-1-\rho},\text{ for
}n\in\mathbb{N},k\in{\mathbb{Z}}.$

This together with (1.6), we get

(1.8) $\displaystyle P_{n}(k)\lesssim\left(n|k|^{-1-\rho}\right)\wedge
n^{-1/\rho}=n^{-1/\rho}\left(|n^{-1/\rho}k|^{-1-\rho}\wedge 1\right),\text{
for }n\in\mathbb{N},k\in{\mathbb{Z}},$

where $a\wedge b:=\min\\{a,b\\}$ for $a,b\in\mathbb{R}$.

It is well known that

$\frac{S_{n}}{n^{1/\rho}}\overset{d}{\longrightarrow}\xi,~{}~{}~{}~{}\mbox{as}~{}~{}n\rightarrow\infty,$

where $\xi$ has the symmetric $\rho$-stable distribution with characteristic
function $\exp\\{-c_{\rho}|\eta|^{\rho}\\}$ for some $c_{\rho}>0$. Letting
$\mathfrak{g}_{\rho}(t,x)$ be the density function of the corresponding
$\rho$-stable process, we have

$\int_{\mathbb{R}}{\rm e}^{\imath\eta
x}\mathfrak{g}_{\rho}(t,x){\mathrm{d}}x={\rm e}^{-c_{\rho}t|\eta|^{\rho}}.$

Throughout the rest of the paper we will omit the subscript $\rho$ and use
$\mathfrak{g}(t,x)$ exclusively to denote the density function of the
$\rho$-stable process $X$. Note that $\mathfrak{g}$ has the following scaling
property

(1.9)
$\mathfrak{g}(t,x)=t^{-\frac{1}{\rho}}\mathfrak{g}(1,t^{-\frac{1}{\rho}}x),$

and the upper bound in parallel with (1.8)

(1.10) $\displaystyle\mathfrak{g}(t,x)\lesssim(t|x|^{-1-\rho})\wedge
t^{-1/\rho}=t^{-1/\rho}\Big{(}|t^{-1/\rho}x|^{-1-\rho}\wedge 1\Big{)},\quad
t\in\mathbb{R}_{+},~{}x\in\mathbb{R}.$

We assume that the disorder in the environment is given by a family of
Gaussian random variables
$\\{{\omega}(n,k),n\in\mathbb{N},k\in{\mathbb{Z}}\\}$ with mean zero and
covariance

(1.11)
${\mathbb{E}}[{\omega}(n,k){\omega}(n^{\prime},k^{\prime})]=\gamma(n-n^{\prime})\delta_{kk^{\prime}},$

where $\delta_{kk^{\prime}}$ is the Kronecker delta function, i.e.,
$\delta_{kk^{\prime}}=1$ if $k=k^{\prime}$ and $\delta_{kk^{\prime}}=0$
otherwise, and $\gamma(n)$ has a power law decay:

(1.12) $\gamma(n)\lesssim|n|^{2H-2}\wedge 1,\text{ for }n\in{\mathbb{Z}},$

where $H\in(1/2,1]$ denotes a fixed constant throughout this paper. We further
assume that for all $t\in\mathbb{R}\backslash\\{0\\}$,

(1.13) $\lim_{N\to\infty}N^{2-2H}\gamma([Nt])=|t|^{2H-2}.$

###  1.3. Main result, strategy and discussions

Consider the stochastic fractional heat equation on $\mathbb{R}$

(1.14) $\begin{cases}\dfrac{\partial u(t,x)}{\partial
t}=-(-\Delta)^{\frac{\rho}{2}}u(t,x)+\beta u(t,x)\dot{W}(t,x),\\\
u(0,x)=1,\end{cases}$

where $\rho\in(1,2]$ and $\dot{W}(t,x)$ is Gaussian noise with covariance
function given by

(1.15)
${\mathbb{E}}\bigg{[}\dot{W}(t,x)\dot{W}(s,y)\bigg{]}=K(t-s,x-y)\overset{\rm{def}}{=}|t-s|^{2H-2}\delta(x-y),$

with $H\in(1/2,1]$ and $\delta(\cdot)$ being the Dirac delta function.

We consider two types of solutions of (1.14)—the Stratonovich solution if the
product $u(t,x)\dot{W}(t,x)$ is an ordinary product (i.e., the associated
stochastic integral is a Stratonovich integral) and the Itô-Skorohod solution
if $u(t,x)\dot{W}(t,x)$ is a Wick product (i.e., the associated stochastic
integral is a Skorohod integral). See Section 2.1 for details.

Now we are ready to present our main results. For the first type of partition
function $Z_{\omega}^{(N)}$ which is given in (1.2), we have the following
result.

######  Theorem 1.2.

Let the random walk $S$ and the disorder ${\omega}$ be given as in Section
1.2. Let $H$ and $\rho$ be parameters satisfying

(1.16) $H\in(1/2,1]~{}\text{ and
}~{}\theta\overset{\rm{def}}{=}H-\frac{1}{2\rho}>\frac{1}{2}.$

Consider the rescaled partition function
$Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$ given in (1.2) under the
scaling

$\beta\to\hat{\beta}_{N}\overset{\rm{def}}{=}\beta N^{-\theta}.$

Let $u(t,x)$ be the Stratonovich solution of (1.14). Then we have

$Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})\overset{d}{\longrightarrow}u(1,x_{0}),\text{
as }N\to\infty.$

The second type of point-to-line partition function is defined by

(1.17)
$\displaystyle\tilde{Z}_{\omega}^{(N)}(\beta,k)\overset{\rm{def}}{=}{\mathbb{E}}_{S}\left[{\rm
e}^{{\beta}\sum_{i=1}^{N}{\omega}(i,S_{i})-\frac{\beta^{2}}{2}\sum_{i,j=1}^{N}\gamma(i-j)\mathbf{1}_{\\{S_{i}=S_{j}\\}}}\bigg{|}S_{N}=k\right],~{}\text{
for }k\in{\mathbb{Z}},$

which is adapted from (1.2). The extra term in the exponential is half of the
variance of ${\beta}\sum_{i=1}^{N}{\omega}(i,S_{i})$ conditional on the random
walk $S$. Indeed, we have

(1.18)
$\displaystyle{\mathbb{E}}_{\omega}\left[\left(\sum_{i=1}^{N}{\omega}(i,S_{i})\right)^{2}\right]=$
$\displaystyle{\mathbb{E}}_{\omega}\left[\sum_{i=1}^{N}\sum_{k\in{\mathbb{Z}}}{\omega}(i,k)\mathbf{1}_{\\{S_{i}=k\\}}\sum_{j=1}^{N}\sum_{l\in{\mathbb{Z}}}{\omega}(j,l)\mathbf{1}_{\\{S_{j}=l\\}}\right]$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{N}\sum_{k,l\in{\mathbb{Z}}}\gamma(i-j)\delta_{kl}\mathbf{1}_{\\{S_{i}=k\\}}\mathbf{1}_{\\{S_{j}=l\\}}$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{N}\gamma(i-j)\mathbf{1}_{\\{S_{i}=S_{j}\\}},$

which can be viewed as a weighted intersection local time of $S$.

For the scaling limit of $\tilde{Z}_{\omega}^{(N)}$ given in (1.17), we have
the following result in parallel with Theorem 1.2.

######  Theorem 1.3.

Let the random walk $S$ and the disorder ${\omega}$ be given as in Section
1.2. Let $H$ and $\rho$ be parameters satisfying

(1.19) $H\in(1/2,1],~{}\rho\in(1,2],\text{ and
}~{}\theta=H-\frac{1}{2\rho}>0.$

Consider the rescaled partition function
$\tilde{Z}_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$ given in (1.17)
under the scaling

$\beta\to\hat{\beta}_{N}\overset{\rm{def}}{=}\beta N^{-\theta}.$

Let $\tilde{u}(t,x)$ be the Itô-Skorohod solution of (1.14). Then we have

$\tilde{Z}_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})\overset{d}{\longrightarrow}\tilde{u}(1,x_{0}),\text{
as }N\to\infty.$

######  Remark 1.4.

In Theorem 1.3, the condition $\rho>1$ in (1.19) arises to guarantee the
existence and uniqueness of the Itô-Skorohod solution of (1.14) (see Remark
2.13 in Section 2.3). In Theorem 1.2, the condition (1.16) also implies
$\rho>\frac{1}{2H-1}\geq 1$. This is why we restrict us on the case
$\rho\in(1,2]$ in this article.

######  Remark 1.5.

For (1.14), it requires less restrictive conditions to have an Itô-Skorohod
solution than to have a Stratonovich solution (see Remark 2.17 and see also
[42] for a more general class of SPDEs). This explains why condition (1.19)
for the Itô-Skorhod case is strictly weaker than condition (1.16) for the
Stratonovich case.

The proofs of Theorems 1.2 and 1.3 will be presented in Section 3. For the
reader’s convenience, here we explain the strategy briefly for Theorem 1.2
(the proof of Theorem 1.3 is similar but easier). By Taylor’s expansion, the
rescaled partition function can be written as

$\displaystyle Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}{\mathbb{S}}_{m}^{(N)},$

where

(1.20)
$\displaystyle{{\mathbb{S}}}_{m}^{(N)}=\hat{\beta}^{m}_{N}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}{\omega}(n_{1},k_{1})\cdots{\omega}(n_{m},k_{m})P_{{\bm{n}}^{*}}(N^{1/\rho}x_{0};k_{1},\dots,k_{m}),$

with $P_{{\bm{n}}^{*}}$ (see (3.4)) being the product of the transition
densities of the random walk $S$. Meanwhile, the solution of the continuum
equation (1.14) has the series representation (see equation (2.27)):

$u(1,x_{0})=\sum_{m=0}^{\infty}\beta^{m}\mathbb{I}_{m}(\mathfrak{g}_{m}(\cdot;1,x_{0})),$

where $\mathbb{I}_{m}(\cdot)$ is a multiple Stratonovich integral (see Section
2.1.2) and $\mathfrak{g}_{m}$ (see (2.28)) is the product of the transition
densities of the limiting stable process $X$. We remind the reader that the
continuum multiple Stratonovich integral
$\mathbb{I}_{m}(\mathfrak{g}_{m}(\cdot;1,x_{0})$ resembles the discrete sum
$\mathbb{S}_{m}^{(N)}$.

Under condition (1.16), one can obtain the $L^{2}$-convergence of the series
of $u(t,x)$(see Proposition 2.12) and the uniform (in $N$) $L^{1}$-convergence
of the series of $Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$ (see
(3.39)). Thus, in order to prove the weak convergence
$Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})\overset{d}{\longrightarrow}u(1,x_{0})$,
in light of Lemma B.1, it suffices to obtain joint weak convergences for
$\mathbb{S}_{m}^{(N)}$’s, that is, for $k\in\mathbb{N}$ and
$l_{1},\dots,l_{k}\in\mathbb{N}$,

$\left(\frac{1}{l_{1}!}\mathbb{S}_{l_{1}}^{(N)},\dots,\frac{1}{l_{k}!}\mathbb{S}_{l_{k}}^{(N)}\right)\overset{d}{\longrightarrow}\Big{(}\beta^{l_{1}}\mathbb{I}_{l_{1}}(\mathfrak{g}_{l_{1}}),\dots,\beta^{l_{k}}\mathbb{I}_{l_{k}}(\mathfrak{g}_{l_{k}})\Big{)},\text{
as }N\to\infty,$

which is proved in Proposition 3.4.

Multiple Wiener integrals $\mathbf{I}_{m}(f)$ and multiple Stratonovich
integrals $\mathbb{I}_{m}(f)$ are linked via the celebrated Hu-Meyer’s formula
(2.19). Usually it is more convenient to deal with multiple Wiener integrals
$\mathbf{I}_{m}(f)$ whose second moment is easier to calculate. For this
reason, in Section 3.1 we first consider $U$-statistics given in (3) which are
multi-linear Wick polynomials of ${\omega}(n_{i},k_{i})$’s, and prove that
they converge weakly to multiple Wiener integrals (see Proposition 3.3). This
together with Hu-Meyer’s formula (2.19) then yields Proposition 3.4.

Note that the exponential term in the modified partition function
$\tilde{Z}_{\omega}^{(N)}$ in (1.17) is actually a Wick exponential (see
(2.15)) of $\beta\sum_{i=1}^{N}{\omega}(i,S_{i})$ conditional on the random
walk $S$. Thus, to prove Theorem 1.3, one only needs Proposition 3.3 on the
weak convergence of multi-linear Wick polynomials.

As can be seen, our approach is very much inspired by [1] where the random
variables of disorder are i.i.d. However, in our setting the disorder is
correlated in time, and it turns out that this difference is critical in the
sense that it prevents us from using the trick of “modified partition
function” (see in [1, equation (4)], which was also employed in [37]), as this
method relies on the temporal independence of the disorder. For the same
reason, the Mayer expansion (see [8, equations (1.5) and (1.7)]), which was
employed in [8] to linearize the exponential while introducing an error term
that comes from the Itô correction, cannot be applied here either, since it
works well only if the disorder is independent in both time and space.

To conclude the introduction, we make some remarks on our main results
Theorems 1.2 and 1.3.

(i) In the polymer model, the temporal independence of the disorder plays a
critical rule. Recently, directed polymer in time-space correlated random
environment was studied in [41] and it was shown that the rescaled partition
function converges weakly to a Stratonovich solution. Assuming that the
disorder is correlated in time and white in space in our setting, the rescaled
partition functions $Z_{\omega}^{(N)}$ and $\tilde{Z}_{\omega}^{(N)}$
converges weakly to Stratonovich and Itô-Skorohod solutions, respectively. In
contrast, if the disorder possesses temporal independence, after a proper
scaling, the limit of the rescaled partition function is an Itô-Skorohod
solution but not a Stratonovich solution, as has been shown in [1, 8, 37, 9].
Among others, a technical explanation for the critical role of the temporal
independence of the disorder is the following: when the temporal independence
of disorder appears, by using the “modified partition function” trick, the
term corresponding to $\mathbb{S}_{m}^{(N)}$ given by (1.20) becomes a multi-
linear polynomial of ${\omega}(n_{i},k_{i})$ for $i=1,\dots,m$ with
$n_{1}<n_{2}<\cdots<n_{m}$, which converges weakly to a multiple Wiener
integral due to the independence (see [34, 8]).

(ii) In contrast to the case that the disorder is independent in time where
one can employ the “modified partition function” trick (or Mayer’s expansion
for the case that the disorder is independent both in time and space) to make
time indices in the summation distinct from each other, we expand the
partition function directly via Taylor’s expansion, and as a consequence the
time indices $n_{1},\dots,n_{m}$ in the summation (1.20) can be repeated. It
turns out that this in fact gives negligible contribution for the Stratonovich
case, due to the condition $\theta>\frac{1}{2}$ in (1.16) which is more
restrictive than the condition $\theta>0$ in (1.19) for the Skorohod case (see
also [1, 8, 37]). For instance, when $m=2$, the expectation of the sum of the
diagonal terms in (1.20) is $\beta\gamma(0)N^{1-2\theta}$ which converges to
zero as $N$ tends to infinity if we assume $\theta>1/2$.

(iii) The study of the scaling limit of partition functions was initiated in
[1] in order to understand the polymer behavior in the so-called intermediate
disorder regime which sits between weak and strong disorder regimes.
Meanwhile, as pointed out in [8], the fact that the scaling limit of partition
functions converges weakly to a non-trivial limit indicates that the directed
polymer model is disorder relevant, since it implies that the presence of
disorder, no matter how small it is, changes the qualitative features of the
underlying homogeneous model.

(iv) As pointed out in [32], it is of interest to study directed polymer in
random environment which has long-range correlations in time and space. In
light of the work of [41], we expect that our results and methodology can be
extended to the model in dimension $1+d$ with space-time long-range correlated
disorder. Note that the corresponding continuum SPDEs have been investigated
in [23, 42]. For instance, consider the limiting Gaussian noise with
covariance

${\mathbb{E}}\bigg{[}\dot{W}(t,x)\dot{W}(s,y)\bigg{]}=|t-s|^{-\beta_{0}}|x-y|^{-\beta},$

where $\beta_{0}\in(0,1)$ and $\beta\in(0,d)$. Then (1.3) has a Stratonovich
solution if $\beta<\rho(1-\beta_{0})$ (see Remark 3.1 in [42]) and an Itô-
Skorohod solution if $\beta<\rho$ (see Remark 5.1 and Theorem 5.3 in [42]). In
particular, this allows to consider the case $d>1,\rho\in(0,1]$ (recall that
the spatial independence forces us to consider only the case
$d=1,\rho\in(1,2]$, see Remarks 1.4 and 2.13).

(v) In our model, we assume the Gaussianity of the disorder ${\omega}$ for
technical reasons. The loss of Gaussian property in ${\omega}$ shall cause a
lot more complexity in computations (see e.g. (A.7) and (A.8) for a comparison
of the computations of moments for Gaussian and non-Gaussian random
variables). The functional analytic approach developed in [41] might be a
choice to circumvent this difficulty. Nevertheless, we conjecture that our
result still holds for sub-Gaussian disorder.

(vi) Assume the disorder $\\{{\omega}(i,k),i\in\mathbb{N},k\in{\mathbb{Z}}\\}$
is a family of i.i.d. standard Gaussian random variables. In this situation,
the exponent in the correction term of the rescaled partition function
$\tilde{Z}_{\omega}^{(N)}(\hat{\beta}_{N},k)$ becomes, recalling (1.18) and
replacing $\gamma(i-j)$ by $\delta_{ij}$ therein,

$\frac{1}{2}{\hat{\beta}_{N}^{2}}\sum_{i,j=1}^{N}\delta_{ij}\mathbf{1}_{\\{S_{i}=S_{j}\\}}=\frac{1}{2}\hat{\beta}_{N}^{2}N=\frac{\beta^{2}}{2}N^{\frac{1}{\rho}},$

which is now independent of the random walk $S$. Thus, the rescaled function
for the Itô-Skorohod case can be written as

$\tilde{Z}_{\omega}^{(N)}(\hat{\beta}_{N},k)={\rm
e}^{-\frac{\beta^{2}}{2}N^{\frac{1}{\rho}}}Z_{\omega}^{(N)}(\hat{\beta}_{N},k),$

which converges weakly to the Itô-Skorohod solution of (1.14) with space-time
white noise. This is consistent with [1, Theorem 2.1], noting that
$N\log{\mathbb{E}}[{\rm
e}^{\hat{\beta}_{N}{\omega}(i,k)}]=\frac{1}{2}\hat{\beta}_{N}^{2}N$ if we
assume the disorder is Gaussian.

(vii) By Skorohod’s representation theorem, if the disorder ${\omega}$ and the
limiting noise $\dot{W}$ live in the same probability space, we can get strong
convergence (a.s. convergence) in Theorems 1.2 and 1.3. In this sense, the
rescaled partition functions $Z_{\omega}^{(N)}$ and $\tilde{Z}_{\omega}^{(N)}$
can be viewed as approximations the Stratonovich and Itô-Skorohod solutions of
equation (1.14), respectively. See e.g. Foondun at el. [15] and Joseph et al.
[28] for related results.

(viii) As a byproduct, we obtain a Jensen type inequality for the integrals
induced by fractional Brownian motion (see Lemmas 2.2 and C.1) which is new in
the literature to our best knowledge. We provide a new approach to prove the
exponential integrability of the weighted self-intersection local time of the
$\rho$-stable process in Proposition 2.7 which plays a critical rule in
obtaining the Feynman-Kac formula for the solution of (1.14); this approach
also simplifies the proof in [23] which works exclusively for the weighted
self-intersection local time of Brownian motion.

The rest of the paper is organized as follows. In Section 2, we provide some
preliminaries on Gaussian spaces and then study the limiting SPDE (1.14). The
main result Theorem 1.2 is proved in Section 3. In Appendices A, B, and C, we
collect some preliminaries on Wick products, convergence of probability
measures, and some other miscellaneous results that are used in the analysis.

##  2\. On the SPDE

In this section, we first recall some preliminaries on Gaussian spaces and
then we study the limiting continuum SPDE (1.14).

###  2.1. Preliminaries on Gaussian spaces

In this subsection, we provide some preliminaries on Gaussian spaces. We refer
to [21, 26, 35] for more details.

####  2.1.1. Banach spaces associated with $\dot{W}$

On probability space $({\Omega},{\mathcal{F}},P)$ satisfying usual conditions,
let $\dot{W}=\\{\dot{W}(t,x):t\in[0,1],x\in\mathbb{R}\\}$ be real-valued
centered Gaussian noise with covariance

(2.1) $\displaystyle{\mathbb{E}}[\dot{W}(t,x)\dot{W}(s,y)]=$ $\displaystyle
K(t-s,x-y)=|t-s|^{2H-2}\delta(x-y),$

where $\delta$ is the Dirac delta function. The Hilbert space $\mathscr{H}$
associated with $\dot{W}$ is the completion of smooth functions with compact
support under the inner product

(2.2) $\displaystyle\langle f,g\rangle_{\mathscr{H}}=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}\int_{\mathbb{R}}f(s,x)K(s-t,x-y)g(t,y){\mathrm{d}}x{\mathrm{d}}y{\mathrm{d}}s{\mathrm{d}}t$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}|s-t|^{2H-2}f(s,x)g(t,x){\mathrm{d}}x{\mathrm{d}}s{\mathrm{d}}t.$

We assume $H\in(1/2,1].$ For the case $H=1$, the noise $\dot{W}(x)$ is indeed
a spatial white noise which does not depend on time.

We remark that the Hilbert space ${\mathscr{H}}$ contains distributions (see
[36]). For our purpose, it suffices to just consider classical measurable
functions. We introduce the following Banach space $\mathcal{B}$ which is a
subset of the set $\mathscr{B}([0,1]\times\mathbb{R})$ of measurable functions
on $[0,1]\times\mathbb{R}$.

######  Definition 2.1.

We define

(2.3)
$\displaystyle\mathcal{B}\overset{\rm{def}}{=}\left\\{f\in\mathscr{B}([0,1]\times\mathbb{R}):\|f\|_{\mathcal{B}}\overset{\rm{def}}{=}\left(\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}|s-t|^{2H-2}|f(s,x)||f(t,x)|{\mathrm{d}}x{\mathrm{d}}s{\mathrm{d}}t\right)^{\frac{1}{2}}<\infty\right\\}.$

Clearly $\mathcal{B}$ is a dense subset of $\mathscr{H}$, and if
$f\in\mathscr{H}$ is a nonnegative function, $f$ also belongs to
$\mathcal{B}$.

For $N\in\mathbb{N}$ and $f\in\mathcal{B}^{\otimes m}$, we denote by
$\mathcal{A}_{N}(f)$ the conditional expectation of $f$ with respect to the
$\sigma$-algebra $\mathscr{B}_{N}^{m}=\sigma({\mathfrak{R}}_{N}^{m})$
generated by the set ${\mathfrak{R}}_{N}^{m}$ of rectangles

(2.4)
${\mathfrak{R}}_{N}^{m}:=\left\\{\left(\frac{{{\bm{i}}}-\overrightarrow{1}_{m}}{N},\frac{{{\bm{i}}}}{N}\right]\times\left(\frac{{{\bm{k}}-{\overrightarrow{1}}_{m}}}{N^{1/\rho}},\frac{{{\bm{k}}}}{N^{1/\rho}}\right]:{{\bm{i}}}\in\llbracket
N\rrbracket^{m},{{\bm{k}}}\in{{\mathbb{Z}}}^{m}\right\\},$

where ${\overrightarrow{1}}_{m}$ is the $m$-dimensional vector of all ones.
That is, $\mathcal{A}_{N}(f)$ is defined by the average values of $f$ on the
blocks $B\in{\mathfrak{R}}_{N}^{m}$:

(2.5)
$\displaystyle{\mathcal{A}_{N}(f)}({{\bm{t}}},{{\bm{x}}})=\frac{1}{|B|}\int_{B}f({{\bm{s}}},{{\bm{y}}}){\mathrm{d}}{{\bm{s}}}{\mathrm{d}}{{\bm{y}}}\cdot\mathbf{1}_{B}({\bm{t}},{\bm{x}}),$

where $\mathbf{1}_{B}$ is an indicator function and $|B|$ is the Lebesgue
measure of $B$.

The following Jensen type of inequality will be used to prove the weak
convergence of U-statistics in Section 3.1.

######  Lemma 2.2.

For $m,N\in\mathbb{N}$, consider $f\in\mathcal{B}^{\otimes m}$ and let
${\mathcal{A}_{N}(f)}$ be given by (2.5). Then, we have

(2.6) $\|{\mathcal{A}_{N}(f)}\|_{\mathcal{B}^{\otimes m}}\leq
C^{m}\|f\|_{\mathcal{B}^{\otimes m}},$

where $C$ is a constant depending only on $H$.

Proof We prove (2.6) for $m=1$ and the general case can be proved in a similar
way. Note that by Fubini’s theorem,
$\mathcal{A}_{N}(f)=\mathcal{A}^{\text{t}}_{N}(\mathcal{A}^{\text{x}}_{N}(f))=\mathcal{A}^{\text{x}}_{N}(\mathcal{A}^{\text{t}}_{N}(f))$,
where $\mathcal{A}^{\text{t}}_{N}$ (resp. $\mathcal{A}^{\text{x}}_{N}$) means
taking average in time (resp. space) only. Therefore, we have

$\|\mathcal{A}_{N}(f)\|_{\mathcal{B}}=\|\mathcal{A}_{N}^{\text{t}}(\mathcal{A}^{\text{x}}_{N}(f))\|_{\mathcal{B}}\leq
C\|\mathcal{A}^{\text{x}}_{N}(f)\|_{\mathcal{B}},$

for some $C$ depending on $H$ only, where the inequality follows from Lemma
C.1.

Now, it suffices to prove
$\|\mathcal{A}^{\text{x}}_{N}(f)\|_{\mathcal{B}}\leq\|f\|_{\mathcal{B}}$.
Using the identity, for $H\in(\frac{1}{2},1)$,

$|s-t|^{2H-2}=c_{H}\int_{\mathbb{R}}|s-\tau|^{H-\frac{3}{2}}|t-\tau|^{H-\frac{3}{2}}{\mathrm{d}}\tau$

where $c_{H}$ is a finite number only depending on $H$, we have

$\displaystyle\|\mathcal{A}^{\text{x}}_{N}(f)\|_{\mathcal{B}}^{2}$
$\displaystyle=c_{H}\int_{\mathbb{R}}{\mathrm{d}}x\int_{0}^{1}{\mathrm{d}}s\int_{0}^{1}{\mathrm{d}}t\int_{\mathbb{R}}{\mathrm{d}}\tau|s-\tau|^{H-\frac{3}{2}}|t-\tau|^{H-\frac{3}{2}}\left|\mathcal{A}^{\text{x}}_{N}(f)\right|(s,x)\left|\mathcal{A}^{\text{x}}_{N}(f)\right|(t,x)$
$\displaystyle=c_{H}\int_{\mathbb{R}}{\mathrm{d}}x\int_{\mathbb{R}}{\mathrm{d}}\tau\left(\int_{0}^{1}{\mathrm{d}}s|s-\tau|^{H-\frac{3}{2}}\left|\mathcal{A}^{\text{x}}_{N}(f)\right|(s,x)\right)^{2}$
$\displaystyle\leq
c_{H}\int_{\mathbb{R}}{\mathrm{d}}x\int_{\mathbb{R}}{\mathrm{d}}\tau\left(\mathcal{A}^{\text{x}}_{N}\left(\int_{0}^{1}{\mathrm{d}}s|s-\tau|^{H-\frac{3}{2}}|f|(s,\cdot)\right)(\tau,x)\right)^{2}$
$\displaystyle\leq
c_{H}\int_{\mathbb{R}}{\mathrm{d}}x\int_{\mathbb{R}}{\mathrm{d}}\tau\left(\int_{0}^{1}{\mathrm{d}}s|s-\tau|^{H-\frac{3}{2}}|f|(s,x)\right)^{2}$
$\displaystyle=\|f\|_{\mathcal{B}}^{2},$

where the last inequality follows from the classical Jensen’s inequality. ∎

####  2.1.2. Chaos expansion, Wick product, multiple integrals, etc

Recall that $\mathscr{H}$ is the Hilbert space associated with Gaussian noise
$\dot{W}$ with covariance (2.1). Let $\\{W(f),f\in\mathscr{H}\\}$ be an
isonormal Gaussian process with covariance

(2.7) ${\mathbb{E}}[W(f)W(g)]=\langle f,g\rangle_{\mathscr{H}}.$

In particular, if $f=\mathbf{1}_{[0,t]\times[0,x]}$, we denote

$W(t,x)\overset{\rm{def}}{=}W(\mathbf{1}_{[0,t]\times[0,x]}).$

Then the fractional noise $\dot{W}(t,x)$ can be viewed as the partial
derivative $\frac{\partial^{2}}{\partial t\partial x}W(t,x)$ in the sense of
distribution. For $f\in\mathscr{H}$, we also use the integral form to denote
the Wiener integral $W(f)$:

$\int_{0}^{1}\int_{\mathbb{R}}f(s,y)W({\mathrm{d}}s,{\mathrm{d}}y)\overset{\rm{def}}{=}W(f).$

For $m\in\mathbb{N}\cup\\{0\\}$, let

(2.8) $H_{m}(x)\overset{\rm{def}}{=}(-1)^{m}{\rm
e}^{x^{2}/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}}x^{m}}{\rm e}^{-x^{2}/2},\quad
x\in{\mathbb{R}}$

be the $m$th Hermite polynomial. For $g\in\mathscr{H}$ with
$\|g\|_{\mathscr{H}}=1$, the multiple Wiener integral of $g^{\otimes
m}\in\mathscr{H}^{\otimes m}$ can be defined via (see e.g. [20, 35])

(2.9) $\mathbf{I}_{m}(g^{\otimes m})\overset{\rm{def}}{=}H_{m}(W(g)).$

In particular, we have $W(g)=\mathbf{I}_{1}(g)$ and $\mathbf{I}_{m}(g^{\otimes
m})=:\\!W(g)^{m}\\!:$, where $:\\!W(g)^{m}\\!:$ is a physical Wick product
(see Section A).

For $f\in\mathscr{H}^{\otimes m}$, let $\hat{f}$ be its symmetrization, i.e.,

$\hat{f}(t_{1},x_{1},\dots,t_{m},x_{m})=\frac{1}{m!}\sum_{\sigma\in\mathcal{P}_{m}}f(t_{\sigma(1)},x_{\sigma(1)},\dots,t_{\sigma(m)},x_{\sigma(m)}),$

where $\mathcal{P}_{m}$ is the set of all permutations of $\llbracket
m\rrbracket=\\{1,2,\dots,m\\}.$ Let ${\mathscr{H}}^{\hat{\otimes}m}$ be the
symmetrization of ${\mathscr{H}}^{\otimes m}$. Then for
$f\in{\mathscr{H}}^{\hat{\otimes}m}$, one can define the $m$th multiple Wiener
integral $\mathbf{I}_{m}(f)$ via (2.9) by a limiting argument. Moreover, for
$f\in{\mathscr{H}}^{\hat{\otimes}m}$ and $g\in{\mathscr{H}}^{\hat{\otimes}n}$,
we have

(2.10) ${\mathbb{E}}[\mathbf{I}_{m}(f)\mathbf{I}_{n}(g)]=m!\langle
f,g\rangle_{\mathscr{H}^{\otimes m}}\delta_{mn},$

where we recall that $\delta_{mn}$ is the Kronecker delta function. For
$f\in{\mathscr{H}}^{\otimes m}$ which is not necessarily symmetric, we simply
let $\mathbf{I}_{m}(f)\overset{\rm{def}}{=}\mathbf{I}_{m}(\hat{f}).$ We also
take the following notation for multiple Wiener integrals:

(2.11)
$\displaystyle\int_{([0,1]\times{\mathbb{R}})^{m}}f({{\bm{t}}},{{\bm{x}}})W({\mathrm{d}}{t_{1}},{\mathrm{d}}{x_{1}})\diamond\cdots\diamond
W({\mathrm{d}}{t_{m}},{\mathrm{d}}{x_{m}})\overset{\rm{def}}{=}\mathbf{I}_{m}(f),\text{
for }f\in\mathscr{H}^{\otimes m}.$

For $f\in\mathscr{H}^{\hat{\otimes}m}$ and $g\in\mathscr{H}^{\hat{\otimes}n}$,
their contraction of $r$ indices for $1\leq r\leq m\land n$ is defined by

$\displaystyle(f\otimes_{r}g)(t_{1},x_{1},\dots,t_{m+n-2r},x_{m+n-2r})$
$\displaystyle\overset{\rm{def}}{=}\int_{[0,1]^{2r}}\int_{{\mathbb{R}}^{2r}}f(t_{1},x_{1},\dots,t_{n-r},x_{n-r},u_{1},y_{1},\dots,u_{r},y_{r})\prod_{i=1}^{r}K(u_{i}-v_{i},y_{i}-z_{i})$
$\displaystyle\qquad\qquad\qquad\times
g(t_{n-r+1},x_{n-r+1},\dots,t_{n+m-2r},x_{n+m-2r},v_{1},z_{1},\dots,v_{r},z_{r}){\mathrm{d}}{{\bm{y}}}{\mathrm{d}}{{\bm{z}}}{\mathrm{d}}{\bm{u}}{\mathrm{d}}{\bm{v}},$

and we have the following recursive formulas:

(2.12) $\displaystyle\mathbf{I}_{m}(f)\mathbf{I}_{n}(g)=\sum_{r=0}^{m\land
n}r!\binom{n}{r}\binom{m}{r}\mathbf{I}_{m+n-2r}(f\otimes_{r}g).$

In particular, when $n=1$ we have,

(2.13)
$\displaystyle\mathbf{I}_{m}(f)\mathbf{I}_{1}(g)=\mathbf{I}_{m+1}(f\otimes
g)+m\mathbf{I}_{m-1}(f\otimes_{1}g).$

Square integrable random variables have a unique chaos expansion as stated
below.

######  Proposition 2.3.

Let $\mathcal{G}$ be the $\sigma$-field generated by $W$. Then for any $F\in
L^{2}({\Omega},\mathcal{G},P)$, it admits a unique chaos expansion

$F=\sum_{m=0}^{\infty}\mathbf{I}_{m}(f_{m})~{}\text{ with
}~{}f_{m}\in\mathscr{H}^{\hat{\otimes}m},$

where the series converges in $L^{2}$. Moreover,

${\mathbb{E}}[F^{2}]=\sum_{m=0}^{\infty}{m!}\|f_{m}\|^{2}_{\mathscr{H}^{\otimes
m}}.$

For $f\in\mathscr{H}^{\otimes m}$ and $g\in\mathscr{H}^{\otimes n}$, the
probabilistic Wick product (or Wick product) of $\mathbf{I}_{m}(f)$ and
$\mathbf{I}_{n}(g)$ is defined by

(2.14)
$\mathbf{I}_{m}(f)\diamond\mathbf{I}_{n}(g)\overset{\rm{def}}{=}\mathbf{I}_{m+n}(f\otimes
g).$

Unlike the physical Wick product, the Wick product defined by (2.14) has the
same properties as the ordinary product. As an example, by (2.14) and (2.13),
we can show that for $f\in\mathscr{H}^{\otimes m}$ and $g\in\mathscr{H}$,

$\mathbf{I}_{m}(f)\diamond\mathbf{I}_{1}(g)=\mathbf{I}_{m}(f)\mathbf{I}_{1}(g)-m\mathbf{I}_{m-1}(f\otimes_{1}g).$

For two square integrable random variables
$F=\sum_{m=0}^{\infty}\mathbf{I}_{m}(f_{m})$ and
$G=\sum_{n=0}^{\infty}\mathbf{I}_{n}(g_{n})$, their Wick product is given by

$F\diamond
G=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\mathbf{I}_{m+n}(f_{m}\otimes g_{n}),$

as long as the right-hand side is well defined. For instance, let
$\varepsilon(u)$ be the exponential vector of $u$ for $u\in\mathscr{H}$:

$\varepsilon(u)\overset{\rm{def}}{=}\exp\left(W(u)-\frac{\|u\|^{2}_{\mathscr{H}}}{2}\right).$

Then $\varepsilon(u)\diamond\varepsilon(v)=\varepsilon(u+v)$ for
$u,v\in\mathscr{H}$. We also recall that for a centered Gaussian random
variable $F$, its Wick exponential is given by

(2.15)
$\exp^{\diamond}(F)\overset{\rm{def}}{=}\exp\left(F-\frac{1}{2}{\mathbb{E}}[F^{2}]\right)=\sum_{m=0}^{\infty}\frac{1}{m!}F^{\diamond
m}.$

As mentioned in Section A, the physical and probabilistic Wick products of a
Gaussian vector coincide. In particular, for
$f_{i}\in\mathscr{H},i=1,\dots,n$, we have

(2.16)
$:\\!\mathbf{I}_{1}(f_{1})\cdots\mathbf{I}_{1}(f_{n})\\!:=\mathbf{I}_{1}(f_{1})\diamond\cdots\diamond\mathbf{I}_{1}(f_{n}).$

In contrast to the multiple Wiener integral given in (2.9), the multiple
Stratonovich integral of $g^{\otimes m}$ for $g\in\mathscr{H}$ is defined by
([19, 21]):

(2.17) $\mathbb{I}_{m}(g^{\otimes m})\overset{\rm{def}}{=}W(g)^{m}.$

By a limiting argument, one can define $\mathbb{I}_{m}(g)$ for $g$ in some
appropriate space. We also take the following notation for $\mathbb{I}_{m}(f)$
(in comparison with $\mathbf{I}_{m}(f)$ in (2.11)):

$\displaystyle\mathbb{I}_{m}(f)=$
$\displaystyle\int_{([0,1]\times{\mathbb{R}})^{m}}f({{\bm{t}}},{\bm{x}})W({\mathrm{d}}{t_{1}},{\mathrm{d}}{x_{1}})\cdots
W({\mathrm{d}}{t_{m}},{\mathrm{d}}{x_{m}})$ $\displaystyle=$
$\displaystyle\int_{([0,1]\times{\mathbb{R}})^{m}}f({{\bm{t}}},{{\bm{x}}})\prod_{i=1}^{m}W({\mathrm{d}}{t_{i}},{\mathrm{d}}{x_{i}}).$

For $f\in\mathscr{H}^{\otimes m}$, define the $k$th order trace
$\mathrm{Tr}^{k}f$ of $f$ by

$\displaystyle\mathrm{Tr}^{k}f(t_{1},x_{1},\dots,t_{m-2k},x_{m-2k})\overset{\rm{def}}{=}$
$\displaystyle\int_{([0,1]\times\mathbb{R})^{2k}}f(s_{1},y_{1},\dots,s_{2k},y_{2k},t_{1},x_{1},\dots,t_{m-2k},x_{m-2k})$
(2.18) $\displaystyle\qquad K(s_{1}-s_{2},y_{1}-y_{2})\cdots
K(s_{2k-1}-s_{2k},y_{2k-1}-y_{2k}){\mathrm{d}}{{\bm{s}}}{\mathrm{d}}{{\bm{y}}}.$

The following Hu-Meyer’s formula (see [19], [14], [27]) connects multiple
Stratonovich integrals $\mathbb{I}_{m}(f)$ with multiple Wiener integrals
$\mathbf{I}_{m}(f)$,

(2.19) $\displaystyle\mathbb{I}_{m}(f)=\sum_{k=0}^{[\frac{m}{2}]}$
$\displaystyle\frac{m!}{k!(m-2k)!2^{k}}\,\mathbf{I}_{m-2k}(\mathrm{Tr}^{k}\hat{f}\,),~{}~{}f\in\mathscr{H}^{\otimes
m},$

as long as the right-hand side is well-defined, i.e.,
$\mathrm{Tr}^{k}\hat{f}\in\mathscr{H}^{\otimes(m-2k)}$ for $k=0,1,\dots[m/2]$.
In this case, we have

(2.20)
$\|f\|^{2}_{\mathbf{S}_{m}}\overset{\rm{def}}{=}{\mathbb{E}}[|\mathbb{I}_{m}(f)|^{2}]=\sum_{k=0}^{[\frac{m}{2}]}\frac{1}{(m-2k)!}\left(\frac{m!}{k!2^{k}}\right)^{2}\|\mathrm{Tr}^{k}\hat{f}\|^{2}_{\mathscr{H}^{\otimes(m-2k)}}.$

To end this section, we introduce the stochastic Fubini theorem which will be
used in the proof of Proposition 2.10. Note that stochastic Fubini theorem has
been proved in different contexts (see e.g. [13, 29]), and here we provide a
version working for multiple Wiener and Stratonovich integrals.

Let $(\mathbf{X},\mathcal{X},\mu)$ be a measurable space with
$\mu(\mathbf{X})<\infty$. For a measurable function
$f:\mathbf{X}\to\mathbb{R}$, we take the notation

$\mu(f)\overset{\rm{def}}{=}\int_{\mathbf{X}}f(x)\mu({\mathrm{d}}x).$

The following result holds for general Gaussian spaces associated with a
separable Hilbert space.

######  Proposition 2.4.

For $m\in\mathbb{N}$, let $h:\mathbf{X}\to\mathscr{H}^{\otimes m}$ be a
measurable mapping such that

$\mu({\mathbb{E}}[|\mathbf{I}_{m}(h)|^{2}])=\mu(\|h\|^{2}_{\mathscr{H}^{\otimes
m}})<\infty.$

Then the following stochastic Fubini theorem holds,

(2.21) $\mu(\mathbf{I}_{m}(h))=\mathbf{I}_{m}(\mu(h)).$

Similarly, if we assume

$\mu\left({\mathbb{E}}[|\mathbb{I}_{m}(h)|^{2}]\right)=\mu(\|h\|^{2}_{\mathbf{S}_{m}})=\sum_{k=0}^{[\frac{m}{2}]}\frac{1}{(m-2k)!}\left(\frac{m!}{k!2^{k}}\right)^{2}\mu\left(\|\mathrm{Tr}^{k}\hat{h}\|^{2}_{\mathscr{H}^{\otimes(m-2k)}}\right)<\infty,$

we also have

(2.22) $\mu(\mathbb{I}_{m}(h))=\mathbb{I}_{m}(\mu(h)).$

Proof Let $\\{e_{i},i\in\mathbb{N}\\}$ be an orthonormal basis of
$\mathscr{H}$. Suppose

$h(x)=\sum_{i_{1},\dots,i_{m}=1}^{\infty}\alpha_{i_{1},\dots,i_{m}}(x)e_{i_{1}}\otimes\cdots\otimes
e_{i_{m}}.$

Then the condition $\mu(\|h\|^{2}_{\mathscr{H}^{\otimes m}})<\infty$ implies
$\sum_{i_{1},\dots,i_{m}}\mu(\alpha_{i_{1},\dots,i_{m}}^{2})<\infty$. This
further implies

(2.23)
$\mu(h)=\sum_{i_{1},\dots,i_{m}}\mu(\alpha_{i_{1},\dots,i_{m}})e_{i_{1}}\otimes\cdots\otimes
e_{i_{m}},$

where the series on the right-hand side converges in the Hilbert space
$\mathscr{H}^{\otimes m}$ due to Jensen’s inequality and the assumption
$\mu(\mathbf{X})<\infty$.

Consider $g\in\mathscr{H}^{\otimes m}$ and suppose

$g=\sum_{i_{1},\dots,i_{m}=1}^{\infty}\beta_{i_{1},\dots,i_{m}}e_{i_{1}}\otimes\cdots\otimes
e_{i_{m}}$

with $\|g\|^{2}_{\mathscr{H}^{\otimes
m}}=\sum_{i_{1},\dots,i_{m}}\beta^{2}_{i_{1},\dots,i_{m}}<\infty$. We have

(2.24) $\displaystyle\mu(\langle h,g\rangle_{\mathscr{H}^{\otimes
m}})=\mu\left(\sum_{i_{1},\dots,i_{m}}\alpha_{i_{1},\dots,i_{m}}\beta_{i_{1},\dots,i_{m}}\right)=\sum_{i_{1},\dots,i_{m}}\mu(\alpha_{i_{1},\dots,i_{m}})\beta_{i_{1},\dots,i_{m}}=\langle\mu(h),g\rangle_{\mathscr{H}^{\otimes
m}},$

where the second equality follows from classical Fubini’s theorem, noting that

$\mu\left(\sum_{i_{1},\dots,i_{m}}|\alpha_{i_{1},\dots,i_{m}}\beta_{i_{1},\dots,i_{m}}|\right)\leq\mu\left(\left(\sum_{i_{1},\dots,i_{m}}\alpha_{i_{1},\dots,i_{m}}^{2}\right)^{1/2}\right)\left(\sum_{i_{1},\dots,i_{m}}\beta_{i_{1},\dots,i_{m}}^{2}\right)^{1/2}<\infty,$

and the last equality follows from (2.23).

By (2.24) we have
${\mathbb{E}}[\mu(\mathbf{I}_{m}(h))\mathbf{I}_{m}(g)]={\mathbb{E}}[\mathbf{I}_{m}(\mu(h))\mathbf{I}_{m}(g)]$
for any $g\in\mathscr{H}^{\otimes m}$, and thus the desired equality (2.21)
holds. Finally, equation (2.22) follows from (2.21) and(2.19). ∎

###  2.2. Mild Stratonovich solution

In this subsection, we consider the fractional SHE (1.14) with initial value
$u(0,x)=u_{0}(x)$, $0\leq t\leq 1,x\in{\mathbb{R}}$. We call
$\\{u(t,x)\\}_{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}}$ a mild Stratonovich
solution of (1.14) if $u(t,x)$ is an adapted process (with respect to the
filtration $(\mathcal{F}_{t})_{t\geq 0}$ where
$\mathcal{F}_{t}=\sigma(W(s,x),0\leq s\leq t,x\in\mathbb{R})$) such that
${\mathbb{E}}[|u(t,x)|^{2}]<\infty$ for all
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}$ and satisfies the following integral
equation

(2.25)
$u(t,x)=\int_{\mathbb{R}}\mathfrak{g}(t,x-y)u_{0}(y){\mathrm{d}}y+{\beta}\int_{0}^{t}\int_{\mathbb{R}}\mathfrak{g}(t-r,x-y)u(r,y)W({\mathrm{d}}r,{\mathrm{d}}y),$

where $\mathfrak{g}(t,x)$ is the density function of the $\rho$-stable process
$\\{X_{t},t\geq 0\\}$ with $X_{0}=0$ and the integral on the right-hand side
is understood in the sense of Stratonovich which will be specified below.

Assuming $u_{0}(x)\equiv 1$, we have

(2.26) $\displaystyle
u(t,x)=1+{\beta}\int_{0}^{t}\int_{\mathbb{R}}\mathfrak{g}({t-r},x-y)u(r,y)W({\mathrm{d}}r,{\mathrm{d}}y).$

Iterate the equation and formally we have a series expansion for a mild
Stratonovich solution:

(2.27) $\displaystyle u(t,x)=$ $\displaystyle
1+\sum_{m=1}^{\infty}{\beta}^{m}\int_{[0,t]_{<}^{m}}\int_{{\mathbb{R}}^{m}}\prod_{i=1}^{m}\mathfrak{g}(t_{i+1}-t_{i},x_{i+1}-x_{i})W({\mathrm{d}}t_{i},{\mathrm{d}}x_{i})$
$\displaystyle=$ $\displaystyle
1+\sum_{m=1}^{\infty}{\beta}^{m}\int_{[0,t]^{m}}\int_{{\mathbb{R}}^{m}}\mathfrak{g}_{m}({\bm{t}},{\bm{x}};t,x)\prod_{i=1}^{m}W({\mathrm{d}}t_{i},{\mathrm{d}}x_{i})$
$\displaystyle=$ $\displaystyle
1+\sum_{m=1}^{\infty}\beta^{m}\mathbb{I}_{m}(\mathfrak{g}_{m}(\cdot;t,x)),$

where

(2.28)
$\displaystyle\mathfrak{g}_{m}({\bm{t}},{\bm{x}};t,x)\overset{\rm{def}}{=}\prod_{i=1}^{m}\mathfrak{g}(t_{i+1}-t_{i},x_{i+1}-x_{i})\mathbf{1}_{[0,1]^{m}_{<}}(t_{1},\dots,t_{m})$

with

$[0,t]_{<}^{m}\overset{\rm{def}}{=}\\{0<t_{1}<\cdots<t_{m}<t\\}\text{ and
}t_{m+1}\overset{\rm{def}}{=}t,x_{m+1}\overset{\rm{def}}{=}x,$

and on the right-hand side are multiple Stratonovich integrals. The series in
(2.27) converges in $L^{2}(\Omega)$ under the condition (1.16) (see
Proposition 2.12). We also refer to [23, 42] for an equivalent description.

Knowing that $u(t,x)$ solves a stochastic heat equation, we now aim to derive
a Feynman-Kac type representation for $u(t,x)$. Given a path of $X$, the
following Wiener integral

(2.29) $\mathcal{I}_{t,x}^{\varepsilon}\overset{\rm{def}}{=}\
\int_{0}^{t}\int_{\mathbb{R}}p_{\varepsilon}(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y),$

is well defined, where $X_{s}^{x}\overset{\rm{def}}{=}X_{s}+x$ and
$p_{\varepsilon}(x)\overset{\rm{def}}{=}\frac{1}{\sqrt{2\pi\varepsilon}}{\rm
e}^{-\frac{x^{2}}{2\varepsilon}}$ is the heat kernel.

As for the discrete model, we use ${\mathbb{E}}_{X}$ and ${\mathbb{E}}_{W}$ to
denote the expectations in the probability space of $X$ and $W$, respectively,
and we abuse the notation
${\mathbb{E}}={\mathbb{E}}_{W}\times{\mathbb{E}}_{X}$ in this section.

######  Proposition 2.5.

Assume the condition (1.16). Then, for each $\varepsilon>0$,
$p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot)$ belongs to $\mathscr{H}$ a.s. and the
family of random variables $\mathcal{I}_{t,x}^{\varepsilon}$ defined by (2.29)
converges in $L^{2}$ to a limit denoted by

(2.30)
$\mathcal{I}_{t,x}\overset{\rm{def}}{=}\int_{0}^{t}\int_{\mathbb{R}}\delta(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y),$

where $\delta(X_{t-\cdot}^{x}-\cdot)$ is an $\mathscr{H}$-valued random
variable given by the $L^{2}$-limit of
$p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot)$. Moreover,
$\delta(X_{t-\cdot}^{x}-\cdot)\in
L^{2}(\Omega_{X},\mathcal{G},P;\mathscr{H})$, where
$(\Omega_{X},\mathcal{G},P)$ is the probability space of $X$.

Conditional on $X$, $\mathcal{I}_{t,x}$ is a Gaussian random variable with
mean $0$ and variance

(2.31)
$\mathrm{Var}[\mathcal{I}_{t,x}|X]=\int_{0}^{t}\int_{0}^{t}|r-s|^{2H-2}\delta(X_{r}-X_{s}){\mathrm{d}}r{\mathrm{d}}s\,.$

Proof For $\varepsilon,\sigma>0$, by (2.2) we have for $t\leq 1$,

$\displaystyle\left\langle
p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot)\mathbf{I}_{[0,t]},p_{{\sigma}}(X_{t-\cdot}^{x}-\cdot)\mathbf{I}_{[0,t]}\right\rangle_{\mathscr{H}}$
(2.32) $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{t}\int_{\mathbb{R}}|s-r|^{2H-2}p_{\varepsilon}(X_{s}^{x}-y)p_{\sigma}(X_{r}^{x}-y){\mathrm{d}}y{\mathrm{d}}s{\mathrm{d}}r$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2}p_{\varepsilon+{\sigma}}(X_{s}-X_{r}){\mathrm{d}}s{\mathrm{d}}r.$

By (1.9), we have

${\mathbb{E}}[p_{\varepsilon+{\sigma}}(X_{s}-X_{r})]=\int_{\mathbb{R}}p_{\varepsilon+\sigma}(y)\mathfrak{g}(|r-s|,y){\mathrm{d}}y\lesssim|r-s|^{-1/\rho}.$

Thus, the condition (1.16) yields

${\mathbb{E}}\left\langle
p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot),p_{{\sigma}}(X_{t-\cdot}^{x}-\cdot)\right\rangle_{\mathscr{H}}\lesssim\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2-\frac{1}{\rho}}{\mathrm{d}}s{\mathrm{d}}r<\infty,$

hence $p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot)$ belongs to $\mathscr{H}$ for
all $\varepsilon>0$ almost surely and

(2.33)
$\displaystyle{\mathbb{E}}[\mathcal{I}_{t,x}^{\varepsilon}\mathcal{I}_{t,x}^{\sigma}]={\mathbb{E}}\left\langle
p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot),p_{\sigma}(X_{t-\cdot}^{x}-\cdot)\right\rangle_{\mathscr{H}}={\mathbb{E}}\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2}p_{\varepsilon+{\sigma}}(X_{s}-X_{r}){\mathrm{d}}s{\mathrm{d}}r.$

As $(\varepsilon,\sigma)\to 0$, by the dominated convergence theorem we have

$\lim_{(\varepsilon,\sigma)\to
0}{\mathbb{E}}[\mathcal{I}_{t,x}^{\varepsilon}\mathcal{I}_{t,x}^{\sigma}]={\mathbb{E}}\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2}\delta(X_{s}-X_{r}){\mathrm{d}}s{\mathrm{d}}r=C\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2-\frac{1}{\rho}}{\mathrm{d}}s{\mathrm{d}}r<\infty$

for some proper constant $C$. As a consequence, we have
$\lim_{(\varepsilon,\sigma)\to
0}{\mathbb{E}}\Big{[}\Big{(}\mathcal{I}_{t,x}^{{\varepsilon}}-\mathcal{I}_{t,x}^{\sigma}\Big{)}^{2}\Big{]}=0$.

This implies that, as $\varepsilon$ converges to zero,
$p_{\varepsilon}(X_{t-\cdot}^{x}-\cdot)$ is a Cauchy sequence in
$L^{2}(\Omega_{X},\mathcal{G},P;\mathscr{H})$, and the limit is denoted by
$\delta(X_{t-\cdot}^{x}-\cdot)$. Therefore, $\mathcal{I}_{t,x}^{\varepsilon}$
converges to $\mathcal{I}_{t,x}$ given in (2.30) in $L^{2}$. Finally, (2.31)
can be proven by a similar argument. ∎

######  Remark 2.6.

By the computation above, it is clear that we have

${\mathbb{E}}[|\mathcal{I}_{t,x}|^{2}]=C\int_{0}^{t}\int_{0}^{t}|s-r|^{2H-2-\frac{1}{\rho}}{\mathrm{d}}s{\mathrm{d}}r,$

which is finite iff (1.16) holds. It turns out that (1.16) is also sufficient
(and necessary, of course) for ${\mathbb{E}}[{\rm
e}^{\mathcal{I}_{t,x}}]<\infty$ by (2.31) and Proposition 2.7 below.

######  Proposition 2.7.

Under the condition (1.16), we have

(2.34)
${\mathbb{E}}\left[\exp\left(\beta\int_{0}^{1}\int_{0}^{1}|r-s|^{2H-2}\delta(X_{r}-X_{s}){\mathrm{d}}r{\mathrm{d}}s\right)\right]<\infty\text{
for all }\beta>0.$

Proof The proof essentially follows from that of [42, Theorem 3.3] with some
minor modifications. Taylor expansion and Fubini’s theorem yield

(2.35)
$\displaystyle{\mathbb{E}}\left[\exp\left(\beta\int_{0}^{1}\int_{0}^{1}|r-s|^{2H-2}\delta(X_{r}-X_{s}){\mathrm{d}}r{\mathrm{d}}s\right)\right]$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{\beta}{2\pi}\right)^{m}\int_{[0,1]^{2m}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|s_{2i}-s_{2i-1}|^{2H-2}{\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}\xi_{i}(X_{s_{2i}}-X_{s_{2i-1}})}\right]{\mathrm{d}}{\bm{\xi}}{\mathrm{d}}{\bm{s}}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{\beta}{2\pi}\right)^{m}\sum_{\sigma\in\mathcal{P}_{2m}}\int_{[0,1]_{<}^{2m}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|s_{\sigma(2i)}-s_{\sigma(2i-1)}|^{2H-2}{\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}\xi_{i}(X_{s_{{\sigma}(2i)}}-X_{s_{{\sigma}(2i-1)}})}\right]{\mathrm{d}}{\bm{\xi}}{\mathrm{d}}{\bm{s}},$

where we recall that $[0,1]_{<}^{2m}=\\{0<s_{1}<s_{2}<\cdots<s_{2m}<1\\}$ and
$\mathcal{P}_{2m}$ is the set of all permutations on $\llbracket
2m\rrbracket$.

Let $\sigma\in\mathcal{P}_{2m}$ be arbitrarily chosen and fixed. Consider
$(s_{1},\dots,s_{2m})\in[0,1]^{2m}_{<}$, and for each pair
$(s_{\sigma(2i-1)},s_{\sigma(2i)})$ , we let
$t^{*}_{2i}\overset{\rm{def}}{=}s_{\sigma(2i-1)}\vee s_{\sigma(2i)}$ and
$t^{*}_{2i-1}$ be the unique $s_{j}$ which is the closest point to
$t^{*}_{2i}$ from the left. Then clearly, we have

$\prod_{i=1}^{m}|s_{\sigma(2i)}-s_{\sigma(2i-1)}|^{2H-2}\leq\prod_{i=1}^{m}|t^{*}_{2i}-t^{*}_{2i-1}|^{2H-2}.$

Noting that

$\sum_{i=1}^{m}\xi_{i}(X_{s_{{\sigma}(2i)}}-X_{s_{{\sigma}(2i-1)}})=\sum_{j=2}^{2m}\eta_{j}(X_{s_{j}}-X_{s_{j-1}})$

where each $\eta_{j}$ is a linear combination of $\xi_{i}$’s for
$i=1,\dots,m.$ Then we have

$\displaystyle{\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}\xi_{i}(X_{s_{{\sigma}(2i)}}-X_{s_{{\sigma}(2i-1)}})}\right]=\prod_{j=2}^{2m}{\rm
e}^{-c_{\rho}|\eta_{j}|^{\rho}(s_{j}-s_{j-1})}\leq\prod_{i=1}^{m}{\rm
e}^{-c_{\rho}|\tilde{\eta}_{i}|^{\rho}(t_{2i}^{*}-t_{2i-1}^{*})},$

where the inequality holds because we only keep the factors resulting from the
characteristic function of $X_{t_{2i}^{*}}-X_{t_{2i-1}^{*}}$ for
$i\in\llbracket m\rrbracket$ and drop all the others. Here, for each
$i\in\llbracket m\rrbracket$, we define $\tilde{\eta}_{i}=\eta_{j}$ where $j$
is the unique index such that $s_{j}=t^{*}_{2i}=s_{\sigma(2i)}\vee
s_{\sigma(2i-1)}$.

Thus, we have

(2.36)
$\displaystyle\int_{[0,1]_{<}^{2m}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|s_{\sigma(2i)}-s_{\sigma(2i-1)}|^{2H-2}{\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}\xi_{i}(X_{s_{{\sigma}(2i)}}-X_{s_{{\sigma}(2i-1)}})}\right]{\mathrm{d}}{\bm{\xi}}{\mathrm{d}}{\bm{s}}$
$\displaystyle\leq\int_{[0,1]_{<}^{2m}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|t^{*}_{2i}-t^{*}_{2i-1}|^{2H-2}\prod_{i=1}^{m}{\rm
e}^{-c_{\rho}|\tilde{\eta}_{i}|^{\rho}(t^{*}_{2i}-t^{*}_{2i-1})}{\mathrm{d}}{\bm{\xi}}{\mathrm{d}}{\bm{s}}$
$\displaystyle=C^{m}\int_{[0,1]_{<}^{2m}}\prod_{i=1}^{m}|t^{*}_{2i}-t^{*}_{2i-1}|^{2H-2-\frac{1}{\rho}}{\mathrm{d}}{\bm{s}}$
$\displaystyle\leq\frac{C^{m}}{\Gamma\left(m(2H-\frac{1}{\rho})+1\right)}.$

where the equality follows from the fact that the Jacobian determinant
$|[\partial\xi_{i}/\partial{\tilde{\eta}_{j}}]_{m\times m}|$ is one and that
$\int_{\mathbb{R}}{\rm e}^{-a|\xi|^{\rho}t}{\mathrm{d}}\xi=Ca^{-1/\rho}$ for
$a>0$, and the last step follows from Lemma C.3.

Therefore, by (2.35) and (2.36), and noting that $|\mathcal{P}_{2m}|=(2m)!$,
we have

${\mathbb{E}}\left[\exp\left(\beta\int_{0}^{1}\int_{0}^{1}|r-s|^{2H-2}\delta(X_{r}-X_{s}){\mathrm{d}}r{\mathrm{d}}s\right)\right]\leq\sum_{m=0}^{\infty}C^{m}\beta^{m}\frac{(2m)!}{m!\Gamma\left(m(2H-\frac{1}{\rho})+1\right)}<\infty,$

where the last step follows from Stirling’s formula and $2H-\frac{1}{\rho}>1$.
∎

######  Remark 2.8.

In the proof of [42, Theorem 3.3], the Markov property instead of the
independent increment property of $X$ was invoked, and as a consequence, [42,
Lemma 2.2] was needed therein. As shown in the proof of Proposition 2.7, if we
utilise the independent increment property of $X$, [42, Lemma 2.2] is no
longer needed, and this is important for the proof of uniform $L^{1}$-bound
for rescaled partition function
${\mathbb{Z}}_{\omega}^{(N)}(\hat{\beta}_{N},k)$ in Section 3.3.

######  Remark 2.9.

Note that when $\rho=2$, the condition $\theta>\frac{1}{2}$ in (1.16) becomes
$H>\frac{3}{4}$. Thus, Proposition 2.7 is consistent with [23, Theorem 6.2].

######  Proposition 2.10.

Under the condition (1.16), the mild Stratonovich solution given in (2.27) has
the following Feynman-Kac representation:

(2.37)
$u(t,x)={\mathbb{E}}_{X}\left[\exp\left(\beta\int_{0}^{t}\int_{{\mathbb{R}}}\delta(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y)\right)\right].$

Proof First, we prove the integrability of the right-hand side of (2.37). For
all $\beta\in\mathbb{R}$,

(2.38)
$\displaystyle{\mathbb{E}}_{W}\left[{\mathbb{E}}_{X}\left[\exp\left(\beta\int_{0}^{t}\int_{{\mathbb{R}}}\delta(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y)\right)\right]\right]$
$\displaystyle={\mathbb{E}}_{X}\left[{\mathbb{E}}_{W}\left[\exp\left(\beta\int_{0}^{t}\int_{{\mathbb{R}}}\delta(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y)\right)\right]\right]$
$\displaystyle={\mathbb{E}}_{X}\left[\exp\left(\frac{1}{2}\beta^{2}t^{2H-\frac{1}{\rho}}\int_{0}^{1}\int_{0}^{1}|r-s|^{2H-2}\delta(X_{r}-X_{s}){\mathrm{d}}r{\mathrm{d}}s\right)\right]$
$\displaystyle<\infty,$

where the second equality follows from (2.31) and the self-similarity of $X$,
and the last step follows from Proposition 2.7. Thus $u(t,x)$ given by (2.37)
is well-defined and is $L^{p}$-integrable for all $p>0$.

The Feynman-Kac formula (2.37) now follows from (2.27) and the following
equation:

(2.39)
$\displaystyle{\mathbb{E}}_{X}\left[\left(\int_{0}^{t}\int_{{\mathbb{R}}}\delta(X_{t-r}^{x}-y)W({\mathrm{d}}r,{\mathrm{d}}y)\right)^{m}\right]$
$\displaystyle=\int_{[0,t]^{m}}\int_{\mathbb{R}^{m}}{\mathbb{E}}_{X}\left[\prod_{i=1}^{m}\delta(X_{t-r_{i}}^{x}-y_{i})\right]W({\mathrm{d}}r_{1},{\mathrm{d}}y_{1})\dots
W({\mathrm{d}}r_{m},{\mathrm{d}}y_{m})$
$\displaystyle=m!\int_{[0,t]^{m}}\int_{{\mathbb{R}}^{m}}\mathfrak{g}_{m}({\bm{r}},{\bm{y}};t,x)\prod_{i=1}^{m}W({\mathrm{d}}r_{i},{\mathrm{d}}y_{i}),$

where the first equality follows from (2.17) and the stochastic Fubini theorem
(Proposition 2.4). ∎

######  Remark 2.11.

A direct corollary of Propositions of 2.7 and 2.10 is

(2.40) $\|\mathfrak{g}_{m}\|_{\mathbf{S}_{m}}<\infty,$

where $\mathfrak{g}_{m}$ is given in (2.28) and $\|\cdot\|_{\mathbf{S}_{m}}$
in (2.20). Indeed,

$\displaystyle\|\mathfrak{g}_{m}\|^{2}_{\mathbf{S}_{m}}={\mathbb{E}}[|\mathbb{I}_{m}(\mathfrak{g}_{m})|^{2}]=\left(\frac{1}{m!}\right)^{2}{\mathbb{E}}_{W}\left[\Big{(}{\mathbb{E}}_{X}[\mathcal{I}_{t,x}^{m}]\Big{)}^{2}\right]\leq\left(\frac{1}{m!}\right)^{2}{\mathbb{E}}[\mathcal{I}_{t,x}^{2m}]\leq\left(\frac{1}{m!}\right)^{2}{\mathbb{E}}[{\rm
e}^{\mathcal{I}_{t,x}}]<\infty,$

where $\mathcal{I}_{t,x}$ is given in (2.30), the second equality follows from
(2.39), and the last inequality follows from (2.38). Thus,
$\mathrm{Tr}^{k}\hat{\mathfrak{g}}_{m}\in\mathscr{H}^{\otimes(m-2k)}$ by
(2.20). This also implies that
$\sum_{m=1}^{\infty}\mathbb{I}_{m}(\mathfrak{g}_{m})$ converges in $L^{2}$,
which is stated in the following proposition.

######  Proposition 2.12.

Under the condition (1.16), the mild Stratonovich solution of (2.26) given in
(2.27) converges in $L^{2}$.

###  2.3. Mild Itô-Skorohod solution

Mild Itô-Skorohod solution is defined similarly as for mild Stratonovich
solution in Section 2.2, and the only difference is that the stochastic
integral in (2.26) is now understood in the Skorohod sense. More precisely,
still assuming $u_{0}(x)\equiv 1$, the Itô-Skorohod solution $\tilde{u}(t,x)$
of (1.14) satisfies

(2.41) $\displaystyle
u(t,x)=1+{\beta}\int_{0}^{t}\int_{\mathbb{R}}\mathfrak{g}({t-r},x-y)u(r,y)\diamond
W({\mathrm{d}}r,{\mathrm{d}}y),$

and its chaos expansion is given by

(2.42) $\displaystyle u(t,x)=$ $\displaystyle
1+\sum_{m=1}^{\infty}{\beta}^{m}\int_{[0,t]^{m}}\int_{{\mathbb{R}}^{m}}\mathfrak{g}_{m}({\bm{t}},{\bm{x}};t,x)W({\mathrm{d}}t_{1},{\mathrm{d}}x_{1})\diamond\cdots\diamond
W({\mathrm{d}}t_{m},{\mathrm{d}}x_{m})$ $\displaystyle=$ $\displaystyle
1+\sum_{m=1}^{\infty}\beta^{m}\mathbf{I}_{m}(\mathfrak{g}_{m}(\cdot;t,x)),$

where $\mathfrak{g}_{m}$ is given in (2.28) and the stochastic integrals above
are in the Skorohod sense. A sufficient and necessary condition for the
existence and uniqueness of the Itô-Skorohod solution is (see, e.g., [22,
42]),

(2.43)
$\sum_{m=1}^{\infty}\beta^{m}m!\left\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\right\|^{2}_{\mathscr{H}^{\otimes
m}}<\infty,$

where $\hat{\mathfrak{g}}_{m}$ is the symmetrization of $\mathfrak{g}_{m}$:

(2.44)
$\hat{\mathfrak{g}}_{m}({\bm{t}},{\bm{x}};t,x)=\frac{1}{m!}\sum_{\sigma\in\mathcal{P}_{m}}\mathfrak{g}_{m}({\bm{t}}_{\sigma},{\bm{x}}_{\sigma};t,x),$

with
${\bm{t}}_{\sigma}\overset{\rm{def}}{=}(t_{\sigma(1)},\dots,t_{\sigma(m)})$
and
${\bm{x}}_{\sigma}\overset{\rm{def}}{=}(x_{\sigma(1)},\dots,x_{\sigma(m)})$.

By [42, Theorem 5.3], there exists a unique mild Itô-Skorohod solution for
(1.14) on $\mathbb{R}^{d}$ if

(2.45)
$\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|^{\rho}}{\mathrm{d}}\xi<\infty\Longleftrightarrow\rho>d.$

######  Remark 2.13.

Note that the requirement $\rho\leq 2$ together with (2.45) forces us to
consider the case $d=1$ only. By (2.45), we also need to assume $\rho>1$ in
Theorem 1.3.

We shall provide an alternative proof for (2.43) under the condition
$H\in(1/2,1],\rho\in(1,2]$, which will be easier to be adapted to estimate the
moments of the terms arising from the modified partition function.

######  Proposition 2.14.

Assume $H\in(1/2,1]$ and $\theta=H-\frac{1}{2\rho}>0$. Then there exists a
constant $C$ such that for all $(t,x)\in[0,1]\times\mathbb{R}$,

(2.46) $\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|_{\mathscr{H}^{\otimes m}}\leq
C^{m}(m!)^{H-1}\left(\frac{\Gamma(\theta/H)^{m}}{\Gamma(m\theta/H+1)}\right)^{H},$

where $\hat{\mathfrak{g}}_{m}$ is given in (2.44). Furthermore, (2.43) holds
if $\rho>1$.

Proof Recalling (2.2), we have

$\displaystyle\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|^{2}_{\mathscr{H}^{\otimes
m}}$
$\displaystyle=\int_{[0,1]^{2m}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|r_{i}-s_{i}|^{2H-2}\hat{\mathfrak{g}}_{m}({\bm{r}},{\bm{x}};t,x)\hat{\mathfrak{g}}_{m}({\bm{s}},{\bm{x}};t,x){\mathrm{d}}{\bm{x}}{\mathrm{d}}{\bm{r}}{\mathrm{d}}{\bm{s}}$
$\displaystyle\leq
C_{H}^{m}\int_{\mathbb{R}^{m}}\left(\int_{[0,1]^{m}}|\hat{\mathfrak{g}}_{m}({\bm{s}},{\bm{x}};t,x)|^{\frac{1}{H}}{\mathrm{d}}{\bm{s}}\right)^{2H}{\mathrm{d}}{\bm{x}}$
$\displaystyle\leq
C_{H}^{m}\left(\int_{[0,1]^{m}}\left(\int_{\mathbb{R}^{m}}|\hat{\mathfrak{g}}_{m}({\bm{s}},{\bm{x}};t,x)|^{2}{\mathrm{d}}{\bm{x}}\right)^{\frac{1}{2H}}{\mathrm{d}}{\bm{s}}\right)^{2H}$

where the first inequality follows from Lemma C.2 and the second one follows
from the Minkowski inequality. Recalling the definition (2.44) of
$\hat{\mathfrak{g}}_{m}$, we have

$\displaystyle\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|^{2}_{\mathscr{H}^{\otimes
m}}$ $\displaystyle\leq
C_{H}^{m}\left(\int_{[0,1]^{m}}(m!)^{-\frac{1}{H}}\left(\int_{\mathbb{R}^{m}}\sum_{\sigma,\tau\in\mathcal{P}_{m}}\mathfrak{g}_{m}({\bm{s}}_{\sigma},{\bm{x}}_{\sigma};t,x)\mathfrak{g}_{m}({\bm{s}}_{\tau},{\bm{x}}_{\tau};t,x){\mathrm{d}}{\bm{x}}\right)^{\frac{1}{2H}}{\mathrm{d}}{\bm{s}}\right)^{2H}$
$\displaystyle=C_{H}^{m}\left(\int_{[0,1]_{<}^{m}}(m!)^{1-\frac{1}{H}}\left(\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|\mathfrak{g}(s_{i+1}-s_{i},x_{i+1}-x_{i})|^{2}{\mathrm{d}}{\bm{x}}\right)^{\frac{1}{2H}}{\mathrm{d}}{\bm{s}}\right)^{2H}.$

Noting that by (1.10), we have that for all $t>0$,

$\int_{\mathbb{R}}|\mathfrak{g}(t,x)|^{2}{\mathrm{d}}x\leq
Ct^{-\frac{1}{\rho}},$

where $C$ is a positive constant depending only on $\rho$. Combining the last
two estimates, we get and then the desired inequality (2.46) follows from
Lemma C.3.

Finally, by (2.46) and Stirling’s formula we get that (2.43) holds if
$\rho>1$. ∎

######  Remark 2.15.

The scaling limit of long-range random walk in i.i.d. random environment was
considered in [8]. In this situation, the condition $\rho>1$ is also a
necessary condition for (2.43). Indeed, now we have

$\displaystyle\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|^{2}_{\mathscr{H}^{\otimes
m}}$
$\displaystyle=\int_{[0,1]^{m}}\int_{\mathbb{R}^{m}}|\hat{\mathfrak{g}}_{m}({\bm{s}},{\bm{x}};t,x)|^{2}{\mathrm{d}}{\bm{x}}{\mathrm{d}}{\bm{s}}$
$\displaystyle=\frac{1}{m!}\int_{[0,1]^{m}_{<}}\int_{\mathbb{R}^{m}}\prod_{i=1}^{m}|\mathfrak{g}(s_{i+1}-s_{i},x_{i+1}-x_{i})|^{2}{\mathrm{d}}{\bm{x}}{\mathrm{d}}{\bm{s}}.$

By the scaling property (1.9), we have

$\int_{\mathbb{R}}|\mathfrak{g}(s,x)|^{2}{\mathrm{d}}x=As^{-\frac{1}{\rho}},$

where $A=\int_{\mathbb{R}}|\mathfrak{g}(1,x)|^{2}{\mathrm{d}}x$ is finite by
(1.10). Thus, we get

$\displaystyle\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|^{2}_{\mathscr{H}^{\otimes
m}}=\frac{A^{m}}{m!}\int_{[0,1]_{<}^{m}}\prod_{i=1}^{m}(s_{i+1}-s_{i})^{-\frac{1}{\rho}}{\mathrm{d}}{\bm{s}},$

which is finite only if $\rho>1$. One can also check that (2.43) holds by
Stirling’s formula if $\rho>1$.

######  Remark 2.16.

From the proof of Proposition 2.14, we can see that the estimation (2.46)
still holds if we replace $\mathfrak{g}(t,x)$ by its upper bound
$C(t\|x\|^{-\rho-1})\wedge t^{-1/\rho}$ (see (1.10)). This fact together with
the upper bound for $P_{n}(k)$ given in (1.8) will be used to estimate the
moments of ${\mathbf{S}}_{m}^{(N)}$ for the Skorohod case in Section 3.3.

######  Remark 2.17.

Assume $H\in(\frac{1}{2},1]$ and $\rho\in(0,2]$. For (1.14), by the analysis
in this section (or by Theorem 4.6 and Theorem 5.3 in [42]), we know that the
condition for the existence of a Stratonovich solution is
$\theta=H-\frac{1}{2\rho}>\frac{1}{2}$ and the condition for the Itô-Skorohod
case is $\rho>1.$ Note that for the Stratonovich case, $\theta>\frac{1}{2}$
yields $\rho>\frac{1}{2H-1}\geq 1$ and moreover $\theta>\frac{1}{2},\rho\leq
2$ implies $H\geq\frac{3}{4}$. Thus, it requires more restrictive condition
for the existence of a Stratonovich solution. This is because the $L^{2}$-norm
of $\mathbb{I}_{m}(\mathfrak{g}_{m})$ is strictly bigger than
$\mathbf{I}_{m}(\mathfrak{g}_{m})$ due to the extra trace terms appearing in
$\|\mathfrak{g}_{m}\|^{2}_{\mathbf{S}_{m}}$ (see (2.20)).

##  3\. On weak convergences

In this section, we aim to prove the main results Theorems 1.2 and 1.3. Recall
that $\theta=H-\frac{1}{2\rho}$. For $N\in\mathbb{N}$, under the scaling
$\beta\rightarrow\hat{\beta}_{N}=\beta N^{-\theta}$, the rescaled point-to-
line partition function is given by

(3.1) $\displaystyle
Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})={\mathbb{E}}_{S}\left[\exp\left(\hat{\beta}_{N}\sum_{n=1}^{N}{\omega}(n,S_{n})\right)\Bigg{|}S_{N}=N^{1/\rho}x_{0}\right],$

where $x_{0}\in\mathbb{R}$ is fixed such that
$N^{1/\rho}x_{0}\in{\mathbb{Z}}$. Taylor expansion yields

(3.2) $\displaystyle Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}{\mathbb{S}}_{m}^{(N)},$

where ${\mathbb{S}}_{m}^{(N)}$ is the $m$th moment of the Hamiltonian (with a
factor $\hat{\beta}_{N}$) given by

(3.3) $\displaystyle{\mathbb{S}}_{m}^{(N)}=$
$\displaystyle{\mathbb{E}}_{S}\left[\left(\hat{\beta}_{N}\sum_{n=1}^{N}{\omega}(n,S_{n})\right)^{m}\Bigg{|}S_{N}=N^{1/\rho}x_{0}\right]$
$\displaystyle=$
$\displaystyle\hat{\beta}_{N}^{m}{\mathbb{E}}_{S}\left[\sum_{{\bm{n}}\in\llbracket
N\rrbracket^{m}}{\omega}(n_{1},S_{n_{1}})\cdots{\omega}(n_{m},S_{n_{m}})\Bigg{|}S_{N}=N^{1/\rho}x_{0}\right].$

For every ${\bm{n}}=(n_{1},\dots,n_{m})\in\llbracket N\rrbracket^{m}$, the
components can be arranged in an increasing order and the resulted sequence is
denoted by ${\bm{n}}^{*}=(n_{1}^{*},\cdots,n_{m}^{*})$ with $n^{*}_{1}\leq
n^{*}_{2}\leq\cdots\leq n^{*}_{m}$. For each ${\bm{n}}\in\llbracket
N\rrbracket^{m}$, there is a permutation $\sigma$ of $\llbracket N\rrbracket$
such that $n_{i}^{*}=n_{\sigma_{i}}$ for $i=1,\dots,m$. Denote

(3.4) $\displaystyle
P_{{\bm{n}}^{*}}=P_{{\bm{n}}^{*}}(N^{1/\rho}x_{0};k_{1},\dots,k_{m})$
$\displaystyle=P_{n^{*}_{2}-n^{*}_{1}}(k_{\sigma_{2}}-k_{\sigma_{1}})\cdots
P_{n_{m}^{*}-n_{m-1}^{*}}(k_{\sigma_{m}}-k_{\sigma_{m-1}})P_{N-n_{m}^{*}}(N^{1/\rho}x_{0}-k_{\sigma_{m}}),$

where we use the convention $P_{0}(0)=1$ and $P_{0}(k)=0$ for $k\neq 0$. We
remark that $P_{{\bm{n}}^{*}}$ is symmetric in its $m$ arguments. The $m$th
moment in (3.3) can be written as

(3.5)
$\displaystyle{{\mathbb{S}}}_{m}^{(N)}=\hat{\beta}^{m}_{N}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}{\omega}(n_{1},k_{1})\cdots{\omega}(n_{m},k_{m})P_{{\bm{n}}^{*}}(N^{1/\rho}x_{0};k_{1},\dots,k_{m}),$

recalling that $S$ is backward random walk with $S_{N}=N^{1/\rho}x_{0}$.

A similar calculation can be done for the modified partition function
$\tilde{\mathbb{Z}}_{\omega}^{(N)}$ given in (1.17). Note that the exponential
is actually a Wick exponential of $\beta\sum_{i=1}^{N}{\omega}(i,S_{i})$
conditional on the random walk $S$. Hence, applying (2.15), we get the
following resemblance of (3.2) and (3.3) respectively,

(3.6) $\displaystyle\tilde{Z}_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}{\mathbf{S}}_{m}^{(N)},$

where

(3.7) $\displaystyle{\mathbf{S}}_{m}^{(N)}=$
$\displaystyle{\mathbb{E}}_{S}\left[\left(\hat{\beta}_{N}\sum_{n=1}^{N}{\omega}(n,S_{n})\right)^{\diamond
m}\Bigg{|}S_{N}=N^{1/\rho}x_{0}\right]$ $\displaystyle=$
$\displaystyle\hat{\beta}_{N}^{m}{\mathbb{E}}_{S}\left[\sum_{{\bm{n}}\in\llbracket
N\rrbracket^{m}}:\\!{\omega}(n_{1},S_{n_{1}})\cdots{\omega}(n_{m},S_{n_{m}})\\!:\Bigg{|}S_{N}=N^{1/\rho}x_{0}\right]$
$\displaystyle=$
$\displaystyle\hat{\beta}^{m}_{N}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}:\\!{\omega}(n_{1},k_{1})\cdots{\omega}(n_{m},k_{m})\\!:P_{{\bm{n}}^{*}}(N^{1/\rho}x_{0};k_{1},\dots,k_{m}).$

Now, we introduce the following general U-statistics $\mathbf{I}_{m}^{(N)}(f)$
which will be proven to converge weakly to a multiple Wiener integral. For
$f\in\mathcal{B}^{\otimes m}$ where we recall that $\mathcal{B}$ is given in
(2.3), we denote

$\displaystyle\mathbf{I}^{(N)}_{m}(f)$
$\displaystyle\overset{\rm{def}}{=}N^{-m(\theta+1/\rho)}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}\bigg{[}:\\!{\omega}(n_{1},k_{1})\cdots{\omega}(n_{m},k_{m})\\!:$
(3.8)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times{\mathcal{A}_{N}(f)}\left(\frac{n_{1}}{N},\frac{k_{1}}{N^{1/\rho}},\cdots,\frac{n_{m}}{N},\frac{k_{m}}{N^{1/\rho}}\right)\bigg{]}$
$\displaystyle\overset{\rm{def}}{=}N^{-m(\theta+1/\rho)}\sum_{{\bm{n}}\in\llbracket
N\rrbracket^{m}}\sum_{{\bm{k}}\in{\mathbb{Z}}^{m}}:\\!{\omega}_{\llbracket
m\rrbracket}\\!:\mathcal{A}_{N}(f)({\bm{t}},{\bm{x}}),$

where $:\\!{\omega}_{\llbracket
m\rrbracket}\\!:\overset{\rm{def}}{=}:\\!{\omega}_{1}\cdots{\omega}_{m}\\!:$
is the physical Wick product of
${\omega}_{i}\overset{\rm{def}}{=}{\omega}(n_{i},k_{i})$,
${\mathcal{A}_{N}(f)}$ is given in (2.5), and

(3.9)
$\displaystyle{\bm{t}}=(t_{1},\dots,t_{m})\overset{\rm{def}}{=}{\bm{n}}/N=(n_{1}/N,\cdots,n_{m}/N),$
$\displaystyle{\bm{x}}=(x_{1},\dots,x_{m})\overset{\rm{def}}{=}{\bm{x}}/N^{1/\rho}=(k_{1}/N^{1/\rho},\cdots,k_{m}/N^{1/\rho}).$

###  3.1. Weak convergence of U-statistics

In this subsection, we shall prove the joint weak convergence for
$U$-statistics $\mathbf{I}_{m}^{(N)}(f)$ defined by (3).

We first introduce the following lemma which provides a uniform (in $N$) bound
for the second moment of $\mathbf{I}_{m}^{(N)}(f)$ with
$f\in\mathcal{B}^{\otimes m}$, which will be used to prove Propositions 3.2
and 3.3.

######  Lemma 3.1.

Assume the disorder ${\omega}$ is given as in Section 1.2. Then, there exists
a constant $C$ such that for all $f\in\mathcal{B}^{\otimes m}$,

(3.10) ${\mathbb{E}}\left[\left(\mathbf{I}_{m}^{(N)}(f)\right)^{2}\right]\leq
C^{m}m!\|\hat{f}\|^{2}_{\mathcal{B}^{\otimes m}},$

where $\hat{f}$ is the symmetrization of $f$.

Proof For $m=1$, we have

(3.11)
$\displaystyle\begin{split}&{\mathbb{E}}\left[\left(\mathbf{I}_{1}^{(N)}(f)\right)^{2}\right]\\\
=&N^{-2(\theta+1/\rho)}\sum_{n,n^{\prime}\in\llbracket
N\rrbracket}\sum_{k,k^{\prime}\in{\mathbb{Z}}}{\mathbb{E}}[{\omega}(n,k){\omega}(n^{\prime},k^{\prime})]{\mathcal{A}_{N}(f)}\left(\frac{n}{N},\frac{k}{N^{1/\rho}}\right){\mathcal{A}_{N}(f)}\left(\frac{n^{\prime}}{N},\frac{k^{\prime}}{N^{1/\rho}}\right)\\\
=&N^{-2(\theta+1/\rho)}\sum_{n,n^{\prime}\in\llbracket
N\rrbracket}\sum_{k\in{\mathbb{Z}}}\gamma(n-n^{\prime}){\mathcal{A}_{N}(f)}\left(\frac{n}{N},\frac{k}{N^{1/\rho}}\right){\mathcal{A}_{N}(f)}\left(\frac{n^{\prime}}{N},\frac{k}{N^{1/\rho}}\right).\end{split}$

For $t\in\mathbb{R}$, let

(3.12)
$\gamma_{N}(t)\overset{\rm{def}}{=}N^{2-2H}\gamma\left(\left[|t|N\right]\right).$

Then, by (1.12) and (1.13), we have for all $t\in\mathbb{R}$

(3.13) $\gamma_{N}(t)\lesssim|t|^{2H-2}\text{ and
}\lim\limits_{N\to\infty}\gamma_{N}(t)=|t|^{2H-2}.$

Then, using the notations $t=n/N,t^{\prime}=n^{\prime}/N,x=k/N^{1/\rho}$, we
have

(3.14)
$\displaystyle{\mathbb{E}}\left[\left(\mathbf{I}_{1}^{(N)}(f)\right)^{2}\right]$
$\displaystyle=N^{-2-1/\rho}\sum_{n,n^{\prime}\in\llbracket
N\rrbracket}\sum_{k\in\mathbb{Z}}N^{2-2H}\gamma\left(n-n^{\prime}\right){\mathcal{A}_{N}(f)}\left(t,x\right){\mathcal{A}_{N}(f)}\left(t^{\prime},x\right)$
$\displaystyle\lesssim\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}|t-t^{\prime}|^{2H-2}\left|{\mathcal{A}_{N}(f)}(t,x){\mathcal{A}_{N}(f)}(t^{\prime},{x})\right|{\mathrm{d}}x{\mathrm{d}}t{\mathrm{d}}t^{\prime}$
$\displaystyle=\|{\mathcal{A}_{N}(f)}\|^{2}_{\mathcal{B}}\lesssim\|{f}\|^{2}_{\mathcal{B}},$

where the last inequality follows from Lemma 2.2.

Recall that $\theta=H-\frac{1}{2\rho}$ given in (1.16). For general $m$,

(3.15)
$\displaystyle{\mathbb{E}}\left[\left(\mathbf{I}_{m}^{(N)}(f)\right)^{2}\right]=N^{-2m(\theta+1/\rho)}\sum_{{\bm{n}},{\bm{n}}^{\prime}}\sum_{{\bm{k}},{\bm{k}}^{\prime}}{\mathbb{E}}\left[:\\!{\omega}_{\llbracket
m\rrbracket}\\!::\\!{\omega}^{\prime}_{\llbracket
m\rrbracket}\\!:\right]\mathcal{A}_{N}(f)({\bm{t}},{\bm{x}})\mathcal{A}_{N}(f)({\bm{t}}^{\prime},{\bm{x}}^{\prime}),$

where $:\\!{\omega}_{\llbracket
m\rrbracket}\\!:=:\\!\prod_{i=1}^{m}{\omega}(n_{i},k_{i})\\!:$ and similarly
for $:\\!{\omega}^{\prime}_{\llbracket m\rrbracket}\\!:$.

Recalling that we have assumed that ${\omega}$ is Gaussian, then by (A.8) we
have

${\mathbb{E}}\left[:\\!{\omega}_{\llbracket
m\rrbracket}\\!::\\!{\omega}^{\prime}_{\llbracket
m\rrbracket}\\!:\right]={\mathbb{E}}\left[:\\!\prod_{i=1}^{m}{\omega}(n_{i},k_{i})\\!::\\!\prod_{i=1}^{m}{\omega}(n_{i}^{\prime},k_{i}^{\prime})\\!:\right]=\sum_{\sigma\in\mathcal{P}_{m}}\prod_{i=1}^{m}\gamma(n_{i}-n_{\sigma(i)}^{\prime})\delta_{k_{i}k^{\prime}_{\sigma(i)}},$

where the summation $\sum_{\sigma\in\mathcal{P}_{m}}$ is taken over the set
$\mathcal{P}_{m}$ of all the permutations of $\llbracket m\rrbracket$. Note
that by the symmetry of the summation
$\sum_{{\bm{n}},{\bm{n}}^{\prime}}\sum_{{\bm{k}},{\bm{k}}^{\prime}}$, we have
$\mathbf{I}_{m}^{(N)}(f)=\mathbf{I}_{m}^{(N)}(\hat{f})$. Thus, by the symmetry
of $\mathcal{A}_{N}(\hat{f})$, we have

${\mathbb{E}}\left[\left(\mathbf{I}_{m}^{(N)}(\hat{f})\right)^{2}\right]=m!\,N^{-2m(\theta+1/\rho)}\sum_{{\bm{n}},{\bm{n}}^{\prime}}\sum_{{\bm{k}}}\left(\prod_{i=1}^{m}\gamma(n_{i}-n_{i}^{\prime})\right)\mathcal{A}_{N}(\hat{f})({\bm{t}},{\bm{x}})\mathcal{A}_{N}(\hat{f})({\bm{t}}^{\prime},{\bm{x}}).$

Then, similarly to the case $m=1$, by Lemma 2.2 we can get
${\mathbb{E}}\left[\left(\mathbf{I}_{m}^{(N)}(f)\right)^{2}\right]\leq
C^{m}\|\hat{f}\|^{2}_{\mathcal{B}^{\otimes m}}.$ ∎

Now, we are ready to prove the weak convergence for $\mathbf{I}_{m}^{(N)}(f)$
defined in (3). We first prove the weak convergence for $m=1$.

######  Proposition 3.2.

Let $f:[0,1]\times\mathbb{R}\to\mathbb{R}$ be a function in $\mathcal{B}$.
Then, $\mathbf{I}_{1}^{(N)}(f)$ converges weakly to the Wiener integral
$\mathbf{I}_{1}(f)$ as $N\to\infty$. Moreover, for any $k\in\mathbb{N}$ and
$f_{1},\dots,f_{k}\in\mathcal{B}$, we have the joint convergence in
distribution:

(3.16)
$\left(\mathbf{I}_{1}^{(N)}(f_{1}),\dots,\mathbf{I}_{1}^{(N)}(f_{k})\right)\overset{d}{\longrightarrow}\left(\mathbf{I}_{1}(f_{1}),\dots,\mathbf{I}_{1}(f_{k})\right),\text{
as }N\to\infty.$

Proof First we consider an indicator function
$f={\mathbf{1}}_{[s,t]\times[y,z]}$ with $0\leq s\leq t\leq 1$ and $y\leq z$.
In this case,

$\displaystyle\mathbf{I}_{1}^{(N)}(f)=N^{-\theta-1/\rho}\sum_{n\in[Ns,Nt]\cap\mathbb{N}}~{}\sum_{k\in[N^{1/\rho}y,N^{1/\rho}z]\cap{\mathbb{Z}}}{\omega}(n,k).$

Clearly it has mean zero and the covariance is

$\displaystyle{\mathbb{E}}\left[\left(\mathbf{I}_{1}^{(N)}(f)\right)^{2}\right]$
$\displaystyle=N^{-2-1/\rho}\sum_{n,n^{\prime}\in[Ns,Nt]\cap\mathbb{N}}\sum_{k\in[N^{1/\rho}y,N^{1/\rho}z]\cap{\mathbb{Z}}}\gamma_{N}(r-r^{\prime}),$

where $\gamma_{N}$ is given by (3.12) and $r=n/N,r^{\prime}=n^{\prime}/N$. By
(3.13) and the dominated convergence theorem, we get

(3.17)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left[\left(\mathbf{I}_{1}^{(N)}(f)\right)^{2}\right]$
$\displaystyle=\int_{s}^{t}\int_{s}^{t}\int_{y}^{z}|r-r^{\prime}|^{2H-2}{\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}r^{\prime}=\|f\|^{2}_{\mathscr{H}}$

where we recall that the inner product
$\langle\cdot,\cdot\rangle_{\mathscr{H}}$ is given in (2.2). Thus, we have
$\mathbf{I}_{1}^{(N)}(f)\overset{d}{\longrightarrow}\mathbf{I}_{1}(f)$ as
$N\to\infty$, noting that $\mathbf{I}_{1}^{(N)}(f),N\in\mathbb{N}$ and
$\mathbf{I}_{1}(f)$ are Gaussian random variables and that $\mathbf{I}_{1}(f)$
has zero mean and variance $\|f\|_{\mathscr{H}}^{2}$.

Similarly, one can show that (3.17) holds for each simple function $f$, i.e.,
$f$ is a linear combination of indicator functions. For a general function
$f\in\mathcal{B}$, there exists an approximation sequence of simple functions
$\\{f^{(n)}\\}_{n\in\mathbb{N}}$ in $\mathcal{B}$. On the one hand,
$\mathbf{I}_{1}^{(N)}(f^{(n)})$ converges weakly to $\mathbf{I}_{1}(f^{(n)})$
as $N\to\infty$ for each fixed $n$. On the other hand, by Lemma 3.1, we have
$\mathbf{I}_{1}^{(N)}(f^{(n)})\to\mathbf{I}_{1}^{(N)}(f)$ in $L^{2}$ uniformly
in $N$ as $n\to\infty$. Combining the above convergences together with the
obvious $L^{2}$-convergence of $\mathbf{I}_{1}(f^{(n)})\to\mathbf{I}_{1}(f)$
as $n\to\infty$, by Lemma B.1 we obtain the weak convergence
$\mathbf{I}_{1}^{(N)}(f)\overset{d}{\longrightarrow}\mathbf{I}_{1}(f)$ as
$N\to\infty$ for $f\in\mathcal{B}$. The argument can be presented in a
diagram:

$\textstyle{\mathbf{I}_{1}^{(N)}(f^{(n)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{N\to\infty}$$\scriptstyle{\mathrm{in\
}L^{2},\mathrm{\ uniformly\ in\
}N}$$\scriptstyle{n\to\infty}$$\textstyle{\mathbf{I}_{1}^{(N)}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{N\to\infty}$$\textstyle{\mathbf{I}_{1}(f^{(n)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n\to\infty}$$\scriptstyle{\mathrm{in\
}L^{2}}$$\textstyle{\mathbf{I}_{1}(f))}$

Finally, the joint weak convergence (3.16) follows from the linearity of
$\mathbf{I}_{1}^{(N)}$ and the Cramér-Wold theorem (see Theorem B.4). ∎

The main result in this section is presented below.

######  Proposition 3.3.

For each $m\in\mathbb{N}$ and $f\in\mathcal{B}^{\otimes m}$, we have

(3.18)
$\displaystyle{\mathbf{I}}_{m}^{(N)}(f)\overset{d}{\longrightarrow}\mathbf{I}_{m}(f),\text{
as }N\to\infty,$

where ${\mathbf{I}}_{m}^{(N)}(f)$ is given in (3) and $\mathbf{I}_{m}(f)$ is
an $m$th multiple Wiener integral. Moreover, for any $k\in\mathbb{N}$ and
$f_{1},\dots,f_{k}$ with $f_{i}\in\mathcal{B}^{\otimes l_{i}}$ with
$l_{i}\in\mathbb{N}$, we have the joint convergence in distribution:

(3.19)
$\left(\mathbf{I}_{l_{1}}^{(N)}(f_{1}),\dots,\mathbf{I}_{l_{k}}^{(N)}(f_{k})\right)\overset{d}{\longrightarrow}\Big{(}\mathbf{I}_{l_{1}}(f_{1}),\dots,\mathbf{I}_{l_{k}}(f_{k})\Big{)},\text{
as }N\to\infty.$

Proof We only prove (3.18) and (3.19) can be proved in a similar way in the
light of the Cramér-Wold theorem (Theorem B.4). We only need to consider
$m\geq 2$. First, we consider $\mathbf{I}_{m}^{(N)}(f)$ for $f=h_{1}^{\otimes
p_{1}}\otimes\cdots\otimes h_{l}^{\otimes p_{l}}$ with
$h_{i}\in\mathcal{B},k_{i}\in\mathbb{N}$ for $i=1,2,\dots,l$ satisfying
$p_{1}+\dots+p_{l}=m$. For such $f$, we have

$\displaystyle{\mathbf{I}}_{m}^{(N)}(f)=$ $\displaystyle
N^{-m(\theta+1/\rho)}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}:{\omega}_{\llbracket
m\rrbracket}:h_{1}^{\otimes p_{1}}\otimes\cdots\otimes h_{l}^{\otimes p_{l}}$
$\displaystyle=$
$\displaystyle:\\!\Big{(}{\mathbf{I}}_{1}^{(N)}(h_{1})\Big{)}^{p_{1}}\cdots\Big{(}{\mathbf{I}}_{1}^{(N)}(h_{l})\Big{)}^{p_{l}}\\!:,$

where the term on the right-hand side is a physical Wick product (see Section
A). By Proposition 3.2, we have
$\big{(}{\mathbf{I}}_{1}^{(N)}(h_{1}),\dots,{\mathbf{I}}_{1}^{(N)}(h_{l})\big{)}\overset{d}{\longrightarrow}\big{(}\mathbf{I}_{1}(h_{1}),\dots,\mathbf{I}_{1}(h_{l})\big{)}$
as $N\to\infty$.

For each $i=1,\dots,l$, $\mathbf{I}_{1}^{(N)}(h_{i})$ is a centered Gaussian
random variable with variance bounded by $\|h_{i}\|^{2}_{\mathcal{B}}$ with
some constant multiplicative factor by Lemma 3.1. Therefore, for any
$q\in\mathbb{N}$, by the Gaussianity of ${\omega}$ and (3.14), we have
${\mathbb{E}}\left[\left(\mathbf{I}_{1}^{(N)}(h_{i})\right)^{2q}\right]\lesssim(2q-1)!!\|h_{i}\|^{2q}_{\mathcal{B}}<\infty$.
Then we can apply Lemma B.3 and get that, as $N\to\infty$,

$\displaystyle{\mathbf{I}}_{m}^{(N)}(f)\overset{d}{\longrightarrow}$
$\displaystyle\Big{(}{\mathbf{I}}_{1}(h_{1})\Big{)}^{\diamond
p_{1}}\diamond\cdots\diamond\Big{(}{\mathbf{I}}_{1}(h_{l})\Big{)}^{\diamond
p_{l}}=\mathbf{I}_{p_{1}}(h_{1}^{\otimes
p_{1}})\diamond\cdots\diamond\mathbf{I}_{p_{l}}(h_{l}^{\otimes p_{l}})$
$\displaystyle=$ $\displaystyle\mathbf{I}_{m}(h_{1}^{\otimes
p_{1}}\otimes\cdots\otimes h_{l}^{\otimes p_{l}})=\mathbf{I}_{m}(f).$

Then, for any $f\in\mathcal{B}^{\otimes m}$, one can find a sequence
$\\{f^{(n)}\\}_{n\in\mathbb{N}}$ which are the linear combinations of the
functions of the form $h_{1}^{\otimes p_{1}}\otimes\cdots\otimes
h_{l}^{\otimes p_{l}}$ such that $f^{(n)}\to f$ in $\mathcal{B}^{\otimes m}$.
By the preceding argument, for each $f^{(n)}$, we have
$\mathbf{I}_{m}^{(N)}(f^{(n)})\overset{d}{\longrightarrow}\mathbf{I}_{m}(f^{(n)})$
as $N\to\infty$. Then similarly as in the proof of Proposition 3.2, we have,
recalling Lemma 3.1,

$\textstyle{\mathbf{I}_{m}^{(N)}(f^{(n)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{N\to\infty}$$\scriptstyle{\mathrm{in\
}L^{2},\mathrm{\ uniformly\ in\
}N}$$\scriptstyle{n\to\infty}$$\textstyle{\mathbf{I}_{m}^{(N)}(f)}$$\textstyle{\mathbf{I}_{m}(f^{(n)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n\to\infty}$$\scriptstyle{\mathrm{in\
}L^{2}}$$\textstyle{\mathbf{I}_{m}(f))}$

This together with Lemma B.1 yields
$\mathbf{I}_{m}^{(N)}(f)\overset{d}{\longrightarrow}\mathbf{I}_{m}(f)$ as
$N\to\infty$ for $f\in\mathcal{B}^{\otimes m}$. ∎

###  3.2. Weak convergence of rescaled partition functions

Recall that the so-called $m$th moment ${\mathbb{S}}_{m}^{(N)}$ appearing in
the rescaled partition function (3.2) has the expression (3.5). We first prove
the joint weak convergence of moments.

######  Proposition 3.4.

Assume condition 1.16. For each $m\in\mathbb{N}$, we have

(3.20)
$\displaystyle\frac{1}{m!}{\mathbb{S}}_{m}^{(N)}\overset{d}{\longrightarrow}\beta^{m}\mathbb{I}_{m}(\mathfrak{g}_{m}),\text{
as }N\to\infty,$

where $\mathfrak{g}_{m}=\mathfrak{g}_{m}({\bm{t}},{\bm{x}};1,x_{0})$ is given
in (2.28) and $\mathbb{I}_{m}(\mathfrak{g}_{m})$ is an $m$th multiple
Stratonovich integral. Moreover, for any $k\in\mathbb{N}$ and
$l_{1},\dots,l_{k}\in\mathbb{N}$, we have the joint convergence in
distribution:

(3.21)
$\left(\frac{1}{l_{1}!}\mathbb{S}_{l_{1}}^{(N)},\dots,\frac{1}{l_{k}!}\mathbb{S}_{l_{k}}^{(N)}\right)\overset{d}{\longrightarrow}\Big{(}\beta^{l_{1}}\mathbb{I}_{l_{1}}(\mathfrak{g}_{l_{1}}),\dots,\beta^{l_{k}}\mathbb{I}_{l_{k}}(\mathfrak{g}_{l_{k}})\Big{)},\text{
as }N\to\infty.$

Proof We first prove (3.20). Note that
$\mathrm{Tr}^{k}\hat{\mathfrak{g}}_{m}\in\mathcal{H}^{\otimes(m-2k)}$ (see
(2.1.2) for the definition of $\mathrm{Tr}^{k}\hat{\mathfrak{g}}_{m}$) for
$k=0,1,\dots,[\frac{m}{2}]$ (see Remark 2.11). Thus, by Proposition 3.3, we
have (we assume $m$ is an odd integer and the analysis for $m$ being an even
number is the same),

(3.22)
$\displaystyle\bigg{(}\mathbf{I}_{1}^{(N)}\left(\mathrm{Tr}^{[\frac{m}{2}]}\hat{\mathfrak{g}}_{m}\right),\mathbf{I}_{3}^{(N)}\left(\mathrm{Tr}^{[\frac{m}{2}]-1}\hat{\mathfrak{g}}_{m}\right),\dots,\mathbf{I}_{m}^{(N)}\left(\hat{\mathfrak{g}}_{m}\right)\bigg{)}$
$\displaystyle\overset{d}{\longrightarrow}\bigg{(}\mathbf{I}_{1}\left(\mathrm{Tr}^{[\frac{m}{2}]}\hat{\mathfrak{g}}_{m}\right),\mathbf{I}_{3}\left(\mathrm{Tr}^{[\frac{m}{2}]-1}\hat{\mathfrak{g}}_{m}\right),\dots,\mathbf{I}_{m}\left(\hat{\mathfrak{g}}_{m}\right)\bigg{)},\text{
as }N\to\infty.$

Recalling (3.5), by the symmetry of the summation, we can write
${\mathbb{S}}_{m}^{(N)}$ as

${\mathbb{S}}_{m}^{(N)}=\hat{\beta}^{m}_{N}\sum_{n_{1},\dots,n_{m}\in\llbracket
N\rrbracket}\sum_{k_{1},\dots,k_{m}\in{\mathbb{Z}}}{\omega}_{\llbracket
m\rrbracket}P_{{\bm{n}}^{*}},$

where $P_{{\bm{n}}^{*}}$ is given in (3.4) and ${\omega}_{\llbracket
m\rrbracket}=\prod_{i=1}^{m}{\omega}_{i}$ with
${\omega}_{i}={\omega}(n_{i},k_{i})$. It follows from (A.5) and (A.4) that

${\omega}_{\llbracket
m\rrbracket}=\sum_{j=0}^{[\frac{m}{2}]}\sum_{\begin{subarray}{c}B\subset{\llbracket
m\rrbracket}\\\
|B|=m-2j\end{subarray}}:\\!{\omega}_{B}\\!:{\mathbb{E}}\left[{\omega}_{{\llbracket
m\rrbracket}\backslash B}\right].$

Therefore,

(3.23) $\displaystyle{\mathbb{S}}_{m}^{(N)}=$
$\displaystyle\hat{\beta}_{N}^{m}\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}\Bigg{(}\sum_{j=0}^{[\frac{m}{2}]}\sum_{\begin{subarray}{c}B\subset{\llbracket
m\rrbracket}\\\
|B|=m-2j\end{subarray}}:\\!{\omega}_{B}\\!:{\mathbb{E}}\left[{\omega}_{{\llbracket
m\rrbracket}\backslash B}\right]\Bigg{)}P_{{\bm{n}}^{*}}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{[\frac{m}{2}]}\sum_{\begin{subarray}{c}B\subset{\llbracket
m\rrbracket}\\\
|B|=m-2j\end{subarray}}\Bigg{(}\hat{\beta}_{N}^{m}\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}:\\!{\omega}_{B}\\!:{\mathbb{E}}\left[{\omega}_{{\llbracket
m\rrbracket}\backslash B}\right]P_{{\bm{n}}^{*}}\Bigg{)}.$

For each $j=0,1,\dots,[\frac{m}{2}]$, all the terms in the summation
$\sum_{\begin{subarray}{c}B\subset{\llbracket m\rrbracket}\\\
|B|=m-2j\end{subarray}}$ are equal by the symmetry. We fix
$j\in\\{0,1,\dots,[\frac{m}{2}]\\}$. Note that there are in total ${m\choose
m-2j}$ subsets in $\llbracket m\rrbracket$ with cardinality $m-2j$. Without
loss of generality, we assume $B=\\{2j+1,\dots,m\\}$. Then by (A.4), we have

(3.24) $\displaystyle{\mathbb{E}}[{\omega}_{\llbracket m\rrbracket\backslash
B}]={\mathbb{E}}[{\omega}_{\llbracket
2j\rrbracket}]=\sum_{V}\prod_{\\{\ell_{1},\ell_{2}\\}\in
V}{\mathbb{E}}[{\omega}_{\ell_{1}}{\omega}_{\ell_{2}}],$

where the sum $\sum_{V}$ is taken over all pair partitions of $\llbracket
2j\rrbracket$. By the symmetry again, the summations

$\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}:\\!{\omega}_{B}\\!:\prod_{\\{\ell_{1},\ell_{2}\\}\in
V}{\mathbb{E}}[{\omega}_{\ell_{1}}{\omega}_{\ell_{2}}]P_{{\bm{n}}^{*}}$

coincide with each other for all the partitions $V$, and note that there are
in total $(2j-1)!!$ pair partitions of $\llbracket 2j\rrbracket$. Thus, we
have the following equation:

$\displaystyle\mathbb{S}_{m}^{(N)}$
$\displaystyle=\sum_{j=0}^{[\frac{m}{2}]}{m\choose{m-2j}}(2j-1)!!~{}\left[\hat{\beta}_{N}^{m}\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}:\\!{\omega}_{B}\\!:\prod_{\ell=1}^{j}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}]P_{{\bm{n}}^{*}}\right]$
(3.25)
$\displaystyle=\sum_{j=0}^{[\frac{m}{2}]}\frac{m!}{j!(m-2j)!2^{j}}~{}\left[\hat{\beta}_{N}^{m}\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}:\\!{\omega}_{B}\\!:\prod_{\ell=1}^{j}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}]P_{{\bm{n}}^{*}}\right],$

where we recall $B=\\{2j+1,\dots,m\\}$.

Compare (3.2) with Hu-Meyer’s formula (2.19) and also note (3.22). It then
follows easily that in order to prove (3.20), it suffices to show, setting
$\beta=1$ without loss of generality,

(3.26) $\displaystyle Y_{N}$
$\displaystyle\overset{\rm{def}}{=}\frac{1}{m!}N^{-m\theta}\sum_{n_{1},\dots,n_{m}}\sum_{k_{1},\dots,k_{m}}:\\!{\omega}_{B}\\!:\prod_{\ell=1}^{j}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}]P_{{\bm{n}}^{*}}-\mathbf{I}_{m-2j}^{(N)}(\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m})$
$\displaystyle=\mathbf{I}_{m-2j}^{(N)}\left(N^{-2j\theta}N^{(m-2j)\frac{1}{\rho}}\sum_{n_{i}:i\in\llbracket
2j\rrbracket}\sum_{k_{i}:i\in\llbracket
2j\rrbracket}\prod_{\ell=1}^{j}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}](P_{{\bm{n}}^{*}}/m!)\right)-\mathbf{I}_{m-2j}^{(N)}(\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m})$
$\displaystyle\to 0\text{ in $L^{2}$ as }N\to\infty.$

for all $j=0,1,\dots,[\frac{m}{2}]$, where we recall that
$\mathbf{I}_{m-2j}^{(N)}(f)$ is given in (3).

We prove (3.26) for $j=0$, and the general case can be proved in a similar
spirit. When $j=0$, we have

$Y_{N}=\mathbf{I}_{m}^{(N)}\left(N^{m/\rho}P_{{\bm{n}}^{*}}/m!-\hat{\mathfrak{g}}_{m}\right).$

Denoting
$\mathfrak{D}_{N}=\mathfrak{D}_{N}({\bm{n}},{\bm{k}})=N^{m/\rho}P_{{\bm{n}}^{*}}/m!-\mathcal{A}_{N}(\hat{\mathfrak{g}}_{m})({\bm{n}}/N,{\bm{k}}/N^{1/\rho}),\mathfrak{D}^{\prime}_{N}=\mathfrak{D}_{N}({\bm{n}}^{\prime},{\bm{k}}^{\prime})$
and
$\tilde{\mathfrak{D}}^{\prime}_{N}=\mathfrak{D}_{N}({\bm{n}}^{\prime},{\bm{k}})$,
we have

(3.27) $\displaystyle{\mathbb{E}}[Y_{N}^{2}]=$ $\displaystyle
N^{-2m(\theta+\frac{1}{\rho})}\sum_{{\bm{n}},{\bm{n}}^{\prime}\in\llbracket
N\rrbracket^{m}}\sum_{{\bm{k}},{\bm{k}}^{\prime}\in{\mathbb{Z}}^{m}}{\mathbb{E}}\left[:\\!{\omega}_{\llbracket
m\rrbracket}\\!::\\!{\omega}^{\prime}_{\llbracket
m\rrbracket}\\!:\right]\mathfrak{D}_{N}\mathfrak{D}^{\prime}_{N}$
$\displaystyle=$ $\displaystyle
m!N^{-2m(\theta+\frac{1}{\rho})}\sum_{{\bm{n}},{\bm{n}}^{\prime}\in\llbracket
N\rrbracket^{m}}\sum_{{\bm{k}}\in{\mathbb{Z}}^{m}}\prod_{i=1}^{m}\gamma(n_{i}-n_{i}^{\prime})\mathfrak{D}_{N}\tilde{\mathfrak{D}}^{\prime}_{N}$
$\displaystyle\lesssim$ $\displaystyle
m!\int_{[0,1]^{2m}\times\mathbb{R}^{m}}\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}\mathfrak{D}_{N}\tilde{\mathfrak{D}}^{\prime}_{N}{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}}$

where the second equality follows from (A.8) and we use the notations that
link the discrete summations and continuum integrals:
$t_{i}=n_{i}/N,t^{\prime}_{i}=n^{\prime}_{i}/N$ and $x_{i}=k_{i}/N^{1/\rho}$.
In light of Lemma 2.2, we have

(3.28) ${\mathbb{E}}[Y_{N}^{2}]\lesssim
m!\int_{[0,1]^{2m}\times\mathbb{R}^{m}}\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}\left|\mathbf{D}_{N}\tilde{\mathbf{D}}^{\prime}_{N}\right|{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}},$

where
$\mathbf{D}_{N}=\mathbf{D}_{N}({\bm{n}},{\bm{k}})=N^{m/\rho}P_{{\bm{n}}^{*}}/m!-\hat{\mathfrak{g}}_{m}({\bm{t}},{\bm{x}})$
and
$\tilde{\mathbf{D}}_{N}^{\prime}=\mathbf{D}_{N}({\bm{n}}^{\prime},{\bm{k}}).$

Inspired by [8], we decompose
$I\overset{\rm{def}}{=}[0,1]^{2m}\times\mathbb{R}^{m}$ as $I\cap(I_{1}\cup
I_{2}\cup I_{3})$, where

$I_{1}\overset{\rm{def}}{=}\bigcap_{i=1}^{m}\left(\Big{\\{}|t_{i}-t^{\prime}_{i}|>\varepsilon\Big{\\}}\cap\\{|x_{i}|<M\\}\right),$
$I_{2}\overset{\rm{def}}{=}\bigcup_{i=1}^{m}\\{|t_{i}-t^{\prime}_{i}|\leq\varepsilon\\},\text{
and }I_{3}\overset{\rm{def}}{=}\bigcup_{i=1}^{m}\\{|x_{i}|\geq M\\},$

for some fixed $\varepsilon,M>0$. On $I\cap I_{1}$,
$\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}|\mathbf{D}_{N}\tilde{\mathbf{D}}^{\prime}_{N}|$
is uniformly bounded and converges to 0 by the local limit theorem, and hence
the integral converges to 0 by the dominated convergence theorem. Then, we can
get $\lim_{N\to\infty}{\mathbb{E}}[Y_{N}^{2}]=0$, if we can show the integral
on $I\cap I_{2}$ (resp. $I\cap I_{3}$) can be arbitrarily small if we choose
$\varepsilon$ sufficiently small (resp. $M$ sufficiently large).

By (1.6) and (1.9), we have

(3.29)
$\displaystyle\left(N^{m/\rho}P_{{\bm{n}}^{*}}\right)\left(N^{m/\rho}P_{({\bm{n}}^{\prime})^{*}}\right)\lesssim\Big{(}(t^{*}_{2}-t^{*}_{1})\cdots(t_{m}^{*}-t_{m-1}^{*})(1-t_{m}^{*})\Big{)}^{-1/\rho}N^{m/\rho}P_{({\bm{n}}^{\prime})^{*}},$
$\displaystyle\hat{\mathfrak{g}}_{m}({\bm{t}},{\bm{x}};1,x_{0})\hat{\mathfrak{g}}_{m}({\bm{t}}^{\prime},{\bm{x}};1,x_{0})\lesssim\Big{(}(t^{*}_{2}-t^{*}_{1})\cdots(t_{m}^{*}-t_{m-1}^{*})(1-t_{m}^{*})\Big{)}^{-1/\rho}\hat{\mathfrak{g}}_{m}({\bm{t}}^{\prime},{\bm{x}};1,x_{0}),$

and similarly for the cross terms
$N^{m/\rho}P_{{\bm{n}}^{*}}\hat{\mathfrak{g}}_{m}({\bm{t}}^{\prime},{\bm{x}})$
and
$N^{m/\rho}P_{({\bm{n}}^{\prime})^{*}}\hat{\mathfrak{g}}_{m}({\bm{t}},{\bm{x}})$,
where $t_{i}^{*}=t_{\sigma(i)}$ for some permutation $\sigma$ on $\llbracket
m\rrbracket$ such that $t_{1}^{*}\leq t_{2}^{*}\leq\cdots\leq t_{m}^{*}$.
These inequalities and the fact for all ${\bm{t}}\in[0,1]^{m}$,

$\int_{\mathbb{R}^{m}}N^{m/\rho}P_{{\bm{n}}^{*}}{\mathrm{d}}{\bm{x}}=\int_{\mathbb{R}^{m}}(m!)\hat{\mathfrak{g}}_{m}{\mathrm{d}}{\bm{x}}=1,$

yield that
$\int_{I}\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}|\mathbf{D}_{N}\tilde{\mathbf{D}}^{\prime}_{N}|{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}}$
is bounded uniformly in $N$ by (up to a multiplicative constant)

(3.30)
$C_{m}\overset{\rm{def}}{=}\frac{1}{m!}\int_{[0,1]^{2m}}\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}\Big{(}(t^{*}_{2}-t^{*}_{1})\cdots(t_{m}^{*}-t_{m-1}^{*})(1-t_{m}^{*})\Big{)}^{-1/\rho}{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime},$

which is finite due to $2H-2>-1$ and $-1/\rho>-1$. This implies that the
integral on $I\cap I_{2}$ can be arbitrarily small by choosing $\varepsilon$
sufficiently small. Similarly, for the integral on $I\cap I_{3}$,

$\displaystyle\int_{I\cap
I_{3}}\prod_{i=1}^{m}|t_{i}-t_{i}^{\prime}|^{2H-2}|\mathbf{D}_{N}\tilde{\mathbf{D}}^{\prime}_{N}|{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}}\lesssim$
$\displaystyle C_{m}\Bigg{(}P\left\\{\sup_{0\leq t\leq 1}|X_{t}|\geq
M\Big{|}X_{1}=x_{0}\right\\}$ $\displaystyle\qquad+P\left\\{\max_{0\leq m\leq
N}|S_{m}|\geq N^{1/\rho}M\Big{|}S_{N}=N^{1/\rho}x_{0}\right\\}\Bigg{)}.$

Noting that $P\left\\{\max_{0\leq m\leq N}|S_{m}|\geq
N^{1/\rho}M\Big{|}S_{N}=N^{1/\rho}x_{0}\right\\}$ converges to
$P\left\\{\sup_{0\leq t\leq 1}|X_{t}|\geq M\Big{|}X_{1}=x_{0}\right\\}$ as $N$
goes to infinity, we can make the integral on $I\cap I_{3}$ as small uniformly
in $N$ as we want by choosing $M$ sufficiently large. This proves (3.26) for
$j=0$.

For general $j=0,1,\dots,[\frac{m}{2}]$, using the same argument leading to
(3.28), we have

(3.31)
${\mathbb{E}}[Y_{N}^{2}]\lesssim(m-2j)!\int_{[0,1]^{2(m-2j)}\times\mathbb{R}^{m-2j}}\prod_{i=1}^{m-2j}|t_{i}-t_{i}^{\prime}|^{2H-2}\left|\mathbf{D}_{N}\tilde{\mathbf{D}}^{\prime}_{N}\right|{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}},$

where now

(3.32) $\displaystyle\mathbf{D}_{N}=\mathbf{D}_{N}({\bm{n}},{\bm{k}})$
$\displaystyle=N^{-2j\theta}N^{(m-2j)\frac{1}{\rho}}\sum_{n_{i}:i\in\llbracket
2j\rrbracket}\sum_{k_{i}:i\in\llbracket
2j\rrbracket}\prod_{\ell=1}^{j}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}](P_{{\bm{n}}^{*}}/m!)-\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m}$
$\displaystyle\overset{\rm{def}}{=}\mathcal{Q}_{N}-\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m}$

and
$\tilde{\mathbf{D}}_{N}^{\prime}=\mathbf{D}_{N}({\bm{n}}^{\prime},{\bm{k}})$.
Then, we estimate the right-hand side of (3.31) in a similar way as for the
case $j=0$, i.e., we split $[0,1]^{2(m-2j)}\times\mathbb{R}^{m-2j}$ as the
union of its restrictions on $I_{1},I_{2}$ and $I_{3}$ and then analyse the
integral restricted on $I_{i}$’s separately for $i=1,2,3$. The analysis for
the integral on $I_{1}$ is the same as for $j=0$. To argue that the integral
on $I_{2}$ and $I_{3}$ can be arbitrarily small if we choose $\varepsilon$
sufficiently small and $M$ sufficiently large, it suffices to find a uniform
(in N) upper bound for

(3.33)
$A\overset{\rm{def}}{=}\int_{[0,1]^{2(m-2j)}\times\mathbb{R}^{m-2j}}\prod_{i=1}^{m-2j}|t_{i}-t_{i}^{\prime}|^{2H-2}\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m}({\bm{t}},{\bm{x}};1,x_{0})\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m}({\bm{t}}^{\prime},{\bm{x}};1,x_{0}){\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}}$

and

(3.34)
$B\overset{\rm{def}}{=}\int_{[0,1]^{2(m-2j)}\times\mathbb{R}^{m-2j}}\prod_{i=1}^{m-2j}|t_{i}-t_{i}^{\prime}|^{2H-2}\mathcal{Q}_{N}\tilde{\mathcal{Q}}^{\prime}_{N}{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{t}}^{\prime}{\mathrm{d}}{\bm{x}}$

where $\mathcal{Q}_{N}$ is given in (3.32) and
$\tilde{\mathcal{Q}}^{\prime}_{N}=\mathcal{Q}_{N}({\bm{n}}^{\prime},{\bm{k}}).$
This is true, noting that Remark 2.11 yields

$A={\mathbb{E}}\left[\left|\mathbf{I}_{m-2j}(\mathrm{Tr}^{j}\hat{\mathfrak{g}}_{m})\right|^{2}\right]\leq{\mathbb{E}}\left[\left|\mathbb{I}_{m}(\mathfrak{g}_{m})\right|^{2}\right]<\infty,$

and similarly, recalling that $\mathbb{S}_{m}^{(N)}$ is given in (3.5), we
have $B\lesssim{\mathbb{E}}[|\mathbb{S}_{m}^{(N)}|^{2}]$ which has a uniform
upper bound by (3.38).

In the above, we have assumed $m$ is an odd integer. If $m$ is even, the
analysis is the same except for the case $j=\frac{m}{2}$. In this case, we
have

$\displaystyle Y_{N}=N^{-m\theta}\sum_{n_{i}:i\in\llbracket
m\rrbracket}\sum_{k_{i}:i\in\llbracket
m\rrbracket}\prod_{\ell=1}^{m/2}{\mathbb{E}}[{\omega}_{2\ell-1}{\omega}_{2\ell}](P_{{\bm{n}}^{*}}/m!)-\mathrm{Tr}^{m/2}\hat{\mathfrak{g}}_{m},$

which is deterministic and converges to 0 by the local limit theorem and a
similar argument proving $\lim_{N\to\infty}{\mathbb{E}}[Y_{N}^{2}]=0$ for
$j=0$.

Finally, one can prove the weak convergence for linear combinations of
$\left(\mathbb{S}_{l_{1}}^{(N)},\dots,\mathbb{S}_{l_{k}}^{(N)}\right)$ in a
similar way, and hence (3.21) holds due to Theorem B.4. ∎

A direct corollary of Proposition 3.4 is its resemblance for the Skorohod case
as follows, recalling that $\mathbf{S}_{m}^{(N)}$ is given in (3.7).

######  Proposition 3.5.

Assume condition 1.19. For each $m\in\mathbb{N}$, we have

$\displaystyle\frac{1}{m!}{\mathbf{S}}_{m}^{(N)}\overset{d}{\longrightarrow}\beta^{m}\mathbf{I}_{m}(\mathfrak{g}_{m}),\text{
as }N\to\infty,$

where $\mathfrak{g}_{m}=\mathfrak{g}_{m}({\bm{t}},{\bm{x}};1,x_{0})$ is given
in (2.28) and $\mathbf{I}_{m}(\mathfrak{g}_{m})$ is an $m$th multiple Wiener
integral. Moreover, for any $k\in\mathbb{N}$ and
$l_{1},\dots,l_{k}\in\mathbb{N}$, we have the joint convergence in
distribution:

$\left(\frac{1}{l_{1}!}\mathbf{S}_{l_{1}}^{(N)},\dots,\frac{1}{l_{k}!}\mathbf{S}_{l_{k}}^{(N)}\right)\overset{d}{\longrightarrow}\Big{(}\beta^{l_{1}}\mathbf{I}_{l_{1}}(\mathfrak{g}_{l_{1}}),\dots,\beta^{l_{k}}\mathbf{I}_{l_{k}}(\mathfrak{g}_{l_{k}})\Big{)},\text{
as }N\to\infty.$

Now we are ready to prove our main results.

Proof of Theorem 1.2. By Proposition 3.4, we have for all $M\in\mathbb{N}$,

$\displaystyle
Z_{\omega}^{(N,M)}\overset{\rm{def}}{=}\sum_{m=0}^{M}\frac{1}{m!}{\mathbb{S}}_{m}^{(N)}\overset{d}{\longrightarrow}\mathcal{Z}^{(M)}\overset{\rm{def}}{=}\sum_{m=0}^{M}\beta^{m}\mathbb{I}_{m}(\mathfrak{g}_{m}(\cdot;1,x_{0})),\text{
as }N\to\infty.$

Recalling Proposition 2.12 which yields the $L^{2}$-convergence of
$\mathcal{Z}^{(M)}$ to $\mathcal{Z}$ as $M\to\infty$, we only need to show
that $Z_{\omega}^{(N,M)}$ converges to
$Z_{\omega}^{(N)}=Z_{\omega}^{(N)}(\hat{\beta}_{N},N^{1/\rho}x_{0})$ given in
(3.2) in probability uniformly in $N$ as $M\to\infty$ by Lemma B.1. This
follows from (3.39) obtained in Section 3.3.

Proof of Theorem 1.3. The proof is the same as the proof of Theorem 1.2 except
Proposition 3.4 and (3.39) are replaced by Proposition 3.5 and (3.40),
respectively.

###  3.3. $L^{p}$-bound of rescaled partition functions

We first deal with the $L^{1}$-bound of
${\mathbb{Z}}_{\omega}^{(N)}(\hat{\beta}_{N},k)$ for the Stratonovich case.
Recalling that ${\omega}$ is Gaussian, we have

$\displaystyle{\mathbb{E}}\left[\exp\left(N^{-\theta}\sum_{n=1}^{N}{\omega}(n,S_{n})\right)\right]$
$\displaystyle={\mathbb{E}}\left[\exp\left(N^{-\theta}\sum_{n=1}^{N}\sum_{k\in{\mathbb{Z}}}{\omega}(n,k)\mathbf{1}_{[S_{n}=k]}\right)\right]$
$\displaystyle={\mathbb{E}}\left[\exp\left(\frac{1}{2}N^{-2\theta}\sum_{n,n^{\prime}=1}^{N}\sum_{k,k^{\prime}\in{\mathbb{Z}}}\gamma(n-n^{\prime})\delta_{kk^{\prime}}\mathbf{1}_{[S_{n}=k]}1_{[S_{n^{\prime}}=k^{\prime}]}\right)\right]$
$\displaystyle={\mathbb{E}}\left[\exp\left(\frac{1}{2}N^{-2\theta}\sum_{n,n^{\prime}=1}^{N}\gamma(n-n^{\prime})\mathbf{1}_{[S_{n}=S_{n^{\prime}}]}\right)\right].$

In this section, we shall prove a discretised version of Proposition 2.7,
i.e., to show that the above exponential moment is uniformly bounded.

Noting that $S_{n}\in{\mathbb{Z}}$ and using the identity

$\mathbf{1}_{[S_{n}=S_{n^{\prime}}]}=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\rm
e}^{{\imath}(S_{n}-S_{n^{\prime}})\lambda}{\mathrm{d}}\lambda,$

we have

$\displaystyle{\mathbb{E}}\left[\exp\left(\frac{1}{2}N^{-2\theta}\sum_{n,n^{\prime}=1}^{N}\gamma(n-n^{\prime})\mathbf{1}_{[S_{n}=S_{n^{\prime}}]}\right)\right]$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}(4\pi)^{-m}N^{-2m\theta}\sum_{n_{1},\dots,n_{m}=1}^{N}\sum_{n_{1}^{\prime},\dots,n^{\prime}_{m}=1}^{N}\int_{[-\pi,\pi]^{m}}\prod_{i=1}^{m}\gamma(n_{i}-n_{i}^{\prime}){\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}(S_{n_{i}}-S_{n_{i}^{\prime}})\lambda_{i}}\right]{\mathrm{d}}\bm{\lambda}.$

Using changes of variables $n_{i}=Nt_{i},n_{i}^{\prime}=Nt_{i}^{\prime}$ and
$\lambda_{i}=u_{i}/N^{1/\rho}$ for $i=1,\dots,m$, we have for each term on the
right-hand side of the above equation,

(3.35) $\displaystyle
A_{m}\overset{\rm{def}}{=}\frac{1}{m!}(4\pi)^{-m}N^{-2m}\sum_{{\bm{n}},{\bm{n}}^{\prime}}$
$\displaystyle\prod_{i=1}^{m}\left[N^{2-2H}\gamma(N(t_{i}-t_{i}^{\prime}))\right]$
$\displaystyle\times\int_{[-\pi N^{1/\rho},\pi
N^{1/\rho}]^{m}}{\mathbb{E}}\left[{\rm
e}^{{\imath}\sum_{i=1}^{m}(S_{n_{i}}-S_{n_{i}^{\prime}})u_{i}/N^{1/\rho}}\right]{\mathrm{d}}{\bm{u}}.$

Recall that $S_{n}=S_{0}+\sum_{j=1}^{n}Y_{j}$ and that
$\psi(u)={\mathbb{E}}[{\rm e}^{{\imath}uY_{j}}]$ is the characteristic
function of $Y_{j}$. The 1-lattice distribution of $Y_{j}$ implies that, for
any $\varepsilon>0$, one can find a positive constant $c$ such that (see [38,
eq. (5.14)])

(3.36) $|\psi(u/N^{1/\rho})|\leq{\rm e}^{-c|u|^{\rho-\varepsilon}/N},\text{
for }1\leq|u|\leq\pi N^{1/\rho}.$

Thus, for $n\in\llbracket N\rrbracket$, we have

(3.37) $\displaystyle\int_{-\pi N^{1/\rho}}^{\pi
N^{1/\rho}}\left|\psi(u/N^{1/\rho})\right|^{n}{\mathrm{d}}u$
$\displaystyle\leq 2+\int_{\\{1\leq|u|\leq\pi
N^{1/\rho}\\}}\left|\psi(u/N^{1/\rho})\right|^{n}{\mathrm{d}}u$
$\displaystyle\leq 2+\int_{\mathbb{R}}{\rm
e}^{-c|u|^{\rho-\varepsilon}n/N}{\mathrm{d}}u$ $\displaystyle\leq
C(n/N)^{-\frac{1}{\rho-\varepsilon}}.$

Under the condition (1.16), we may choose $\varepsilon>0$ sufficiently small
such that $2H-\frac{1}{\rho-\varepsilon}>1$. To estimate $A_{m}$ given in
(3.35), we combine the estimate (3.37) with the argument used in the proof of
Proposition 2.7 which leads to (2.36), and we can get for all
$N\in\mathbb{N}$,

$\displaystyle A_{m}\leq$
$\displaystyle\frac{(2m)!}{m!}C^{m}\int_{[0,1]_{<}^{2m}}\prod_{i=1}^{m}|t^{*}_{2i}-t^{*}_{2i-1}|^{2H-2-\frac{1}{\rho-\varepsilon}}{\mathrm{d}}{\bm{s}}$
$\displaystyle\leq$
$\displaystyle\frac{(2m)!C^{m}}{m!\Gamma\left(m(2H-\frac{1}{\rho-\varepsilon})+1\right)}$

Therefore, we have uniformly in $N$,

$\sum_{m=0}^{\infty}A_{m}\leq\sum_{m=0}^{\infty}\frac{(2m)!C^{m}}{m!\Gamma\left(m(2H-\frac{1}{\rho-\varepsilon})+1\right)}<\infty,$

where the finiteness follows from Stirling’s formula and
$2H-\frac{1}{\rho-\varepsilon}>1$. This implies

(3.38)
$\sup_{N\in\mathbb{N}}{\mathbb{E}}\left[\exp\left(N^{-\theta}\sum_{n=1}^{N}{\omega}(n,S_{n})\right)\right]<\infty.$

Finally, the above analysis also yields

(3.39)
$\lim_{M\to\infty}\sup_{N\in\mathbb{N}}\sum_{m=M+1}^{\infty}{\mathbb{E}}\left[\left|N^{-\theta}\sum_{n=1}^{N}{\omega}(n,S_{n})\right|^{m}\right]=0,$

which shall be used to prove the weak convergence of rescaled partition
functions.

Now we consider the Skorohod case. Recall the modified partition function
(3.6). Noting that $\mathbf{S}_{m}^{(N)}$ and $\mathbf{S}_{n}^{(N)}$ given in
(3.7) are orthogonal in $L^{2}(\Omega)$ if $m\neq n$, we have

${\mathbb{E}}\left[\left|\sum_{m=0}^{\infty}\frac{1}{m!}{\mathbf{S}}_{m}^{(N)}\right|^{2}\right]=\sum_{m=0}^{\infty}\frac{1}{(m!)^{2}}{\mathbb{E}}_{\omega}\left[\left|\mathbf{S}_{m}^{(N)}\right|^{2}\right].$

Applying (A.8), one can calculate
${\mathbb{E}}_{\omega}\left[\left|\mathbf{S}_{m}^{(N)}\right|^{2}\right]$ to
get the same upper bound uniformly in $N$ (up to a multiplicative constant
$C^{m}$) as $\|\hat{\mathfrak{g}}_{m}(\cdot;t,x)\|_{\mathscr{H}^{\otimes m}}$
(see (2.46) in Proposition 2.14 and see also Remark 2.16). Thus, we have

(3.40)
$\lim_{M\to\infty}\sup_{N\in\mathbb{N}}{\mathbb{E}}\left[\left|\sum_{m=M+1}^{\infty}\frac{1}{m!}{\mathbf{S}}_{m}^{(N)}\right|^{2}\right]=0,$

##  Appendix A Physical Wick product

In order to expand the partition function (1.1) in a proper way to obtain its
weak convergence, we shall invoke the notion of physical Wick product for
general random variables (in contrast to the probabilistic Wick product
defined via Wiener chaos in Section 2.1). The physical Wick product (also
known as Wick power, Wick polynomial or Wick renormalization) was introduced
by Wick [45] in the study of quantum field theory. We collect some facts on
physical Wick products in this subsection. The reader is referred to [2, 3,
16, 17, 31, 43] for more readable account.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a family of real random variables with
finite moments of all orders. The physical Wick product $:\\!X_{1}\cdots
X_{n}\\!:$ is defined recursively as follows. For $n=0$, we set $:~{}:=1$, and
for $n\geq 1$,

$\frac{\partial}{\partial X_{i}}:\\!X_{1}\cdots
X_{n}\\!:=:\\!X_{1}\cdots\hat{X}_{i}\cdots
X_{n}\\!:\,,\quad{\mathbb{E}}\left[:\\!X_{1}\cdots X_{n}\\!:\right]=0,$

where $\hat{X}_{i}$ means the absence of $X_{i}$ in the product. For example,

$\displaystyle:\\!X_{1}\\!:$ $\displaystyle=X_{1}-{\mathbb{E}}[X_{1}];$
$\displaystyle:\\!X_{1}X_{2}\\!:$
$\displaystyle=X_{1}X_{2}-X_{1}{\mathbb{E}}[X_{2}]-X_{2}{\mathbb{E}}[X_{1}]+2{\mathbb{E}}[X_{1}]{\mathbb{E}}[X_{2}]-{\mathbb{E}}[X_{1}X_{2}].$

We remark that different indices may refer to the same random variable. In
this situation, as an example, we also write
$:\\!X^{n}Y^{m}\\!:\overset{\rm{def}}{=}:\\!X_{1}\cdots X_{n}Y_{1}\cdots
Y_{m}\\!:$, if $X_{1}=\cdots=X_{n}=X$ and $Y_{1}=\cdots=Y_{m}=Y$.

If we assume ${\mathbb{E}}\left[e^{\beta\sum_{i=1}^{n}|X_{i}|}\right]<\infty$
for some $\beta>0$, the physical Wick product can be equivalently defined by

(A.1) $\displaystyle:\\!X_{1}\cdots
X_{n}\\!:\overset{\rm{def}}{=}\frac{\partial^{n}}{\partial z_{1}\cdots\partial
z_{n}}G(z_{1},\cdots,z_{n};X_{1},\cdots,X_{n})\bigg{|}_{z=0}\,,$

where

$G(z_{1},\cdots,z_{n};X_{1},\cdots,X_{n})\overset{\rm{def}}{=}\frac{{\rm
e}^{\sum_{i=1}^{n}z_{i}X_{i}}}{{\mathbb{E}}\left[{\rm
e}^{\sum_{i=1}^{n}z_{i}X_{i}}\right]}$

is called the generating function (or Wick exponential). If
$\\{X_{i}\\}_{i\in\mathbb{N}}$ is a centred Gaussian family, the generating
function is simply $G(z,{\bm{X}})={\rm e}^{z\cdot{\bm{X}}-z\cdot Qz/2}$ where
$Q=({\mathbb{E}}[X_{i}X_{j}])_{n\times n}$ is the covariance matrix of
${\bm{X}}=(X_{1},\cdots,X_{n})$, and the resulting Wick products are related
to Hermite polynomials (see (2.8) in Section 2.1). In particular, for a
Gaussian random vector $(X_{1},\dots,X_{n})$, the physical Wick product
$:\\!X_{1}\cdots X_{n}\\!:$ coincides with probabilistic Wick product
$X_{1}\diamond X_{2}\diamond\cdots\diamond X_{n}$ defined in Section 2.1.

We collect some basic properties of physical Wick products. Clearly
$:\\!X_{1}X_{2}\dots X_{n}\\!:$ only involves the random variables
$X_{1},\dots,X_{n}$ and their joint moments up to order $n$, and it is
symmetric and multilinear in $(X_{1},\dots,X_{n})$ (multilinear means linear
in terms of each $X_{i},i=1,\dots,n$). If two groups of random variables
$\\{X_{1},X_{2},\dots,X_{m}\\}$ and $\\{X_{m+1},\dots,X_{n}\\}$ are
independent of each other, then $:\\!X_{1}\cdots X_{n}\\!:=:\\!X_{1}\cdots
X_{m}\\!::\\!X_{m+1}\cdots X_{n}\\!:$ (see [2, eq. (2.4)]). We remind that the
physical Wick product is no longer associative, which is different from the
ordinary product. For instance, assuming ${\mathbb{E}}[X]=0$, we have
$:\\!XXX\\!:=X^{3}-3X{\mathbb{E}}[X^{2}]-{\mathbb{E}}[X^{3}]$ which is
different from
$:\\!XY\\!:|_{Y=:X^{2}:}=X^{3}-X{\mathbb{E}}[X^{2}]-{\mathbb{E}}[X^{3}]$. Thus
a physical Wick product is a single term whose value is determined by the
definition, and cannot be viewed as a composition of two (or several) physical
Wick products. For instance, all the physical Wick products
$:\\!X^{3}\\!:,:\\!X^{2}X\\!:$ and $:\\!XX^{2}\\!:$ mean the same
$:\\!XXX\\!:$, and in particular $:\\!XX^{2}\\!:\neq:\\!XY\\!:|_{Y=:X^{2}:}$.

For any finite index set $A=\\{i_{1},\dots,i_{n}\\}\subset\mathbb{N}$, we
denote $:\\!X_{A}\\!:\overset{\rm{def}}{=}:\\!X_{i_{1}}\cdots X_{i_{n}}\\!:$
and similarly, we take the notation $X_{A}\overset{\rm{def}}{=}\prod_{i\in
A}X_{i}$ for the ordinary product. We use $X^{A}$ to denote the set
$\\{X_{i},i\in A\\}$ of random variables.

We recall some facts about cumulants. Let $\kappa(X^{A})$ denote the joint
cumulant of $X^{A}=\\{X_{i},i\in A\\}$. Then

(A.2)
$\kappa(X^{A})=\sum_{V}(|V|-1)!(-1)^{|V|-1}\prod_{i=1}^{|V|}{\mathbb{E}}[X_{V_{i}}]$

and

(A.3) ${\mathbb{E}}[X_{A}]=\sum_{V}\prod_{i=1}^{|V|}\kappa(X^{V_{i}}),$

where the sum $\sum_{V}$ is taken over all partitions
$V=\\{V_{1},\dots,V_{k}\\},k\geq 1$ of $A$, and $|V|=k$ is the partition
number. For instance,

$\kappa(X_{1})={\mathbb{E}}[X_{1}],~{}\kappa(X_{1},X_{2})={\mathbb{E}}[X_{1}X_{2}]-{\mathbb{E}}[X_{1}]{\mathbb{E}}[X_{2}],$

and

$\kappa(X_{1},X_{2},X_{3})={\mathbb{E}}[X_{1}X_{2}X_{3}]-{\mathbb{E}}[X_{1}X_{2}]{\mathbb{E}}[X_{3}]-{\mathbb{E}}[X_{1}X_{3}]{\mathbb{E}}[X_{2}]-{\mathbb{E}}[X_{2}X_{3}]{\mathbb{E}}[X_{1}]+2{\mathbb{E}}[X_{1}]{\mathbb{E}}[X_{2}]{\mathbb{E}}[X_{3}].$

If $\\{X_{i}\\}_{i\in\mathbb{N}}$ is Gaussian, we have $\kappa(X^{A})=0$ if
$|A|\geq 3$. If we assume further ${\mathbb{E}}[X_{i}]=0$, the formula (A.3)
reduces to the Wick’s theorem:

(A.4) ${\mathbb{E}}[X_{A}]=\begin{cases}0,&\text{ if $|A|$ is
odd},\vspace{0.2cm}\\\ \displaystyle\sum_{V}\prod_{\\{i,j\\}\in
V}{\mathbb{E}}[X_{i}X_{j}],&\text{ if $|A|$ is even},\end{cases}$

where the summation $\sum_{V}$ is taken over all pair partitions
$V=\\{V_{1},\dots,V_{|A|/2}\\}$ of $A$.

Ordinary products and physical Wick products are connected by the following
formula (see [43, Prop. 1] or [3, Appendix B]),

(A.5) $X_{A}=\sum_{B\subset
A}:\\!X_{B}\\!:\sum_{V}\prod_{i=1}^{|V|}\kappa(X^{V_{i}})=\sum_{B\subset
A}:\\!X_{B}\\!:{\mathbb{E}}[X_{A\backslash B}],$

and

(A.6) $:\\!X_{A}\\!:=\sum_{B\subset
A}X_{B}\sum_{V}(-1)^{|V|}\prod_{i=1}^{|V|}\kappa(X^{V_{i}}),$

where the sum $\sum_{B\subset A}$ is taken over all subsets $B\subset A$
including $B=\emptyset$, and the sum $\sum_{V}$ is over all partitions
$V=\\{V_{1},\dots,V_{k}\\},k\geq 1$ of the set $A\backslash B$. We use the
convention $X_{\emptyset}=:\\!X_{\emptyset}\\!:=\kappa(X_{\emptyset})=1.$

The following formula (see [3, Appendix B] or [18, Lemma 4.5]) will be used

(A.7)
${\mathbb{E}}[:\\!X_{A}\\!::\\!X_{B}\\!:]=\sum_{V}\prod_{i=1}^{|V|}\kappa(X^{V_{i}}),$

where the summation $\sum_{V}$ is taken over all partitions
$V=\\{V_{1},\dots,V_{k}\\},k\geq 1$ of $A\cup B$ satisfying $V_{i}\cap
A\neq\emptyset\neq V_{i}\cap B$ for each $V_{i}$. In particular, if we assume
$\\{X_{i}\\}_{i\in\mathbb{N}}$ is a centered Gaussian family, equations (A.7)
and (A.4) yield

(A.8) ${\mathbb{E}}[:\\!X_{A}\\!::\\!X_{B}\\!:]=\begin{cases}0,&\text{ if
}|A|\neq|B|\vspace{0.2cm}\\\ \displaystyle\sum_{V}\prod_{\\{i,j\\}\in
V}{\mathbb{E}}[X_{i}X_{j}],&\text{ if }|A|=|B|,\end{cases}$

where the summation $\sum_{V}$ is taken over all pair partitions
$V=\\{V_{1},\dots,V_{|A|}\\}$ of $A\cup B$ such that $V_{k}=\\{i,j\\}$ with
$i\in A,j\in B$ for $k=1,\dots,|A|$.

##  Appendix B Some preliminaries on convergence of probability measures

The following result can be found in [7, Theorem 3.2, Chapter 1].

######  Lemma B.1.

Consider random variables $Y_{n}^{(N)},Y^{(N)},Y_{n}$ and $Y$, such that
$Y^{(N)}_{n}\overset{d}{\longrightarrow}Y_{n}$ as $N\to\infty$,
$Y_{n}\overset{d}{\longrightarrow}Y$ as $n\to\infty$, and $Y_{n}^{(N)}$
converges in probability to $Y^{(N)}$ uniformly in $N$ as $n\to\infty$, then
we have $Y^{(N)}\overset{d}{\longrightarrow}Y$ as $N\to\infty$. That is,
assuming

$\textstyle{Y_{n}^{(N)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{N\to\infty}$$\scriptstyle{\mathrm{in\
probability,\ uniformly\ in\
}N}$$\scriptstyle{n\to\infty}$$\textstyle{Y^{(N)}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{n\to\infty}$$\textstyle{Y,}$

we have $Y^{(N)}\overset{d}{\longrightarrow}Y$ as $N\to\infty.$

######  Lemma B.2.

Let $k\geq 1$ be an integer. If
$(X_{1}^{(n)},\dots,X_{k}^{(n)})\overset{d}{\longrightarrow}(X_{1},\dots,X_{k})$
as $n\to\infty$, we have

$f(X_{1}^{(n)},\dots,X_{k}^{(n)})\overset{d}{\longrightarrow}f(X_{1},\dots,X_{k})$

for any continuous function $f$.

######  Lemma B.3.

Let $k\geq 1$ be an integer. Suppose
$(X_{1}^{(n)},\dots,X_{k}^{(n)})\overset{d}{\longrightarrow}(X_{1},\dots,X_{k})$
as $n\to\infty$ and assume that for any subset $A$ of $\\{1,\dots,k\\}$,
$\left\\{\prod_{i\in A}X_{i}^{(n)}\right\\}_{n\in\mathbb{N}}$ is uniformly
integrable. Then $:\\!X_{1}^{(n)}\cdots
X_{k}^{(n)}\\!:\overset{d}{\longrightarrow}:\\!X_{1}\cdots X_{k}\\!:$ as
$n\to\infty$.

Proof Lemma B.2 yields the weak convergence of the ordinary product
$\prod_{i\in A}X_{i}^{(n)}\overset{d}{\longrightarrow}\prod_{i\in A}X_{i}$ for
any $A\subset\\{1,\dots,k\\}$ as $n\to\infty$. By (A.6), it suffices to prove
the convergence of the cumulants

$\lim_{n\to\infty}\kappa(X_{i}^{(n)},i\in A)=\kappa(X_{i},i\in A).$

This follows from the Skorohod representation theorem, equation (A.2), and the
assumption of uniform integrability for the products of
$X_{1}^{(n)},\dots,X_{k}^{(n)}$. ∎

######  Theorem B.4.

[Cramér-Wold Theorem] As $n\to\infty$,
$(X_{1}^{(n)},\dots,X_{k}^{(n)})\overset{d}{\longrightarrow}(X_{1},\dots,X_{k})$
if and only if

$\sum_{i=1}^{k}a_{i}X_{i}^{(n)}\overset{d}{\longrightarrow}\sum_{i=1}^{k}a_{i}X_{i},~{}\text{
for all $(a_{1},\dots,a_{k})\in\mathbb{R}^{k}$.}$

##  Appendix C Miscellaneous results

Let $f:[0,1]^{m}\to\mathbb{R}$ be an integrable function. For a fixed
$N\in\mathbb{N}$, let $t_{i}=i/N$ for $i=0,1,\dots,N$ and denote
$I_{i}=(t_{i-1},t_{i}]$. Define

(B.1)
$f_{N}(t_{1},\dots,t_{m})\overset{\rm{def}}{=}N^{m}\int_{I_{\ell_{1}}\times\cdots\times
I_{\ell_{m}}}f(s_{1},\dots,s_{m}){\mathrm{d}}{\bm{s}},$

if $(t_{1},\dots,t_{m})\in I_{\ell_{1}}\times\dots\times I_{\ell_{m}}$ for
some $\ell_{i}\in\llbracket N\rrbracket.$ That is, $f_{N}$ is the conditional
expectation of $f$ with respect to the the $\sigma$-field generated by the
rectangles of the form $I_{\ell_{1}}\times\dots\times I_{\ell_{m}}$. Then we
have the following inequality of Jensen type.

######  Lemma C.1.

Assume $H\in(1/2,1]$ and suppose $f,g:[0,1]^{m}\to\mathbb{R}$ is an integrable
function. Then, for each $N\in\mathbb{N}$,

(B.2)
$\displaystyle\int_{[0,1]^{2m}}\prod_{i=1}^{m}|s_{i}-t_{i}|^{2H-2}|f_{N}(s_{1},\dots,s_{m})||g_{N}(t_{1},\dots,t_{m})|{\mathrm{d}}{\bm{s}}{\mathrm{d}}{\bm{t}}$
$\displaystyle\leq
C^{m}\int_{[0,1]^{2m}}\prod_{i=1}^{m}|s_{i}-t_{i}|^{2H-2}|f(s_{1},\dots,s_{m})||g(t_{1},\dots,t_{m})|{\mathrm{d}}{\bm{s}}{\mathrm{d}}{\bm{t}},$

for some constant $C$ depending on $H$ only.

Proof We only prove the case $m=1$, and the other cases $m>1$ follows from an
induction argument together with Fubini’s theorem. It suffices to prove (B.2)
with $m=1$ for nonnegative functions, i.e., for $f,g\geq 0$,

(B.3) $\int_{I_{i}\times
I_{j}}|s-t|^{2H-2}f_{N}(s)g_{N}(t){\mathrm{d}}s{\mathrm{d}}t\leq
C\int_{I_{i}\times
I_{j}}|s-t|^{2H-2}f(s)g(t){\mathrm{d}}s{\mathrm{d}}t,~{}1\leq i,j\leq N,$

where recalling $I_{i}=(t_{i-1},t_{i}]$. We shall prove (B.3) for simple
nonnegative functions. The general case can be proved by a limiting argument
and thus is omitted.

For $i=j$, we assume that on $I_{i}$,

$f(s)=\sum_{\ell=1}^{k}a_{\ell}\mathbf{1}_{A_{\ell}}(s),~{}g(t)=\sum_{\ell=1}^{k}b_{\ell}\mathbf{1}_{A_{\ell}}(t),$

where $A_{\ell}=(t_{i-1}+(\ell-1)/Nk,t_{i-1}+\ell/Nk]$ with $\ell=1,\dots,k$
form a uniform partition of the interval $I_{i}$, and $a_{\ell},b_{\ell}$ are
nonnegative numbers. Then, denoting
$\bar{a}=\frac{1}{k}\sum_{\ell=1}^{k}a_{\ell},\bar{b}=\frac{1}{k}\sum_{\ell=1}^{k}b_{\ell}$,

$f_{N}(s)=\bar{a},~{}g_{N}(t)=\bar{b},\text{ on }I_{i}.$

The left-hand side of (B.3) is

(B.4) $\bar{a}\bar{b}\int_{I_{i}\times
I_{i}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t=\sum_{\ell=1}^{k}\sum_{m=1}^{k}a_{\ell}b_{m}\left(\frac{1}{k^{2}}\int_{I_{i}\times
I_{i}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\right),$

and the right-hand side without the constant $C$ is

(B.5) $\sum_{\ell=1}^{k}\sum_{m=1}^{k}a_{\ell}b_{m}\left(\int_{A_{\ell}\times
A_{m}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\right).$

Then, (B.3) follows from (B.4), (B.5) and the following inequality

$\frac{1}{k^{2}}\int_{I_{i}\times
I_{i}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\leq C\min_{\ell,m\in\llbracket
k\rrbracket}\int_{A_{\ell}\times
A_{m}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t,$

where $C$ is a constant depending on $H$ only. Indeed, by change of variables,
the above inequality is equivalent to

$\frac{1}{k^{2}}\int_{0}^{1}\int_{0}^{1}|s-t|^{2H-2}dsdt\leq
C\min_{\ell,m\in\llbracket
k\rrbracket}\int_{(m-1)/k}^{m/k}\int_{(\ell-1)/k}^{\ell/k}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t,$

which holds for
$C=\int_{0}^{1}\int_{0}^{1}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t$, noting
that for all $\ell,m\in\llbracket k\rrbracket$, we have $|s-t|\leq 1$ on
$[(\ell-1)/k,\ell/k]\times[(m-1)/k,m/k]$, and hence
$\int_{(m-1)/k}^{m/k}\int_{(\ell-1)/k}^{\ell/k}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\geq
1/k^{2}$. This proves (B.3) for $i=j$.

Now we prove (B.3) for $i\neq j$. By symmetry, we only need to consider $j>i.$
Let

$f(s)=\sum_{\ell=1}^{k}a_{\ell}\mathbf{1}_{A_{\ell}}(s)\text{ on
}I_{i}=(t_{i-1},t_{i}],$

and

$g(t)=\sum_{\ell=1}^{k}b_{\ell}\mathbf{1}_{B_{\ell}}(t)\text{ on
}I_{j}=(t_{j-1},t_{j}],$

where $a_{\ell},b_{\ell}$ are nonnegative numbers, and
$\\{A_{\ell},\ell=1,\dots,k\\}$ and $\\{B_{\ell},\ell=1,\dots,k\\}$ are
uniform partitions of $I_{i}$ and $I_{j}$, respectively. Then,

$f_{N}(s)=\bar{a}\text{ on }I_{i}\text{ and }g_{N}(t)=\bar{b}\text{ on
}I_{j},$

where $\bar{a}=\frac{1}{k}\sum_{\ell=1}^{k}a_{\ell}$ and
$\bar{b}=\frac{1}{k}\sum_{\ell=1}^{k}b_{\ell}$. The left-hand side of (B.3) is

(B.6) $\bar{a}\bar{b}\int_{I_{i}\times
I_{j}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t=\sum_{\ell=1}^{k}\sum_{m=1}^{k}a_{\ell}b_{m}\left(\frac{1}{k^{2}}\int_{I_{i}\times
I_{j}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\right),$

and the right-hand side of (B.3) without $C$ is

(B.7) $\sum_{\ell=1}^{k}\sum_{m=1}^{k}a_{\ell}b_{m}\left(\int_{A_{\ell}\times
B_{m}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\right).$

To get (B.3), it suffices to show

(B.8) $\frac{1}{k^{2}}\int_{I_{i}\times
I_{j}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\leq C\min_{\ell,m\in\llbracket
k\rrbracket}\int_{A_{\ell}\times B_{m}}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t$

for some constant $C$ depending on $H$ only. Note that by change of variables,
(B.8) is equivalent to the following equality

(B.9)
$\frac{1}{k^{2}}\int_{j-i}^{j-i+1}\int_{0}^{1}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\leq
C\min_{\ell,m\in\llbracket
k\rrbracket}\int_{j-i+(m-1)/k}^{j-i+m/k}\int_{(\ell-1)/k}^{\ell/k}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t$

We shall prove (B.9) for $j=i+1$ and $j\geq i+2$ separately. If $j=i+1$, (B.9)
follows directly by choosing

$C=2^{2-2H}\int_{1}^{2}\int_{0}^{1}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t,$

noting that

$\int_{1+(m-1)/k}^{1+m/k}\int_{(\ell-1)/k}^{\ell/k}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\geq
2^{2H-2}/k^{2},\text{ for all }\ell,m\in\llbracket k\rrbracket.$

When $n\overset{\rm{def}}{=}j-i\geq 2$, to prove (B.9), it suffices to prove
that the biggest integral among the integrals over the regions
$\\{(n+(m-1)/k,n+m/k]\times((\ell-1)/k,\ell/k],\ell\in\llbracket
k\rrbracket\\}$ can be dominated by the smallest one, i.e.,

$\int_{n}^{n+1/k}\int_{1-1/k}^{1}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t\leq
C\int_{n+1-1/k}^{n+1}\int_{0}^{1/k}|s-t|^{2H-2}{\mathrm{d}}s{\mathrm{d}}t$

for some constant $C$ only depending on $H$. This is true because there exists
a finite constant depending on $H$ only such that

$|n-1|^{2H-2}\leq 3^{2-2H}|n+1|^{2H-2}\text{ for all }n\geq 2.$

This proves (B.9), and hence completes the proof of (B.3) for $i\neq j$. ∎

The following Hardy-Littlewood-Sobolev inequality is taken from [4, Lemma B.3
] (see [33] for the one-dimensional version).

######  Lemma C.2.

For $H\in(\frac{1}{2},1]$, the following inequality holds:

$\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{m}}f({\bm{t}})f({\bm{s}})\prod_{i=1}^{m}|t_{i}-s_{i}|^{2H-2}{\mathrm{d}}{\bm{t}}{\mathrm{d}}{\bm{s}}\leq
C_{H}^{n}\left(\int_{\mathbb{R}^{m}}\left|f({\bm{t}})\right|^{1/H}{\mathrm{d}}{\bm{t}}\right)^{2H},$

where $C_{H}>0$ is a constant depending on $H$, and we denote
${\bm{t}}=(t_{1},\dots,t_{n})$ and ${\bm{s}}=(s_{1},\dots,s_{n})$.

The following result can be calculated by a direct calculation.

######  Lemma C.3.

Suppose $\alpha_{i}<1$ for $i=1,\dots,m$ and let
$\alpha=\sum_{i=1}^{m}\alpha_{i}$. Then

$\int_{[0<r_{1}<\dots<r_{n}<r_{m+1}=t]}~{}\prod_{i=1}^{m}(r_{i+1}-r_{i})^{-\alpha_{i}}{\mathrm{d}}{\bm{r}}=\frac{\prod_{i=1}^{m}\Gamma(1-\alpha_{i})}{\Gamma(m-\alpha+1)}\,t^{m-\alpha},$

where $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}{\mathrm{d}}t$ is the Gamma
function.

Acknowledgement The authors would like to thank Rongfeng Sun for his helpful
comments. G. Rang is partially supported by the National Natural Science
Foundation of China (nos. 11971361, 11731012 and 12131019). J. Song is
partially supported by the National Natural Science Foundation of China (nos.
12071256 and 12226001), the Major Basic Research Program of the Natural
Science Foundation of Shandong Province in China (nos. ZR2019ZD42 and
ZR2020ZD24).

Declaration of Conflicting Interest The Authors declare that there is no
conflict of interest.

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