# SMALL BALL PROBABILITIES FOR THE STOCHASTIC HEAT EQUATION WITH COLORED NOISE

Jiaming Chen  Dept. of Mathematics
University of Rochester
Rochester, NY 14627<EMAIL_ADDRESS>
###### Abstract.

We consider the stochastic heat equation on the 1-dimensional torus
$\mathbb{T}:=\left[-1,1\right]$ with periodic boundary conditions:

$\partial_{t}u(t,x)=\partial^{2}_{x}u(t,x)+\sigma(t,x,u)\dot{F}(t,x),\quad
x\in\mathbb{T},t\in\mathbb{R}_{+},$

where $\dot{F}(t,x)$ is a generalized Gaussian noise which is white in time
and colored in space. Assuming that $\sigma$ is Lipschitz in $u$ and uniformly
bounded, we estimate small ball probabilities for the solution $u$ when
$u(0,x)\equiv 0$.

###### Key words and phrases:

Stochastic heat equation; Colored noise; Small ball probabilities.

###### 2020 Mathematics Subject Classification:

Primary, 60H15; Secondary, 60G60.

##  1\. Introduction

In this paper, we estimate small ball probabilities for the solution of the
stochastic heat equation of the type:

(1.1) $\partial_{t}u(t,x)=\partial^{2}_{x}u(t,x)+\sigma(t,x,u)\dot{F}(t,x)$

where $t\in\mathbb{R}_{+}$, $x\in\mathbb{T}:=\left[-1,1\right]$, which denotes
the 1-dimensional torus in the usual way. The centered Gaussian noise
$\dot{F}$ is white in time and homogeneous in space, i.e.,

(1.2)
$\mathbb{E}\left[\dot{F}(t,x)\dot{F}(s,y)\right]=\delta_{0}(t-s)\Lambda(x-y),$

where $\delta_{0}$ is the Dirac delta generalized function and
$\Lambda:\mathbb{T}\to\mathbb{R}_{+}$ is a non-negative generalized function
whose Fourier series is given by

(1.3) $\Lambda(x)=\sum_{n=0}^{\infty}q_{n}\exp(n\pi ix).$

In this article, we focus on an example where $\Lambda(0)$ is infinite while
all other values are finite (Riesz-kernel case). To be more precise, for
$\gamma\in(0,1)$, we have

(1.4) $\Lambda(r)=|r|^{-\gamma},\quad\forall r\in\mathbb{T}.$

[NV09] shows that $\Lambda(r)$ is a bonafide homogeneous spatial covariance
function, that is, $\\{q_{n}\\}_{n=0}^{\infty}$ is a sequence of non-negative
real numbers. Furthermore, there exist constants $C$, $C^{\prime}>0$ such that

(1.5) $Cn^{\gamma-1}\leq q_{n}\leq C^{\prime}n^{\gamma-1},\quad\forall n\geq
1.$

Small ball problems have a long history, and one can see [LS01] for more
surveys. In short, we are interested in the probability that a stochastic
process $X_{t}$ starting at 0 stays in a small ball for a long time period,
i.e.,

$P\left(\sup_{0\leq t\leq T}|X_{t}|<\varepsilon\right)$

where $\varepsilon>0$ is small. A recent paper [AJM21] studied small ball
probabilities for the solution to this type of stochastic heat equation driven
by a space-time white noise and another paper [FJK22] considered small ball
probabilities for the solution to the same stochastic heat equation but under
a Hölder semi-norm. Thanks to the Da Prato-Kwapien-Zabczyk’s factorization, we
may estimate the noise term and control the error in approximation by choosing
a proper time interval. The objective of this paper is to continue this topic
with colored noise.

##  2\. Main Result

We write $u_{0}(x):=u(0,x)$ and assume that $\sigma$ is Lipschitz continuous
in the third variable, i.e., there exists a constant $\mathcal{D}>0$ such that
for all $t\geq 0$, $x\in\mathbb{T}$, $u,v\in\mathbb{R}$,

(2.1) $|\sigma(t,x,u)-\sigma(t,x,v)|\leq\mathcal{D}|u-v|,$

and $\sigma$ is uniformly bounded, i.e., there exist constants
$\mathcal{C}_{1}$, $\mathcal{C}_{2}>0$ such that for all $t\geq 0$,
$x\in\mathbb{T}$, $u\in\mathbb{R}$,

(2.2) $\mathcal{C}_{1}\leq\sigma(t,x,u)\leq\mathcal{C}_{2}.$

Assumption (2.2) assures that $\sigma$ is non-zero, which is helpful for us to
control the variance of noise terms. In fact, (1.1) is not well-posed since
the solution $u$ is not differentiable and $\dot{F}$ exists as a generalized
function. However, the existence and uniqueness of the mild solution $u(t,x)$
to (1.1) under the above assumptions are well known as follow,

(2.3) $\begin{split}u(t,x)&=\int_{\mathbb{T}}G(t,x-y)u_{0}(y)dy\\\ &\hskip
85.35826pt+\int_{0}^{t}\int_{\mathbb{T}}G(t-s,x-y)\sigma(s,y,u(s,y))F(dyds)\end{split}$

where $G:\mathbb{R}_{+}\times\mathbb{T}\to\mathbb{R}$ is the fundamental
solution of the heat equation on $\mathbb{T}$,

$\displaystyle\partial_{t}$ $\displaystyle G(t,x)=\partial_{x}^{2}G(t,x),$
$\displaystyle G(0,x)=\delta_{0}(x).$

See for example [Wal86], [Dal99], [DKM+09], and [TV99] for the proofs and
various other properties. Now we are ready to state the main theorem.

######  Theorem 2.1.

Under the assumptions (2.1), (2.2), and a given parameter $\gamma$ of Riesz
kernel defined in (1.4), if $u(t,x)$ is the solution to (1.1) with
$u_{0}(x)\equiv 0$, then there are positive constants
$\textbf{C}_{0},\textbf{C}_{1},\textbf{C}_{2},\textbf{C}_{3},\mathcal{D}_{0}$
depending only on $\mathcal{C}_{1},\mathcal{C}_{2}$ and $\gamma$, such that
for all $\mathcal{D}<\mathcal{D}_{0},\varepsilon_{0}>\varepsilon>0$, $T>1$, we
have

$\textbf{C}_{0}\exp\left(-\frac{\textbf{C}_{1}T}{\varepsilon^{\frac{2\gamma+4}{\gamma(2-\gamma)}}}\right)\leq
P\left(\sup_{\begin{subarray}{c}0\leq t\leq T\\\
x\in\mathbb{T}\end{subarray}}|u(t,x)|\leq\varepsilon\right)<\textbf{C}_{2}\exp\left(-\frac{\textbf{C}_{3}T}{\varepsilon^{\frac{2\gamma+4}{2-\gamma}}}\right).$

Here we make a couple of remarks. These could be of independent interests.

######  Remark 2.1.

  1. (a)

$\Lambda(x)=\delta_{0}(x)$ and $\dot{F}$ becomes the white noise when
$\gamma=1$. Can we get this case by taking a limit as $\gamma\uparrow 1$?
$\Lambda(x)$ is a constant and $\dot{F}$ becomes a smooth noise when
$\gamma=0$ and $x\neq 0$. Can we also get this case by taking a limit as
$\gamma\downarrow 0$?

  2. (b)

When $\gamma=1$, both lower and upper bounds include a term $\varepsilon^{-6}$
in the exponent. However, in other cases, there exists a discrepancy between
the exponents of $\varepsilon$, due to the fact that $c_{0}$ depends on
$\varepsilon$, as defined in (3.1). It requires an alternative technique to
fill the gap.

  3. (c)

We may extend this result to other colored noise examples, such as Bessel
kernel and Poisson kernel. However, the problem gets more complicated if the
noise becomes fractional in time since we can not use the Markov property,
which plays an essential role in the proof.

Here is the organization of the rest of this chapter. In section 3, we present
Proposition 3.1 and its corresponding relationship to Theorem 2.1. In section
4, we provide some useful estimates. In section 5, we prove Proposition 3.1
and complete the chapter.

Throughout the entire chapter, $C$ and $C^{\prime}$ denote positive constants
whose values may vary from line to line. The dependence of constants on
parameters will be denoted by mentioning the parameters in parentheses.

##  3\. Key Proposition

We decompose $[-1,1]$ into subintervals of length $\varepsilon^{2}$ and divide
$[0,T]$ into intervals of length $c_{0}\varepsilon^{4}$ where $c_{0}$ denotes
the temporal coefficient satisfying

(3.1) $c_{0}=\mathcal{C}_{0}\varepsilon^{\frac{4-4\gamma}{\gamma}},$

where $0<\mathcal{C}_{0}<\mathcal{C}$ and $\mathcal{C}$ is specified in Lemma
5.2.

######  Remark 3.1.

Apart from the space-time white noise case in [AJM21], $c_{0}$ needs to be
selected depending on $\varepsilon$ due to the large correlation between pairs
of points. Indeed, $c_{0}$ does not appear in either the upper bound nor lower
bound for small ball probabilities.

We define $t_{i}=ic_{0}\varepsilon^{4},x_{j}=j\varepsilon^{2}$ where
$i\in\mathbb{N}$ and $j\in\mathbb{Z}$, and

$n_{1}:=\min\left\\{n\in\mathbb{Z}:n\varepsilon^{2}>1\right\\}.$

It is clear that $x_{n_{1}}>1$ and $x_{n_{1}-1}\leq 1$, therefore, by
symmetry, the point $x_{j}$ lies in $\left[-1,1\right]$ if

(3.2) $-n_{1}+1\leq j\leq n_{1}-1.$

For $n\geq 0$, we define the following sequence of events to determine the
upper bound in Theorem 2.1,

(3.3) $F_{n}=\left\\{|u(t_{n},x_{j})|\leq
t_{1}^{\frac{2-\gamma}{4}}\text{~{}for all $-n_{1}+1\leq j\leq
n_{1}-1$}\right\\},$

which is the event that the magnitude of the solution is small (recall that
$t_{1}=c_{0}\varepsilon^{4}$) at time $t_{n}$ for each grid point. In
addition, let $E_{-1}=\Omega$. For $n\geq 0$, we define the following sequence
of events to determine the lower bound in Theorem 2.1,

(3.4)
$E_{n}=\left\\{|u(t_{n+1},x)|\leq\frac{1}{3}\varepsilon^{2-\gamma}~{}\text{and}~{}|u(t,x)|\leq\varepsilon^{2-\gamma}~{}\forall
t\in[t_{n},t_{n+1}],x\in[-1,1]\right\\},$

which is the event that the magnitude of the solution is small at any time in
$[t_{n},t_{n+1}]$ for any points within $[-1,1]$ and even smaller at time
$t_{n+1}$.

######  Proposition 3.1.

Consider the solution to (1.1) with $u_{0}(x)\equiv 0$. Then, there exist
positive constants
$\textbf{C}_{4},\textbf{C}_{5},\textbf{C}_{6},\textbf{C}_{7},\mathcal{D}_{0}$
and $\varepsilon_{1}$ depending solely on $\mathcal{C}_{1}$, $\mathcal{C}_{2}$
and $\gamma$ such that for any $0<\varepsilon<\varepsilon_{1}$ and
$\mathcal{D}<\mathcal{D}_{0}$ in (2.2), we have

  1. (a)

$P\left(F_{n}\bigg{|}\bigcap_{k=0}^{n-1}F_{k}\right)\leq{\bf
C_{4}}\exp\left(-\frac{{\bf C_{5}}}{t_{1}^{\gamma/2}}\right),$

  2. (b)

$P\left(E_{n}\bigg{|}\bigcap_{k=-1}^{n-1}E_{k}\right)\geq{\bf
C_{6}}\exp\left(-\frac{{\bf C_{7}}}{t_{1}^{\gamma/2}}\right).$

We then demonstrate how Theorem 2.1 results from Proposition 3.1.

Proof of Theorem 2.1: The event $F_{n}$ in $\eqref{Fn}$ deals with the
behavior of $u(t,x)$ at time $t_{n}$, thus combining these events together
indicates

$F:=\bigcap_{n=0}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}F_{n}\supset\left\\{|u(t,x)|\leq
t_{1}^{\frac{2-\gamma}{4}},t\in[0,T],x\in\mathbb{T}\right\\},$

and

$P(F)=P\left(\bigcap_{n=0}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}F_{n}\right)=P(F_{0})\prod_{n=1}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}P\left(F_{n}\bigg{|}\bigcap_{k=0}^{n-1}F_{k}\right).$

With $u_{0}(x)\equiv 0$, $F_{0}=\Omega$, and Proposition 3.1(a) immediately
yields

$\displaystyle P(F)$
$\displaystyle\leq\left[\textbf{C}_{4}\exp\left(-\frac{\textbf{C}_{5}}{t_{1}^{\gamma/2}}\right)\right]^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}\leq\textbf{C}_{2}\exp\left(-\frac{\textbf{C}_{3}T}{t_{1}^{1+\gamma/2}}\right).$

Therefore we have

$P\left(\left\\{|u(t,x)|\leq
t_{1}^{\frac{2-\gamma}{4}},t\in[0,T],x\in\mathbb{T}\right\\}\right)<\textbf{C}_{2}\exp\left(-\frac{\textbf{C}_{3}T}{t_{1}^{1+\gamma/2}}\right),$

and replacing $t_{1}^{\frac{2-\gamma}{4}}$ with $\varepsilon_{2}$ gives

(3.5)
$P\left(\left\\{|u(t,x)|\leq\varepsilon_{2},t\in[0,T],x\in\mathbb{T}\right\\}\right)<\textbf{C}_{2}\exp\left(-\frac{\textbf{C}_{3}T}{\varepsilon_{2}^{\frac{4+2\gamma}{2-\gamma}}}\right).$

We derive that, from (3.1),

$\varepsilon_{2}=t_{1}^{\frac{2-\gamma}{4}}=\left(c_{0}\varepsilon^{4}\right)^{\frac{2-\gamma}{4}}=\left(\mathcal{C}_{0}\varepsilon^{\frac{4}{\gamma}}\right)^{\frac{2-\gamma}{4}}<\left(\mathcal{C}_{0}\varepsilon_{1}^{\frac{4}{\gamma}}\right)^{\frac{2-\gamma}{4}},$

so (3.5) provides the upper bound in Theorem 2.1 whenever
$\varepsilon_{0}<\left(\mathcal{C}_{0}\varepsilon_{1}^{\frac{4}{\gamma}}\right)^{\frac{2-\gamma}{4}}$.

On the other hand, the event $E_{n}$ in $\eqref{En}$ deals with the behavior
of $u(t,x)$ at any time between $t_{n}$ and $t_{n+1}$, thus combining these
events together indicates

$E:=\bigcap_{n=-1}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}E_{n}\subset\left\\{|u(t,x)|\leq\varepsilon^{2-\gamma},t\in[0,T],x\in\mathbb{T}\right\\},$

and

$P(E)=P\left(\bigcap_{n=-1}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}E_{n}\right)=P(E_{-1})\prod_{n=0}^{\left\lfloor\frac{T}{t_{1}}\right\rfloor}P\left(E_{n}\bigg{|}\bigcap_{k=-1}^{n-1}E_{k}\right).$

With $u_{0}(x)\equiv 0$ and $E_{-1}=\Omega$, Proposition 3.1(b) immediately
yields

$P(E)\geq\left[\textbf{C}_{6}\exp\left(-\frac{\textbf{C}_{7}}{t_{1}^{\gamma/2}}\right)\right]^{\left\lfloor\frac{T}{t_{1}}\right\rfloor+1}\geq\textbf{C}_{0}\exp\left(-\frac{\textbf{C}_{1}T}{t_{1}^{1+\gamma/2}}\right).$

From the definition of $c_{0}$ in (3.1), we get

$P\left(\left\\{|u(t,x)|\leq\varepsilon^{2-\gamma},t\in[0,T],x\in\mathbb{T}\right\\}\right)>\textbf{C}_{0}\exp\left(-\frac{\textbf{C}_{1}T}{\varepsilon^{\frac{4+2\gamma}{\gamma}}}\right),$

and replacing $\varepsilon^{2-\gamma}$ with $\varepsilon_{3}$ gives

(3.6)
$P\left(\left\\{|u(t,\textbf{x})|\leq\varepsilon_{3},t\in[0,T],\textbf{x}\in\mathbb{T}^{d}\right\\}\right)>\textbf{C}_{0}\exp\left(-\frac{\textbf{C}_{1}T}{\varepsilon_{3}^{\frac{4+2\gamma}{\gamma(2-\gamma)}}}\right).$

It is straightforward to check that

$\varepsilon_{3}=\varepsilon^{2-\gamma}<\varepsilon_{1}^{2-\gamma},$

so (3.6) provides the lower bound in Theorem 2.1 whenever
$\varepsilon_{0}<\varepsilon_{1}^{2-\gamma}$.

With such an
$\varepsilon_{0}<\min\left\\{\mathcal{C}_{0}^{\frac{2-\gamma}{4}}\varepsilon_{1}^{\frac{2-\gamma}{\gamma}},\varepsilon_{1}^{2-\gamma}\right\\}$,
we conclude Theorem 2.1 from Proposition 3.1, and the rest of this chapter is
devoted to the proof of Proposition 3.1.

##  4\. Preliminary

In this section, we provide some preliminary results that are used to prove
Proposition 3.1.

###  4.1. Heat Kernel Estimates

In this subsection, we prove some results related to the heat kernel $G(t,x)$.
To simplify our computation, we write $G_{t}(x):=G(t,x)$ and recall that the
explicit form for the heat kernel on $\mathbb{T}$ is given by

$G(t,x)=\sum_{n\in\mathbb{Z}}(4\pi
t)^{-1/2}\exp\left(-\frac{(x+2n)^{2}}{4t}\right).$

The following statement has been proven for $x\in\mathbb{R}$ in the paper
[SSS02]. Here we prove for the case when $x\in\mathbb{T}$.

######  Lemma 4.1.

There exist positive constants $C(\alpha,\xi)$ and $C(\alpha,\zeta)$ such that
for all $\alpha\in(0,1)$, $\xi\in(0,2\alpha)$, $\zeta\in(0,\alpha)$,
$0<s<t\leq 1$ and $x,y\in\mathbb{T}$, we have

(4.1)
$\int_{0}^{t}\int_{-1}^{1}|G_{t-r}(x-z)-G_{t-r}(y-z)|(t-r)^{\alpha-1}dzdr\leq
C(\alpha,\xi)|x-y|^{\xi},$ (4.2)
$\int_{0}^{s}\int_{-1}^{1}\left|G_{t-r}(x-z)(t-r)^{\alpha-1}-G_{s-r}(x-z)(s-r)^{\alpha-1}\right|dzdr\leq
C(\alpha,\zeta)|t-s|^{\zeta}.$

###### Proof.

For any $\xi\in(0,2\alpha)$, we have

$\displaystyle\int_{-1}^{1}|G_{t-r}(x-z)-G_{t-r}(y-z)|(t-r)^{\alpha-1}dz$
$\displaystyle\leq
2^{1-\xi}(t-r)^{\alpha-1-\frac{\xi}{2}}\left(\int_{-1}^{1}\left|\sum_{n\in\mathbb{Z}}\left[\exp\left(-\frac{(x+2n-z)^{2}}{4(t-r)}\right)\right.\right.\right.$
$\displaystyle\hskip
199.16928pt\left.\left.\left.-\exp\left(-\frac{(y+2n-z)^{2}}{4(t-r)}\right)\right]\right|dz\right)^{\xi}$
$\displaystyle\leq
2^{1-\xi}(t-r)^{\alpha-1-\frac{\xi}{2}}\left(\sum_{n\in\mathbb{Z}}\int_{-1}^{1}\left|\exp\left(-\frac{(x+2n-z)^{2}}{4(t-r)}\right)\right.\right.$
$\displaystyle\hskip
199.16928pt\left.\left.-\exp\left(-\frac{(y+2n-z)^{2}}{4(t-r)}\right)\right|dz\right)^{\xi}$
$\displaystyle=2^{1-\xi}(t-r)^{\alpha-1-\frac{\xi}{2}}\left(\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(y-z)^{2}}{4(t-r)}\right)\right|dz\right)^{\xi}$
$\displaystyle\leq C(\xi)(t-r)^{\alpha-1-\frac{\xi}{2}}|x-y|^{\xi}.$

The first inequality follows that the integral
$\int_{-1}^{1}|G_{t-r}(x-z)-G_{t-r}(y-z)|dz$ is bounded above by $2$. The last
inequality is obtained by applying the Mean Value theorem to the function
$x\to\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)$, so there exists
$x^{\prime}\in(x,y)$ such that

$\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(y-z)^{2}}{4(t-r)}\right)\right|=\frac{1}{2}|x-y|\exp\left(-\frac{(x^{\prime}-z)^{2}}{4(t-r)}\right)\cdot\frac{|x^{\prime}-z|}{(t-r)}.$

Because $x\leq\exp\left(\frac{x^{2}}{2}\right)$ for $\forall x\in\mathbb{R}$,
we have

$\frac{|x^{\prime}-z|}{2(t-r)}=\frac{|x^{\prime}-z|}{2(t-r)^{\frac{1}{2}}}\cdot\frac{1}{(t-r)^{\frac{1}{2}}}\leq\exp\left(\frac{(x^{\prime}-z)^{2}}{8(t-r)}\right)\cdot\frac{1}{(t-r)^{\frac{1}{2}}},$

which implies

$\begin{split}&\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(y-z)^{2}}{4(t-r)}\right)\right|dz\\\
&\hskip
113.81102pt=\int_{\mathbb{R}}|x-y|\exp\left(-\frac{(x^{\prime}-z)^{2}}{4(t-r)}\right)\cdot\frac{|x^{\prime}-z|}{2(t-r)}dz\\\
&\hskip
113.81102pt\leq|x-y|\int_{\mathbb{R}}\exp\left(-\frac{(x^{\prime}-z)^{2}}{8(t-r)}\right)\cdot\frac{1}{(t-r)^{\frac{1}{2}}}dz=C|x-y|.\end{split}$

Therefore, we derive

$\int_{0}^{t}\int_{-1}^{1}|G_{t-r}(x-z)-G_{t-r}(y-z)|(t-r)^{\alpha-1}dzdr\\\
\leq\int_{0}^{t}C(\xi)(t-r)^{\alpha-1-\frac{\xi}{2}}|x-y|^{\xi}dr\leq
C(\alpha,\xi)t^{\alpha-\frac{\xi}{2}}|x-y|^{\xi}\leq
C(\alpha,\xi)|x-y|^{\xi},$

which completes the proof of (4.1).

For (4.2), we decompose the integral into two parts,

(4.3)
$\begin{split}&\int_{0}^{s}\int_{-1}^{1}\left|\sum_{n\in\mathbb{Z}}\left[\exp\left(-\frac{(x+2n-z)^{2}}{4(t-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}\right.\right.\\\
&\hskip
170.71652pt\left.\left.-\exp\left(-\frac{(x+2n-z)^{2}}{4(s-r)}\right)(s-r)^{\alpha-1-\frac{1}{2}}\right]\right|dzdr\\\
&\leq\int_{0}^{s}\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(s-r)^{\alpha-1-\frac{1}{2}}\right|dzdr\\\
&\leq\int_{0}^{s}\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}\right|dzdr\\\
&+\int_{0}^{s}\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(s-r)^{\alpha-1-\frac{1}{2}}\right|dzdr.\end{split}$

For any $\zeta\in(0,\alpha)$, we have

(4.4)
$\begin{split}&\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)\right|\\\
&\hskip
142.26378pt\leq\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)\right|^{\zeta}.\end{split}$

Apply the Mean Value theorem to function
$t\to\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)$, and there exists
$t^{\prime}\in(s,t)$ such that

$\displaystyle\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)\right|=(t-s)\exp\left(-\frac{(x-z)^{2}}{4(t^{\prime}-r)}\right)\cdot\frac{(x-z)^{2}}{4(t^{\prime}-r)^{2}}$
$\displaystyle\hskip
142.26378pt=2(t-s)(t^{\prime}-r)^{-1}\exp\left(-\frac{(x-z)^{2}}{4(t^{\prime}-r)}\right)\cdot\frac{(x-z)^{2}}{8(t^{\prime}-r)}$
$\displaystyle\hskip 142.26378pt\leq
2(t-s)(t^{\prime}-r)^{-1}\exp\left(-\frac{(x-z)^{2}}{8(t^{\prime}-r)}\right).$

The last inequality follows from the property $x\leq\exp(x)$ for $\forall
x\in\mathbb{R}$. Plug the above into the RHS of (4.4) to parametrize the upper
bound in term of $\zeta$, thus we have

$\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)\right|\leq
C(\zeta)(t-s)^{\zeta}(s-r)^{-\zeta}\exp\left(-\zeta\frac{(x-z)^{2}}{8(t-r)}\right).$

As a result,

(4.5)
$\begin{split}&\int_{0}^{s}\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(t-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}\right|dzdr\\\
&\leq
C(\zeta)(t-s)^{\zeta}\int_{0}^{s}(s-r)^{-\zeta}(t-r)^{\alpha-1}\int_{\mathbb{R}}\exp\left(-\zeta\frac{(x-z)^{2}}{8(t-r)}\right)(t-r)^{-\frac{1}{2}}dzdr\\\
&\leq C(\zeta)(t-s)^{\zeta}\int_{0}^{s}(s-r)^{\alpha-1-\zeta}dr\leq
C_{\alpha,\zeta}s^{\alpha-\zeta}(t-s)^{\zeta}\leq
C_{\alpha,\zeta}(t-s)^{\zeta}.\end{split}$

Similarly, for any $\zeta\in(0,\alpha),r<s\leq t$, we have

$\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|^{1-\zeta}\leq
2(s-r)^{(\alpha-1-\frac{1}{2})(1-\zeta)}.$

If we multiply
$\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|^{\zeta}$
to both sides, we obtain

(4.6)
$\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|\leq
2(s-r)^{(\alpha-1-\frac{1}{2})(1-\zeta)}\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|^{\zeta}.$

Applying the Mean Value theorem to the function
$t\to(t-r)^{\alpha-1-\frac{1}{2}}$ yields

$\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|=C(\alpha)(t-s)(t^{\prime}-r)^{\alpha-2-\frac{1}{2}}$

for some $t^{\prime}\in(s,t)$. Plug the above into the RHS of (4.6) to
parametrize the upper bound in term of $\zeta$, thus we have

$\displaystyle\left|(t-r)^{\alpha-1-\frac{1}{2}}-(s-r)^{\alpha-1-\frac{1}{2}}\right|$
$\displaystyle\leq
C(\alpha,\zeta)(s-r)^{(\alpha-1-\frac{1}{2})(1-\zeta)+(\alpha-2-\frac{1}{2})\zeta}(t-s)^{\zeta}$
$\displaystyle=C(\alpha,\zeta)(s-r)^{(\alpha-1-\frac{1}{2}-\zeta)}(t-s)^{\zeta}.$

Consequently,

(4.7)
$\begin{split}&\int_{0}^{s}\int_{\mathbb{R}}\left|\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(t-r)^{\alpha-1-\frac{1}{2}}-\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)(s-r)^{\alpha-1-\frac{1}{2}}\right|dzdr\\\
&\leq
C(\alpha,\zeta)(t-s)^{\zeta}\int_{0}^{s}(s-r)^{(\alpha-1-\zeta)}\int_{\mathbb{R}}(s-r)^{-\frac{1}{2}}\exp\left(-\frac{(x-z)^{2}}{4(s-r)}\right)dzdr\\\
&\leq C(\alpha,\zeta)s^{\alpha-\zeta}(t-s)^{\zeta}\leq
C(\alpha,\zeta)(t-s)^{\zeta}.\end{split}$

Combining (4.3), (4.5) and (4.7) completes the proof of (4.2). ∎

###  4.2. Noise Term Estimates

Define the second integral of (2.3), i.e. noise term, by

(4.8) $N(t,x):=\int_{0}^{t}\int_{-1}^{1}G(t-s,x-y)\sigma(s,y,u(s,y))F(dyds).$

When $\sigma\equiv 1$, we may express $N(t,x)$ in terms of random Fourier
series decomposition as

(4.9) $\begin{split}&N(t,x)=\sum_{n=0}^{\infty}\sqrt{q_{n}}\left(\cos(\pi
nx)\int_{0}^{t}e^{-\pi^{2}n^{2}(t-s)}\beta_{n}(ds)\right.\\\ &\hskip
142.26378pt\left.+\sin(\pi
nx)\int_{0}^{t}e^{-\pi^{2}n^{2}(t-s)}\tilde{\beta}_{n}(ds)\right),\end{split}$

where the sequence $\\{\beta_{n}\\}_{n=0}^{\infty}$ and
$\\{\tilde{\beta}_{n}\\}_{n=0}^{\infty}$ are independent and each formed of
independent 1-dimensional real-valued $(\mathcal{F}_{t})$-adapted Brownian
motions. [TTV03] Corollary 1 shows that the necessary and sufficient condition
for existence of (4.9) in $L^{2}(\Omega\times[0,T]\times[-1,1])$ is

$\sum_{n=1}^{\infty}\frac{q_{n}}{n^{2}}<\infty.$

We could apply a factorization method (see [DPZ14] Theorem 5.10) to $N(t,x)$
under assumptions (2.1) and (2.2) of $\sigma$. By the semigroup property of
$G$ and a stochastic Fubini’s theorem (see [Wal86] Theorem 2.6), for all
$\alpha\in(0,1)$, we have

$N(t,x)=\frac{\sin(\pi\alpha)}{\pi}\int_{0}^{t}\int_{-1}^{1}G_{t-r}(x-z)(t-r)^{\alpha-1}Y_{\alpha}(r,z)dzdr,$

where

$Y_{\alpha}(r,z)=\int_{0}^{r}\int_{-1}^{1}G_{r-s}(z-y)\sigma(s,y,u(s,y))(r-s)^{-\alpha}F(dyds).$

Thanks to this factorization, we could estimate the noise term. Before that,
we prove the following lemma which is similar to [SSS02] Lemma 2.2.

######  Lemma 4.2.

For any $T\geq 0$, $\alpha\in\left(0,\frac{2-\gamma}{4}\right)$,

$\sup_{0\leq r\leq
T}\sup_{z\in\mathbb{T}}\mathbb{E}\left[Y_{\alpha}^{2}(r,z)\right]<+\infty$

###### Proof.

By assumption (2.2) of $\sigma$, we have

$\displaystyle\mathbb{E}\left[Y_{\alpha}^{2}(r,z)\right]=\mathbb{E}\left[\left(\int_{0}^{r}\int_{-1}^{1}G_{r-s}(z-y)\sigma(s,y,u(s,y))(r-s)^{-\alpha}F(dyds)\right)^{2}\right]$
$\displaystyle\leq\sup_{s,y}\mathbb{E}\left[\sigma^{2}(s,y,u(s,y))\right]\cdot\mathbb{E}\left[\left(\int_{0}^{r}\int_{-1}^{1}G_{r-s}(z-y)(r-s)^{-\alpha}F(dyds)\right)^{2}\right]$
$\displaystyle\leq\mathcal{C}_{2}^{2}\mathbb{E}\left[\left(\sum_{n=0}^{\infty}\sqrt{q_{n}}\left(\cos(n\pi
z)\int_{0}^{r}(r-s)^{-\alpha}e^{-\pi^{2}n^{2}(r-s)}\beta_{n}(ds)\right.\right.\right.$
$\displaystyle\hskip 142.26378pt\left.\left.\left.+\sin(n\pi
z)\int_{0}^{r}(r-s)^{-\alpha}e^{-\pi^{2}n^{2}(r-s)}\tilde{\beta}_{n}(ds)\right)\right)^{2}\right]$
$\displaystyle=\mathcal{C}_{2}^{2}\sum_{n=0}^{\infty}q_{n}\left(\sin^{2}(n\pi
z)+\cos^{2}(n\pi
z)\right)\int_{0}^{r}(r-s)^{-2\alpha}e^{-2\pi^{2}n^{2}(r-s)}ds$
$\displaystyle=\mathcal{C}_{2}^{2}\sum_{n=0}^{\infty}q_{n}\int_{0}^{r}(r-s)^{-2\alpha}e^{-2\pi^{2}n^{2}(r-s)}ds$
$\displaystyle\leq
C\mathcal{C}_{2}^{2}\left[q_{0}+\sum_{n=1}^{\infty}q_{n}n^{4\alpha-2}\Gamma(1-2\alpha)\right].$

The second inequality follows the Fourier series decomposition of a spatial
convolution, and we substitute $2\pi^{2}n^{2}(r-s)$ in the last inequality to
get an Euler gamma function. Recall $q_{n}\leq Cn^{\gamma-1}$ in (1.5), and
the summation converges so that the lemma follows. ∎

We will now estimate the regularity of $N(t,x)$ in space,

$\displaystyle\mathbb{E}[(N(t,x)-N(t,y))^{2}]$
$\displaystyle=C(\alpha)\mathbb{E}\left[\left(\int_{0}^{t}\int_{-1}^{1}[G_{t-r}(x-z)-G_{t-r}(y-z)](t-r)^{\alpha-1}Y_{\alpha}(r,z)dzdr\right)^{2}\right]$
$\displaystyle\leq
C(\alpha)\mathbb{E}\left[\left(\int_{0}^{t}\int_{-1}^{1}\left|[G_{t-r}(x-z)-G_{t-r}(y-z)](t-r)^{\alpha-1}Y_{\alpha}(r,z)\right|dzdr\right)^{2}\right]$
$\displaystyle\leq
C(\alpha)\sup_{r,z}\mathbb{E}\left[Y_{\alpha}^{2}(r,z)\right]\left(\int_{0}^{t}\int_{-1}^{1}\left|[G_{t-r}(x-z)-G_{t-r}(y-z)](t-r)^{\alpha-1}\right|dzdr\right)^{2}.$

By Lemma 4.1 and Lemma 4.2, we conclude that there exists a constant $C>0$
depending only on $\alpha,\gamma$ and $\xi\in(0,2\alpha)$ such that

(4.10) $\mathbb{E}[(N(t,x)-N(t,y))^{2}]\leq C\mathcal{C}_{2}^{2}|x-y|^{2\xi}.$

For regularity in time,

$\displaystyle\mathbb{E}[(N(t,x)-N(s,x))^{2}]$
$\displaystyle=C(\alpha)\mathbb{E}\left[\left(\int_{0}^{s}\int_{-1}^{1}\left[G_{t-r}(x-z)(t-r)^{\alpha-1}-G_{s-r}(x-z)(s-r)^{\alpha-1}\right]Y_{\alpha}(r,z)dzdr\right.\right.$
$\displaystyle\hskip
142.26378pt\left.\left.+\int_{s}^{t}\int_{-1}^{1}G_{t-r}(x-z)(t-r)^{\alpha-1}Y_{\alpha}(r,z)dzdr\right)^{2}\right]$
$\displaystyle=C(\alpha)\left(\mathbb{E}\left[\left(\int_{0}^{s}\int_{-1}^{1}\left[G_{t-r}(x-z)(t-r)^{\alpha-1}-G_{s-r}(x-z)(s-r)^{\alpha-1}\right]Y_{\alpha}(r,z)dzdr\right)^{2}\right]\right.$
$\displaystyle\hskip
142.26378pt\left.+\mathbb{E}\left[\left(\int_{s}^{t}\int_{-1}^{1}G_{t-r}(x-z)(t-r)^{\alpha-1}Y_{\alpha}(r,z)dzdr\right)^{2}\right]\right)$
$\displaystyle\leq
C(\alpha)\sup_{r,z}\mathbb{E}\left[Y_{\alpha}^{2}(r,z)\right]\left(I_{1}^{2}+I_{2}^{2}\right)$

where

$I_{1}=\int_{0}^{s}\int_{-1}^{1}\left|G_{t-r}(x-z)(t-r)^{\alpha-1}-G_{s-r}(x-z)(s-r)^{\alpha-1}\right|dzdr,$

and

(4.11)
$I_{2}=\int_{s}^{t}\int_{-1}^{1}\left|G_{t-r}(x-z)(t-r)^{\alpha-1}\right|dzdr=C_{\alpha}(t-s)^{\alpha}<C_{\alpha}(t-s)^{\zeta}.$

By Lemma 4.1, we derive

(4.12) $I_{1}\leq C_{\alpha,\zeta}(t-s)^{\zeta}.$

Therefore, we conclude that there exists a constant $C>0$ depending only on
$\alpha,\gamma$ and $\zeta\in(0,\alpha)$ such that

$\mathbb{E}[(N(t,x)-N(s,x))^{2}]\leq C\mathcal{C}_{2}^{2}(t-s)^{2\zeta}.$

######  Lemma 4.3.

There exist constants $C_{1},C_{2}>0$ depending only on $\alpha,\gamma$ and
$\xi$, and $C_{3},C_{4}>0$ depending only on $\alpha,\gamma$ and $\zeta$ such
that for all $0\leq s<t\leq 1$, $x,y\in\mathbb{T}$,
$\xi\in(0,2\alpha),\zeta\in(0,\alpha)$, and $\lambda>0$,

(4.13) $P(|N(t,x)-N(t,y)|>\lambda)\leq
C_{1}\exp\left(-\frac{C_{2}\lambda^{2}}{\mathcal{C}_{2}^{2}|x-y|^{2\xi}}\right),$
(4.14) $P(|N(t,x)-N(s,x)|>\lambda)\leq
C_{3}\exp\left(-\frac{C_{4}\lambda^{2}}{\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}\right).$

###### Proof.

Define

$N_{t}(s,x):=\int_{0}^{s}\int_{-1}^{1}G_{t-r}(x-y)\sigma(r,y,u(r,y))F(dydr).$

Note that $N_{t}(t,x)=N(t,x)$ and $N_{t}(s,x)$ is a continuous
$\mathcal{F}_{s}^{F}$-adapted martingale in $s$ since the integrand does not
depend on $s$. Consider

$M_{s}:=N_{t}(s,x)-N_{t}(s,y)=\int_{0}^{s}\int_{-1}^{1}[G_{t-r}(x-z)-G_{t-r}(y-z)]\sigma(r,z,u(r,z))F(dzdr),$

so that $M_{t}=N(t,x)-N(t,y)$. As $M_{s}$ is a continuous
$\mathcal{F}_{s}^{F}$-adapted martingale with $M_{0}=0$, it is a time changed
Brownian motion. In particular, we have

$M_{t}=B_{\langle M\rangle_{t}},$

and (4.10) gives an upper bound on the time change as

$\langle M\rangle_{t}\leq C\mathcal{C}_{2}^{2}|x-y|^{2\xi}.$

Therefore, by the reflection principle for the Brownian motion $B_{\langle
M\rangle_{t}}$,

$\displaystyle P(N(t,x)-N(t,y)>\lambda)$
$\displaystyle=P(M_{t}>\lambda)=P(B_{\langle M\rangle_{t}}>\lambda)$
$\displaystyle\leq P\left(\sup_{s\leq
C\mathcal{C}_{2}^{2}|x-y|^{2\xi}}B_{s}>\lambda\right)=2P\left(B_{C\mathcal{C}_{2}^{2}|x-y|^{2\xi}}>\lambda\right)$
$\displaystyle\leq
C_{1}\exp\left(-\frac{C_{2}\lambda^{2}}{\mathcal{C}_{2}^{2}|x-y|^{2\xi}}\right).$

Switching $x$ and $y$ gives

$\displaystyle P(-N(t,x)+N(t,y)>\lambda)$ $\displaystyle=P(M_{t}<-\lambda)\leq
2P\left(B_{C\mathcal{C}_{2}^{2}|x-y|^{2\xi}}<-\lambda\right)$
$\displaystyle\leq
C_{1}\exp\left(-\frac{C_{2}\lambda^{2}}{\mathcal{C}_{2}^{2}|x-y|^{2\xi}}\right).$

Consequently, for $\forall\xi\in(0,2\alpha)$,

$P(|N(t,x)-N(t,y)|>\lambda)\leq
C_{1}\exp\left(-\frac{C_{2}\lambda^{2}}{\mathcal{C}_{2}^{2}|x-y|^{2\xi}}\right),$

which completes the proof of (4.13).

Define

$R_{q_{1}}=\int_{0}^{q_{1}}\int_{-1}^{1}[G_{t-r}(x-y)-G_{s-r}(x-y)]\sigma(r,y,u(r,y))F(dydr)$

where $0\leq q_{1}\leq s$. Note $R_{q_{1}}$ is a continuous
$\mathcal{F}_{q_{1}}^{F}$-adapted martingale with $R_{0}=0$. Also define

$S_{q_{2}}=\int_{s}^{s+{q_{2}}}\int_{-1}^{1}G_{t-r}(x-y)\sigma(r,y,u(r,y))F(dydr)$

where $0\leq q_{2}\leq t-s$. Note $S_{q_{2}}$ is a continuous
$\mathcal{F}_{q_{2}}^{F}$-adapted martingale with $S_{0}=0$. Thus, both
$R_{q_{1}}$ and $S_{q_{2}}$ are time changed Brownian motions, i.e.,

$R_{s}=B_{\langle R\rangle_{s}}\quad\text{and}\quad
S_{t-s}=B^{\prime}_{\langle S\rangle_{t-s}}$

where $B$, $B^{\prime}$ are two different Brownian motions. Note that
$N(t,x)-N(s,x)=R_{s}+S_{t-s}$, then

$P(N(t,x)-N(s,x)>\lambda)\leq P(R_{s}>\lambda/2)+P(S_{t-s}>\lambda/2).$

(4.11) and (4.12) give an upper bound on the time changes as

$\langle R\rangle_{s}\leq
C\mathcal{C}_{2}^{2}(t-s)^{2\zeta}\quad\text{and}\quad\langle
S\rangle_{t-s}\leq C\mathcal{C}_{2}^{2}(t-s)^{2\zeta}.$

By the reflection principle for the Brownian motions $B_{\langle
R\rangle_{s}}$ and $B^{\prime}_{\langle S\rangle_{t-s}}$,

$\displaystyle P(N(t,x)-N(s,x)>\lambda)$ $\displaystyle\leq P\left(B_{\langle
R\rangle_{s}}>\lambda/2\right)+P\left(B^{\prime}_{\langle
S\rangle_{t-s}}>\lambda/2\right)$ $\displaystyle\leq 2P\left(\sup_{r\leq
C\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}B_{r}>\frac{\lambda}{2}\right)=4P\left(B_{C\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}>\frac{\lambda}{2}\right)$
$\displaystyle\leq
C_{3}\exp\left(-\frac{C_{4}\lambda^{2}}{\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}\right).$

In addition,

$\displaystyle P(-N(t,x)+N(s,x)>\lambda)$ $\displaystyle\leq
P(R_{s}<-\lambda/2)+P(S_{t-s}<-\lambda/2)$ $\displaystyle\leq
4P\left(B_{C\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}<-\frac{\lambda}{2}\right)$
$\displaystyle\leq
C_{3}\exp\left(-\frac{C_{4}\lambda^{2}}{\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}\right).$

Consequently, for $\forall\zeta\in(0,\alpha)$,

$P(|N(t,x)-N(s,x)|>\lambda)\leq
C_{3}\exp\left(-\frac{C_{4}\lambda^{2}}{\mathcal{C}_{2}^{2}|t-s|^{2\zeta}}\right),$

which completes the proof of (4.14). ∎

######  Definition 4.1.

Given a grid

$\mathbb{G}_{n}=\left\\{\left(\frac{j}{2^{2n}},\frac{k}{2^{n}}\right):0\leq
j\leq 2^{2n},0\leq k\leq 2^{n},j,k\in\mathbb{Z}\right\\},$

and we write

$\left(t_{j}^{(n)},x_{k}^{(n)}\right)=\left(\frac{j}{2^{2n}},\frac{k}{2^{n}}\right).$

Two points $\left(t_{j_{i}}^{(n)},x_{k_{i}}^{(n)}\right):i=1,2$ are called
nearest neighbors if either

$j_{1}=j_{2}\quad\text{and}\quad|k_{1}-k_{2}|=1\quad\textbf{or}\quad|j_{1}-j_{2}|=1\quad\text{and}\quad
k_{1}=k_{2}.$

The following lemma is analogous to Lemma 3.4 in [AJM21] and Lemma 2.5 in
[FJK22], which is crucial for estimating small ball probabilities.

######  Lemma 4.4.

There exist constants $C_{5},C_{6}>0$ depending on $\gamma$ such that for all
$\beta,\lambda,\varepsilon>0$ and $\beta\varepsilon^{4}\leq 1$, we have

$P\left(\sup_{\begin{subarray}{c}0\leq t\leq\beta\varepsilon^{4}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\lambda\varepsilon^{2-\gamma}\right)\leq\frac{C_{5}}{1\wedge\sqrt{\beta}}\exp\left(-\frac{C_{6}\lambda^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

###### Proof.

Fix $\beta\geq 1$, and consider the grid

$\mathbb{G}_{n}=\left\\{\left(\frac{j}{2^{2n}},\frac{k}{2^{n}}\right):0\leq
j\leq\beta\varepsilon^{4}2^{2n},0\leq
k\leq\varepsilon^{2}2^{n},j,k\in\mathbb{Z}\right\\}.$

Let

(4.15)
$n_{0}=\left\lceil\log_{2}\left(\beta^{-1/2}\varepsilon^{-2}\right)\right\rceil,$

and for $n<n_{0}$, $\mathbb{G}_{n}$ contains only the origin. For $n\geq
n_{0}$, the grid $\mathbb{G}_{n}$ has at most
$\left(\beta\varepsilon^{4}2^{2n}+1\right)\cdot\left(\varepsilon^{2}2^{n}+1\right)\leq
2^{5}\cdot 2^{3(n-n_{0})}$ many points. We will choose two parameters
$0<\delta_{1}(\gamma)<\delta_{2}(\gamma)<\frac{2-\gamma}{2}$, and define

$\delta:=\delta_{2}-\delta_{1},$

which satisfies the following constraint

(4.16) $4\zeta\wedge 2\xi=2-\gamma-2\delta,$

where $\xi\in\left(0,2\alpha\right)$ and $\zeta\in\left(0,\alpha\right)$.

We now define another constant

(4.17) $\mathcal{M}=\frac{1-2^{-\delta_{1}}}{4\cdot 2^{\delta n_{0}}},$

and consider the event

$A(n,\lambda)=\left\\{|N(p)-N(q)|\leq\lambda\mathcal{M}\varepsilon^{2-\gamma}2^{-\delta_{1}n}2^{\delta_{2}n_{0}}\text{
for all nearest neighbors $p,q\in\mathbb{G}_{n}$}\right\\}.$

If $p,q\in\mathbb{G}_{n}$ have different spatial variables, (4.13) implies

$P\left(|N(p)-N(q)|>\lambda\mathcal{M}\varepsilon^{2-\gamma}2^{-\delta_{1}n}2^{\delta_{2}n_{0}}\right)\leq
C_{1}\exp\left(-\frac{C_{2}\lambda^{2}\mathcal{M}^{2}\varepsilon^{4-2\gamma}}{2^{-2n\xi}\mathcal{C}_{2}^{2}}2^{-2\delta_{1}n}2^{2\delta_{2}n_{0}}\right).$

If $p,q\in\mathbb{G}_{n}$ have different time variables, (4.14) implies

$P\left(|N(p)-N(q)|>\lambda\mathcal{M}\varepsilon^{2-\gamma}2^{-\delta_{1}n}2^{\delta_{2}n_{0}}\right)\leq
C_{3}\exp\left(-\frac{C_{4}\lambda^{2}\mathcal{M}^{2}\varepsilon^{4-2\gamma}}{2^{-4n\zeta}\mathcal{C}_{2}^{2}}2^{-2\delta_{1}n}2^{2\delta_{2}n_{0}}\right).$

Therefore, a union bound gives

$\displaystyle
P(A^{c}(n,\lambda))\leq\sum_{\begin{subarray}{c}p,q\in\mathbb{G}_{n}\\\
\text{nearest
neighbors}\end{subarray}}P\left(|N(p)-N(q)|>\lambda\mathcal{M}\varepsilon^{2-\gamma}2^{-\delta_{1}n}2^{\delta_{2}n_{0}}\right)$
$\displaystyle\leq
C2^{3(n-n_{0})}\exp\left(-\frac{C^{\prime}\lambda^{2}\mathcal{M}^{2}\varepsilon^{4-2\gamma}}{\mathcal{C}_{2}^{2}}2^{n(4\zeta\wedge
2\xi)}2^{-2\delta_{1}n}2^{2\delta_{2}n_{0}}\right)$
$\displaystyle=C2^{3(n-n_{0})}\exp\left(-\frac{C^{\prime}\lambda^{2}\mathcal{M}^{2}}{\mathcal{C}_{2}^{2}}\left(\varepsilon^{4-2\gamma}2^{n_{0}(2-\gamma)}\right)2^{n(4\zeta\wedge
2\xi)}2^{-2\delta_{1}n}2^{-n_{0}(2-\gamma)}2^{2\delta_{2}n_{0}}\right)$
$\displaystyle\leq
C2^{3(n-n_{0})}\exp\left(-\frac{C^{\prime}\lambda^{2}\mathcal{M}^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}2^{(4\zeta\wedge
2\xi-2\delta_{1})(n-n_{0})}\right),$

where $C,C^{\prime}$ are positive constants depending only on $\gamma$. The
last inequality follows from that
$\varepsilon^{4-2\gamma}2^{n_{0}(2-\gamma)}\geq\beta^{-\frac{(2-\gamma)}{2}}$
by the definition of $n_{0}$ in (4.15), and our choices of
$\delta_{1},\delta_{2}$ and $\delta$ in (4.16).

Let $A(\lambda)=\bigcap_{n\geq n_{0}}A(n,\lambda)$ and we can bound
$P(A^{c}(\lambda))$ by summing $P(A^{c}(n,\lambda))$ for all $n\geq n_{0}$,

$\displaystyle P\left(A^{c}(\lambda)\right)$ $\displaystyle\leq\sum_{n\geq
n_{0}}P\left(A^{c}(n,\lambda)\right)$ $\displaystyle\leq\sum_{n\geq
n_{0}}C2^{3(n-n_{0})}\exp\left(-\frac{C^{\prime}\lambda^{2}\mathcal{M}^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}2^{(4\zeta\wedge
2\xi-2\delta_{1})(n-n_{0})}\right)$ $\displaystyle\leq
C_{5}\exp\left(-\frac{C^{\prime}\lambda^{2}\mathcal{M}^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

From definitions (4.15) and (4.17), we have

$\displaystyle P\left(A^{c}(\lambda)\right)$ $\displaystyle\leq
C_{5}\exp\left(-\frac{C^{\prime}\lambda^{2}2^{-2\delta
n_{0}}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right)$
$\displaystyle\leq
C_{5}\exp\left(-\frac{C_{6}\lambda^{2}(\beta\varepsilon^{4})^{\delta}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right)$

for any $\delta\in\left(0,\frac{2-\gamma}{2}\right)$, we thus obtain

$\displaystyle P\left(A^{c}(\lambda)\right)$
$\displaystyle\leq\inf_{\delta\in\left(0,\frac{2-\gamma}{2}\right)}C_{5}\exp\left(-\frac{C_{6}\lambda^{2}(\beta\varepsilon^{4})^{\delta}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right)$
$\displaystyle\leq
C_{5}\exp\left(-\frac{C_{6}\lambda^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

Now we consider a point $(t,x)$, which is in a grid $\mathbb{G}_{n}$ for some
$n\geq n_{0}$. From arguments similar to page 128 of [DKM+09], we can find a
sequence of points from the origin to $(t,x)$ as
$(0,0)=p_{0},p_{1},...,p_{k}=(t,x)$ such that each pair is the nearest
neighbor in some grid $G_{m},n_{0}\leq m\leq n$, and at most 4 such pairs are
nearest neighbors in any given grid. On the event $A(\lambda)$, we have

$\displaystyle|N(t,x)|$
$\displaystyle\leq\sum_{j=0}^{k-1}|N(p_{j})-N(p_{j+1})|\leq 4\sum_{n\geq
n_{0}}\lambda\mathcal{M}\varepsilon^{2-\gamma}2^{-\delta_{1}n}2^{\delta_{2}n_{0}}$
$\displaystyle\leq\lambda\varepsilon^{2-\gamma}.$

Points in $\bigcup\limits_{n}\mathbb{G}_{n}$ are dense in
$\left[0,\beta\varepsilon^{4}\right]\times\left[0,\varepsilon^{2}\right]$, and
we may extend $N(t,x)$ to a continuous version. Therefore, for $\beta\geq 1$,

$P\left(\sup_{\begin{subarray}{c}0\leq t\leq\beta\varepsilon^{4}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\lambda\varepsilon^{2-\gamma}\right)\leq
C_{5}\exp\left(-\frac{C_{6}\lambda^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

For $0<\beta<1$, we divide the interval $[0,\varepsilon^{2}]$ into
$\frac{1}{\sqrt{\beta}}$ pieces each of length $\sqrt{\beta}\varepsilon^{2}$,
and then a union bound and stationarity in $x$ imply that

$P\left(\sup_{\begin{subarray}{c}0\leq t\leq\beta\varepsilon^{4}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\lambda\varepsilon^{2-\gamma}\right)\leq\frac{1}{\sqrt{\beta}}P\left(\sup_{\begin{subarray}{c}0\leq
t\leq\beta\varepsilon^{4}\\\
x\in\left[0,\sqrt{\beta}\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\lambda\varepsilon^{2-\gamma}\right)$
$=\frac{1}{\sqrt{\beta}}P\left(\sup_{\begin{subarray}{c}0\leq
t\leq(\sqrt{\beta}\varepsilon^{2})^{2}\\\
x\in\left[0,\sqrt{\beta}\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\frac{\lambda}{\beta^{\frac{2-\gamma}{4}}}\left(\beta^{1/4}\varepsilon\right)^{2-\gamma}\right)\leq\frac{C_{5}}{\sqrt{\beta}}\exp\left(-\frac{C_{6}\lambda^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

As a result,

$P\left(\sup_{\begin{subarray}{c}0\leq t\leq\beta\varepsilon^{4}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}|N(t,x)|>\lambda\varepsilon^{2-\gamma}\right)\leq\frac{C_{5}}{1\wedge\sqrt{\beta}}\exp\left(-\frac{C_{6}\lambda^{2}}{\mathcal{C}_{2}^{2}\beta^{\frac{2-\gamma}{2}}}\right).$

∎

######  Remark 4.1.

If we assume that $\sigma$ in (4.8) satisfies $|\sigma(s,y,u)|\leq
C\left(\beta\varepsilon^{4}\right)^{\frac{2-\gamma}{4}}$, then the probability
in Lemma 4.4 is bounded above by

$\frac{C_{5}}{1\wedge\sqrt{\beta}}\exp\left(-\frac{C_{6}\lambda^{2}}{(\beta\varepsilon^{2})^{2-\gamma}}\right),$

which can be proved similarly to the above lemma.

##  5\. Proof of Proposition 3.1

The following lemma gives a lower bound for variance of the noise term
$N(t_{1},x_{k})$ and an upper bound on the decay of covariance between two
random variables $N(t_{1},x_{k}),N(t_{1},x_{k^{\prime}})$ as $|k-k^{\prime}|$
increases.

######  Lemma 5.1.

Consider noise terms $N(t_{1},x_{k}),N(t_{1},x_{k^{\prime}})$ with a
deterministic $\sigma(t,x,u)=\sigma(t,x)$, then there exist constants
$C_{7},C_{8}>0$ depending only on $\mathcal{C}_{1},\mathcal{C}_{2}$ and
$\gamma$, such that

$C_{7}t_{1}^{\frac{2-\gamma}{2}}\leq{\rm Var}(N(t_{1},x_{k}))$

and

${\rm Cov}(N(t_{1},x_{k}),N(t_{1},x_{k^{\prime}}))\leq
C_{8}t_{1}\left|(k-k^{\prime})\varepsilon^{2}\right|^{-\gamma}.$

###### Proof.

We use the Burkholder-Davis-Gundy inequality and the assumption of $\sigma$ in
(2.2) to achieve

$\displaystyle\textrm{Var}(N(t_{1},x_{k}))$
$\displaystyle\geq\int_{0}^{t_{1}}\int_{-1}^{1}\int_{-1}^{1}G_{t_{1}-s}(x_{k}-y)G_{t_{1}-s}(x_{k}-z)\sigma(s,y)\sigma(s,z)\Lambda(y-z)dydzds$
$\displaystyle\geq\mathcal{C}_{1}^{2}\int_{0}^{t_{1}}\sum_{n=0}^{\infty}q_{n}e^{-2\pi^{2}n^{2}(t_{1}-s)}ds$
$\displaystyle=\mathcal{C}_{1}^{2}\int_{0}^{t_{1}}\int_{0}^{\infty}x^{\gamma-1}e^{-2\pi^{2}x^{2}(t_{1}-s)}dxds.$

The last inequality follows from that $q_{n}\geq Cn^{\gamma-1}$ in (1.5) and
the fact that $e^{-2\pi^{2}n^{2}(t_{1}-s)}$ decreases as $n$ increases. Using
Fubini’s theorem and changing variable with $w=2\pi^{2}x^{2}t_{1}$, we
continue as follows,

$\textrm{Var}(N(t_{1},x_{k}))\geq\int_{0}^{\infty}\frac{1-e^{-2\pi^{2}x^{2}t_{1}}}{2\pi^{2}x^{3-\gamma}}dx=Ct_{1}^{\frac{2-\gamma}{2}}\int_{0}^{\infty}\frac{1-e^{-w}}{w^{\frac{4-\gamma}{2}}}dw\geq
Ct_{1}^{\frac{2-\gamma}{2}}\int_{0}^{\infty}\frac{1-e^{-w}}{w^{\frac{4-\gamma}{2}}}dw.$

The last integral converges with $\gamma\in(0,1)$, which completes the proof
of the first part.

In addition, we use the definition of $\Lambda(x)$ and the fact $1-e^{-x}\leq
x$ to derive the upper bound of covariance between $N(t_{1},x_{k})$ and
$N(t_{1},x_{k^{\prime}})$ when $k\neq k^{\prime}$,

$\displaystyle\textrm{Cov}(N(t_{1},x_{k}),N(t_{1},x_{k^{\prime}}))$
$\displaystyle\leq\mathcal{C}_{2}\left(\sum_{n=0}^{\infty}\mathbb{E}\left[\left(\int_{0}^{t_{1}}e^{-\pi^{2}n^{2}(t_{1}-s)}\beta_{n}(ds)\right)^{2}\right]q_{n}\cos\left(n\pi(k-k^{\prime})\varepsilon^{2}\right)\right)$
$\displaystyle\leq
C\left(t_{1}q_{0}+\sum_{n=1}^{\infty}\frac{1-e^{-2\pi^{2}n^{2}t_{1}}}{2\pi^{2}n^{2}}q_{n}\cos\left(n\pi(k-k^{\prime})\varepsilon^{2}\right)\right)$
$\displaystyle\leq
Ct_{1}\sum_{n=0}^{\infty}q_{n}\cos(n\pi(k-k^{\prime})\varepsilon^{2})=Ct_{1}|(k-k^{\prime})\varepsilon^{2}|^{-\gamma}.$

∎

Proof of Proposition 3.1(a) Recall $F_{n}$ in (3.3). The Markov property of
$u(t,\cdot)$ (see page 247 in [DPZ14]) implies

$P\left(F_{j}|\sigma\\{u(t_{i},\cdot)\\}_{0\leq
i<j}\right)=P\left(F_{j}|u(t_{j-1},\cdot)\right).$

If we can show that $P\left(F_{j}|u(t_{j-1},\cdot)\right)$ has a uniform bound
$\textbf{C}_{4}\exp\left(-\frac{\textbf{C}_{5}}{t_{1}^{\gamma/2}}\right)$ that
does not depend on $j$, then the same bound holds for the conditional
probability $P\left(F_{j}\Big{|}\bigcap\limits_{k=0}^{j-1}F_{k}\right)$, which
is conditioned on a realization of $u(t_{k},\cdot),0\leq k<j$. Thus, it is
enough to prove

(5.1)
$P\left(F_{1}\right)\leq\textbf{C}_{4}\exp\left(-\frac{\textbf{C}_{5}}{t_{1}^{\gamma/2}}\right),$

where $\textbf{C}_{4}$, $\textbf{C}_{5}$ do not depend on $u_{0}$.

Consider the truncated function

$f_{\varepsilon}(x)=\begin{cases}x&|x|\leq t_{1}^{\frac{2-\gamma}{4}}\\\
\frac{x}{|x|}\cdot
t_{1}^{\frac{2-\gamma}{4}}&|x|>t_{1}^{\frac{2-\gamma}{4}}\end{cases},$

and, particularly, we have $|f_{\varepsilon}(x)|\leq
t_{1}^{\frac{2-\gamma}{4}}$. Consider the following two equations

$\partial_{t}v(t,x)=\partial_{x}^{2}v(t,x)+\sigma(t,x,f_{\varepsilon}(v(t,x)))\dot{F}(t,x)$

and

(5.2)
$\partial_{t}v_{g}(t,x)=\partial_{x}^{2}v_{g}(t,x)+\sigma(t,x,f_{\varepsilon}(u_{0}(x)))\dot{F}(t,x)$

with the same initial $u_{0}(x)$. We can decompose $v(t,x)$ by

$v(t,x)=v_{g}(t,x)+D(t,x)$

with

$D(t,x)=\int_{0}^{t}\int_{-1}^{1}G(t-s,x-y)[\sigma(s,y,f_{\varepsilon}(v(s,y)))-\sigma(s,y,f_{\varepsilon}(u_{0}(y)))]F(dyds).$

We define a new sequence of events for $n=1$,

$H_{j}=\left\\{|v(p_{1j})|\leq t_{1}^{\frac{2-\gamma}{4}}\right\\}.$

Clearly, the property of $f_{\varepsilon}(x)$ implies

$F_{1}=\bigcap_{j=-n_{1}+1}^{n_{1}-1}H_{j}.$

Also, define another two sequences of events

$A_{j}=\left\\{|v_{g}(p_{1j})|\leq
2t_{1}^{\frac{2-\gamma}{4}}\right\\}\quad\text{and}\quad
B_{j}=\left\\{|D(p_{1j})|>t_{1}^{\frac{2-\gamma}{4}}\right\\}.$

If $|D(p_{1j})|\leq t_{1}^{\frac{2-\gamma}{4}}$ and
$|v_{g}(p_{1j})|>2t_{1}^{\frac{2-\gamma}{4}}$, then
$|v(p_{1j})|>t_{1}^{\frac{2-\gamma}{4}}$. As a result,

$H_{j}^{c}\supset A_{j}^{c}\cap B_{j}^{c},$

which leads to

(5.3)
$\begin{split}P(F_{1})&=P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}H_{j}\right)\leq
P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}[A_{j}\cup B_{j}]\right)\\\ &\leq
P\left(\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)\bigcup\left(\bigcup_{j=-n_{1}+1}^{n_{1}-1}B_{j}\right)\right)\\\
&\leq
P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)+P\left(\bigcup_{j=-n_{1}+1}^{n_{1}-1}B_{j}\right)\\\
&\leq
P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)+\sum_{j=-n_{1}+1}^{n_{1}-1}P(B_{j}),\end{split}$

where the second inequality can be shown by using induction.

The Lipschitz property on the third variable of $\sigma(t,x,u)$ and Remark 4.1
give

$P(B_{j})\leq\frac{C_{5}}{1\wedge\sqrt{c_{0}}}\exp\left(-\frac{1}{4}\mathcal{D}^{-2}C_{6}t_{1}^{-\frac{2-\gamma}{2}}\right),$

so that a bound (3.2) on $j$ yields

(5.4)
$\sum_{j=-n_{1}+1}^{n_{1}-1}P(B_{j})\leq\frac{C_{5}}{(1\wedge\sqrt{c_{0}})\varepsilon^{2}}\exp\left(-\frac{C_{6}}{4\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right).$

Define a sequence of events involving $v_{g}$ to compute the upper bound for
$P\left(\bigcap\limits_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)$,

$I_{j}=\left\\{|v_{g}(p_{1k})|\leq 2t_{1}^{\frac{2-\gamma}{4}}\text{~{}for all
$-n_{1}+1\leq k\leq j$}\right\\},\quad I_{-n_{1}}=\Omega.$

Then we can write $P\left(\bigcap\limits_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)$
in terms of conditional probabilities,

(5.5)
$P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)=P(I_{n_{1}-1})=P(I_{-n_{1}})\prod_{j=-n_{1}+1}^{n_{1}-1}\frac{P(I_{j})}{P(I_{j-1})}=\prod_{j=-n_{1}+1}^{n_{1}-1}P(I_{j}|I_{j-1}).$

Let $\mathcal{G}_{j}$ be the $\sigma-$algebra generated by

$N_{\varepsilon}(t_{1},x_{k}):=\int_{0}^{t_{1}}\int_{-1}^{1}G(t_{1}-s,x_{k}-y)\sigma(s,y,f_{\varepsilon}(u_{0}(y)))F(dyds),-n_{1}+1\leq
k\leq j$

where $N_{\varepsilon}(t_{1},x_{k})$ is the noise term of $v_{g}(t_{1},x_{k})$
in (5.2). If we can show that there is a uniform bound for
$P(I_{j}|\mathcal{G}_{j-1})$, then the bound holds for the conditional
probability $P(I_{j}|I_{j-1})$, which is conditioned on a realization of
$N_{\varepsilon}(t_{1},x_{k}),k=-n_{1}+1,...,j-1$. Notice that
$\sigma(s,y,f_{\varepsilon}(u_{0}(y)))$ is deterministic and $\sigma$ is
uniformly bounded. By Lemma 5.1, we have

(5.6) $\textrm{Var}(N_{\varepsilon}(t_{1},x_{k}))\geq
C_{7}t_{1}^{\frac{2-\gamma}{2}},$

and one can decompose

$v_{g}(t_{1},x_{j})=\int_{-1}^{1}G_{t_{1}}(x-y)u_{0}(y)dy+X+Y$

where $X=\mathbb{E}[N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1}]$ is a
Gaussian random variable, which can be written as

$X=\sum_{k=-n_{1}+1}^{j-1}\eta_{k}^{(j)}N_{\varepsilon}(t_{1},x_{k})$

for some coefficients $\left(\eta_{k}^{(j)}\right)_{k=-n_{1}+1}^{j-1}$. Then
the conditional variance equals

$\displaystyle\textrm{Var}(Y|\mathcal{G}_{j-1})=\mathbb{E}[(N_{\varepsilon}(t_{1},x_{j})-X|\mathcal{G}_{j-1})^{2}]-(\mathbb{E}[N_{\varepsilon}(t_{1},x_{j})-X|\mathcal{G}_{j-1}])^{2}$
$\displaystyle=\mathbb{E}[(N_{\varepsilon}(t_{1},x_{j})-\mathbb{E}[N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1}]|\mathcal{G}_{j-1})^{2}]=\textrm{Var}(N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1}).$

Since $Y=N_{\varepsilon}(t_{1},x_{j})-X$ is independent of
$\mathcal{G}_{j-1}$, we write $\textrm{Var}(Y)$ as

$\textrm{Var}(Y)=\textrm{Var}(Y|\mathcal{G}_{j-1})=\textrm{Var}(N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1}).$

In fact, for a Gaussian random variable $Z\sim N(\mu,\sigma^{2})$ and any
$a>0$, the probability $P(|Z|\leq a)$ is maximized when $\mu=0$, thus

$P(I_{j}|\mathcal{G}_{j-1})\leq P\left(|v_{g}(t_{1},x_{j})|\leq
2t_{1}^{\frac{2-\gamma}{4}}\bigg{|}\mathcal{G}_{j-1}\right)\leq
P\left(|Z^{\prime}|\leq\frac{2t_{1}^{\frac{2-\gamma}{4}}}{\sqrt{\textrm{Var}(N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1})}}\right)$

where $Z^{\prime}\sim N(0,1)$. Let’s use the notation SD to denote the
standard deviation of a random variable. By the Minkowski inequality,

$\textrm{SD}(X)\leq\sum_{k=-n_{1}+1}^{j-1}\left|\eta_{k}^{(j)}\right|\cdot\textrm{SD}(N_{\varepsilon}(t_{1},x_{k})),$

and

$\textrm{SD}(N_{\varepsilon}(t_{1},x_{j}))\leq\textrm{SD}(X)+\textrm{SD}(Y).$

If we can control coefficients by restricting

$\sum_{k=-n_{1}+1}^{j-1}\left|\eta_{k}^{(j)}\right|\leq\frac{1}{2},$

then the standard deviation of $X$ is less than one half the standard
deviation of $N_{\varepsilon}(t_{1},x_{j})$,

$\textrm{SD}(N_{\varepsilon}(t_{1},x_{j}))\leq\textrm{SD}(X)+\textrm{SD}(Y)\leq\frac{1}{2}\textrm{SD}(N_{\varepsilon}(t_{1},x_{j}))+\textrm{SD}(Y).$

From (5.6), $\textrm{Var}(Y)$ is bounded below by
$Ct_{1}^{\frac{2-\gamma}{2}}$, so that we can derive the uniform upper bound
of $P(I_{j}|\mathcal{G}_{j-1})$,

$\begin{split}P(I_{j}|\mathcal{G}_{j-1})&\leq
P\left(|Z^{\prime}|\leq\frac{2t_{1}^{\frac{2-\gamma}{4}}}{\sqrt{\textrm{Var}(N_{\varepsilon}(t_{1},x_{j})|\mathcal{G}_{j-1})}}\right)\\\
&\leq
P\left(|Z^{\prime}|\leq\frac{2t_{1}^{\frac{2-\gamma}{4}}}{\sqrt{Ct_{1}^{\frac{2-\gamma}{2}}}}\right)\\\
&=P(|Z^{\prime}|\leq C^{\prime}).\end{split}$

A bound (3.2) on $j$ and (5.5) together yield

(5.7) $P\left(\bigcap_{j=-n_{1}+1}^{n_{1}-1}A_{j}\right)\leq
C^{\varepsilon^{-2}}=C\exp\left(-\frac{C^{\prime}}{\varepsilon^{2}}\right).$

The following lemma shows how to select $c_{0}$ to make
$\sum\limits_{k=-n_{1}+1}^{j-1}\left|\eta_{k}^{(j)}\right|\leq\frac{1}{2}$,
which completes the proof.

######  Lemma 5.2.

For a given $\varepsilon>0$, we may choose $c_{0}>0$ in (3.1) such that

$\sum_{k=-n_{1}+1}^{j-1}\left|\eta_{k}^{(j)}\right|\leq\frac{1}{2}.$

###### Proof.

Since $Y$ and $\mathcal{G}_{j-1}$ are independent,

$\textrm{Cov}(Y,N_{\varepsilon}(t_{1},x_{k}))=0\text{~{}for
$k=-n_{1}+1,...,j-1,$}$

and for $l=-n_{1}+1,...,j-1$, we have

$\textrm{Cov}(N_{\varepsilon}(t_{1},x_{j}),N_{\varepsilon}(t_{1},x_{l}))=\textrm{Cov}(X,N_{\varepsilon}(t_{1},x_{l}))=\sum_{k=-n_{1}+1}^{j-1}\eta^{(j)}_{k}\textrm{Cov}(N_{\varepsilon}(t_{1},x_{k}),N_{\varepsilon}(t_{1},x_{l})).$

Consider the vector
${\bf\eta}=\left(\eta^{(j)}_{-n_{1}+1},...,\eta^{(j)}_{j-1}\right)^{T}$ and
vector

${\bf
x}=(\textrm{Cov}(N_{\varepsilon}(t_{1},x_{j}),N_{\varepsilon}(t_{1},x_{-n_{1}+1})),...,\textrm{Cov}(N_{\varepsilon}(t_{1},x_{j}),N_{\varepsilon}(t_{1},x_{j-1})))^{T},$

and let
${\bf\Sigma}=[\textrm{Cov}\left(N_{\varepsilon}(t_{1},x_{k}),N_{\varepsilon}(t_{1},x_{l})\right)]_{-n_{1}+1\leq
k,l\leq j-1}$ be the covariance matrix of $N_{\varepsilon}(t_{1},\cdot)$, then
we get

${\bf\eta}={\bf\Sigma}^{-1}{\bf x}.$

Let $||\cdot||_{1,1}$ be the operator norm on matrices, then

$||{\bf\eta}||_{l^{1}}=||{\bf\Sigma}^{-1}{\bf
x}||_{l^{1}}\leq||{\bf\Sigma}^{-1}||_{1,1}||{\bf x}||_{l^{1}}.$

We rewrite ${\bf\Sigma}={\bf D}{\bf T}{\bf D}$, where ${\bf D}$ is a diagonal
matrix with diagonal entries
$\sqrt{\textrm{Var}(N_{\varepsilon}(t_{1},x_{k}))}$, and ${\bf T}$ is the
correlation matrix with entries

$t_{kl}=\frac{\textrm{Cov}(N_{\varepsilon}(t_{1},x_{k}),N_{\varepsilon}(t_{1},x_{l}))}{\sqrt{\textrm{Var}(N_{\varepsilon}(t_{1},x_{k}))}\sqrt{\textrm{Var}(N_{\varepsilon}(t_{1},x_{l}))}}.$

Thanks to Lemma 5.1, for $k\neq l$, $t_{kl}$ can be bounded above by

$|t_{kl}|\leq\frac{C_{7}c_{0}\varepsilon^{4}\left|(k-l)\varepsilon^{2}\right|^{-\gamma}}{C_{8}(c_{0}\varepsilon^{4})^{\frac{2-\gamma}{2}}}=\frac{C_{7}}{C_{8}}\cdot\left(\frac{\sqrt{c_{0}}}{|k-l|}\right)^{\gamma}.$

Define ${\bf A}={\bf I-T}$. Because ${\bf A}$ has zero diagonal entries, we
can bound $||{\bf A}||_{1,1}$ by

$\displaystyle||{\bf A}||_{1,1}$ $\displaystyle\leq
2\sum_{k-l=1}^{\left[\varepsilon^{-2}\right]}|t_{kl}|\leq\frac{2C_{7}\left(\sqrt{c_{0}}\right)^{\gamma}}{C_{8}}\int_{0}^{\varepsilon^{-2}}x^{-\gamma}dx$
$\displaystyle=\frac{2C_{7}\left(\sqrt{c_{0}}\right)^{\gamma}}{C_{8}}\cdot\frac{\left(\varepsilon^{2}\right)^{\gamma-1}}{1-\gamma}=\frac{2C_{7}}{C_{8}(1-\gamma)}\cdot\frac{\left(c_{0}\varepsilon^{4}\right)^{\gamma/2}}{\varepsilon^{2}}.$

For any $\varepsilon>0$, $c_{0}$ is selected to satisfy

$||{\bf
A}||_{1,1}\leq\frac{2C_{7}}{C_{8}(1-\gamma)}\cdot\frac{(c_{0}\varepsilon^{4})^{\gamma/2}}{\varepsilon^{2}}<\frac{1}{3},$

which becomes

$c_{0}<\left(\frac{C_{8}(1-\gamma)}{6C_{7}}\right)^{\frac{2}{\gamma}}\varepsilon^{4/\gamma-4}.$

Let’s denote
$\mathcal{C}=\left(\frac{C_{8}(1-\gamma)}{6C_{7}}\right)^{\frac{2}{\gamma}}$
in (3.1). Choosing such $c_{0}$ will make

$||{\bf T}^{-1}||_{1,1}=||{\bf(I-A)}^{-1}||_{1,1}\leq\frac{1}{1-||{\bf
A}||_{1,1}}\leq\frac{3}{2},$

and $||{\bf\Sigma}^{-1}||_{1,1}\leq||{\bf D}^{-1}||_{1,1}\cdot||{\bf
T}^{-1}||_{1,1}\cdot||{\bf
D}^{-1}||_{1,1}\leq\frac{3}{2}C_{8}^{-1}(c_{0}\varepsilon^{4})^{\frac{\gamma-2}{2}}$,
which leads to

$||{\bf\eta}||_{l^{1}}\leq\frac{3}{2}C_{8}^{-1}(c_{0}\varepsilon^{4})^{\frac{\gamma-2}{2}}||{\bf
x}||_{l^{1}}<\frac{3}{2}\cdot\frac{1}{3}=\frac{1}{2}.$

∎

Combining (5.3), (5.4) and (5.7) yields

$\begin{split}P(F_{1})&\leq\frac{C_{5}}{(1\wedge\sqrt{{c_{0}}})\varepsilon^{2}}\exp\left(-\frac{C_{6}}{4\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right)+C\exp\left(-\frac{C^{\prime}}{\varepsilon^{2}}\right)\\\
&\leq C_{5}^{\prime}\exp\left(-\frac{1}{2}\ln
t_{1}-\frac{C_{6}}{4\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right)+C\exp\left(-\frac{C^{\prime}}{\varepsilon^{2}}\right),\end{split}$

and we may choose a constant $\mathcal{D}_{1}(\gamma)>0$ such that for any
$0<\mathcal{D}<\mathcal{D}_{1}$,

$\displaystyle P(F_{1})$ $\displaystyle\leq
C_{5}^{\prime}\exp\left(-\frac{C_{6}^{\prime}}{\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right)+C\exp\left(-\frac{C^{\prime}}{\varepsilon^{2}}\right)$
$\displaystyle\leq
C\exp\left(-\frac{C^{\prime}}{\varepsilon^{2}+\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right).$

Since $\varepsilon^{2}>\mathcal{C}^{-\gamma/2}t_{1}^{\gamma/2}$ from (3.1), we
finally attain

$P(F_{1})\leq\textbf{C}_{4}\exp\left(-\frac{\textbf{C}_{5}}{t_{1}^{\gamma/2}}\right),$

which completes the proof of (5.1) as well as Proposition 3.1(a).

Proof of Proposition 3.1 (b) We state the Gaussian correlation inequality,
which is crucial to prove Proposition 3.1.

######  Lemma 5.3.

For any convex symmetric sets $K,L$ in $\mathbb{R}$ and any centered Gaussian
measure $\mu$ on $\mathbb{R}$, we have

$\mu(K\cap L)\geq\mu(K)\mu(L).$

###### Proof.

See in paper [Roy14], [LM17]. ∎

By the Markov property of $u(t,\cdot)$, the behavior of $u(t,\cdot)$ in the
interval $[t_{n},t_{n+1}]$ depends only on $u(t_{n},\cdot)$ and $\dot{F}(t,x)$
on $[t_{n},t]\times[-1,1]$. Therefore, it is enough to show that

(5.8) $P\left(E_{0}\right)\geq{\bf C_{6}}\exp\left(-\frac{{\bf
C_{7}}}{t_{1}^{\gamma/2}}\right),$

where ${\bf C_{6}}$, ${\bf C_{7}}$ do not depend on $u_{0}$ and
$|u_{0}(x)|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}$.

We start with the Gaussian case that $\sigma(t,x,u)=\sigma(t,x)$, which does
not depend on $u$. And we assume that $\sigma$ is a smooth and deterministic
function. For $n\geq 0$, we define a sequence of events

$D_{n}=\left\\{|u(t_{n+1},x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}~{}\text{and}~{}|u(t,x)|\leq\frac{2}{3}\varepsilon^{2-\gamma}~{}\forall
t\in[t_{n},t_{n+1}],x\in[-1,1]\right\\}.$

Denote

$G_{t}(u_{0})(x)=G_{t}*u_{0}(x)=\int_{-1}^{1}G_{t}(x-y)u_{0}(y)dy,$

and we have

(5.9)
$|G_{t}(u_{0})(x)|\leq\sup_{x}|u_{0}(x)|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}.$

We consider the measure $Q$ given by

$\frac{dQ}{dP}=\exp\left(Z_{t_{1}}-\frac{1}{2}\langle Z\rangle_{t_{1}}\right)$

where

$Z_{t_{1}}=-\int_{0}^{t_{1}}\int_{-1}^{1}f(s,y)F(dyds).$

If $Z_{t_{1}}$ satisfies Novikov’s condition in [All98], then for each fixed
$T\in[0,\infty)$, $0\leq t\leq T$, and $\forall A\in\mathcal{B}(\mathbb{T})$,

$\widetilde{F}_{t}(A):=F_{t}(A)-\int_{[0,t]\times
A^{2}}f(s,y)\Lambda(x-y)dxdyds$

is a centered spatially homogeneous Wiener process under the measure $Q$ (see
[All98] for more details). Moreover, for $x\in\mathbb{T}$, the covariance
structure of $\dot{F}(t,x)$ provides the formal form of
$\dot{\widetilde{F}}(t,x)$ as follows,

$\displaystyle\dot{\widetilde{F}}(t,x)=\dot{F}(t,x)+\int_{-1}^{1}f(t,y)\Lambda(x-y)dy,$

which is a spatially homogeneous noise under measure $Q$.

We consider the function $\frac{G_{t}(u_{0})(x)}{t_{1}\sigma(t,x)}$, which is
smooth and bounded on $[0,t_{1}]\times\mathbb{T}$. [RS16] provided a pointwise
formula for the fractional Laplacian of a certain class of functions on
$\mathbb{T}$, so that one may assume that there is a continuous function
$f(t,x)$ on $[0,t_{1}]\times\mathbb{T}$ such as

(5.10)
$\begin{split}f(t,x)&=(-\Delta)^{\frac{1-\gamma}{2}}\left(\frac{G_{t}(u_{0})(x)}{t_{1}\sigma(t,x)}\right)\\\
&=\int_{-1}^{1}\left(\frac{G_{t}(u_{0})(x)}{t_{1}\sigma(t,x)}-\frac{G_{t}(u_{0})(y)}{t_{1}\sigma(t,y)}\right)K_{1-\gamma}(x-y)dy\\\
&\leq
2\sup_{s,x}\left|\frac{G_{t}(u_{0})(x)}{t_{1}\sigma(s,x)}\right|\int_{-1}^{1}K_{1-\gamma}(x-y)dy,\end{split}$

where $K_{1-\gamma}$ is the positive kernel function on $\mathbb{T}$ defined
in [RS16], Theorem 1.5.

We now check whether $Z_{t_{1}}$ satisfies Novikov’s condition. When $f$ is a
deterministic function defined in (5.10), it is equivalent to verify that, for
a constant $K(t_{1})$,

(5.11)
$\int_{0}^{t_{1}}\int_{-1}^{1}\int_{-1}^{1}f(s,y)f(s,z)\Lambda(y-z)dzdyds<K(t_{1}).$

Given (5.9) and (5.10), we derive

(5.12)
$\begin{split}&\int_{0}^{t_{1}}\int_{-1}^{1}\int_{-1}^{1}f(s,y)f(s,z)\Lambda(y-z)dzdyds=\int_{0}^{t_{1}}\int_{-1}^{1}f(s,y)\frac{G_{t}(u_{0})(y)}{t_{1}\sigma(s,y)}dyds\\\
&\leq\int_{0}^{t_{1}}\int_{-1}^{1}|f(s,y)|\left(\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}\right)\cdot\frac{1}{t_{1}\mathcal{C}_{1}}dyds=\frac{C(\mathcal{C}_{1})}{t_{1}^{\frac{2+\gamma}{4}}}\int_{0}^{t_{1}}\int_{-1}^{1}|f(s,y)|dyds\\\
&\leq\frac{C(\gamma,\mathcal{C}_{1})}{t_{1}^{\frac{2+\gamma}{4}}}\int_{0}^{t_{1}}\sup_{s,x}\left|\frac{G_{t}(u_{0})(x)}{t_{1}\sigma(s,x)}\right|ds\leq
C(\gamma,\mathcal{C}_{1})t_{1}^{-\gamma/2},\end{split}$

which satisfies (5.11). Thus, we can rewrite equation $(1.1)$ with
deterministic $\sigma$ as

$\displaystyle u(t,x)$
$\displaystyle=G_{t}(u_{0})(x)+\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)\sigma(s,y)\left[\widetilde{F}(dyds)-\frac{G_{s}(u_{0})(y)}{t_{1}\sigma(s,y)}dyds\right]$
$\displaystyle=G_{t}(u_{0})(x)-\frac{G_{t}(u_{0})(x)}{t_{1}}+\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)\sigma(s,y)\widetilde{F}(dyds)$
$\displaystyle=\left(1-\frac{t}{t_{1}}\right)G_{t}(u_{0})(x)+\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)\sigma(s,y)\widetilde{F}(dyds).$

The first term is $0$ at $t_{1}$, and since
$|u_{0}(x)|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}$, we have

$\left|\left(1-\frac{t}{t_{1}}\right)G_{t}(u_{0})(x)\right|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}<\frac{1}{3}\varepsilon^{2-\gamma},x\in[-1,1],t<t_{1}.$

Define

$\widetilde{N}(t,x)=\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)\sigma(s,y)\widetilde{F}(dyds),$

and applying Lemma 4.4 to $\widetilde{F}$ gives

$Q\left(\sup_{\begin{subarray}{c}0\leq t\leq c_{0}\varepsilon^{4}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}\left|\widetilde{N}(t,x)\right|>\frac{1}{6}\varepsilon^{2-\gamma}\right)\leq\frac{C_{5}}{\sqrt{c_{0}}}\exp\left(-\frac{C_{6}}{36\mathcal{C}_{2}^{2}c_{0}^{\frac{2-\gamma}{2}}}\right).$

where the right-hand side is strictly less than 1 when $c_{0}(\varepsilon)$ is
small enough. By the Gaussian correlation inequality (Lemma 5.3), we obtain

$\displaystyle Q\left(\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}\left|\widetilde{N}(t,x)\right|\leq\frac{1}{6}t_{1}^{\frac{2-\gamma}{4}}\right)$
$\displaystyle\geq Q\left(\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in\left[0,\varepsilon^{2}\right]\end{subarray}}\left|\widetilde{N}(t,x)\right|\leq\frac{1}{6}t_{1}^{\frac{2-\gamma}{4}}\right)^{\frac{2}{\varepsilon^{2}}}$
$\displaystyle\geq\left[1-\frac{C_{5}}{\sqrt{c_{0}}}\exp\left(-\frac{C_{6}}{36\mathcal{C}_{2}^{2}c_{0}^{\frac{2-\gamma}{2}}}\right)\right]^{\frac{2}{\varepsilon^{2}}}.$

Observe that

$Q(D_{0})\geq Q\left(\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}\left|\widetilde{N}(t,x)\right|\leq\frac{1}{6}t_{1}^{\frac{2-\gamma}{4}}\right),$

and if we replace $f(s,y)$ with $2f(s,y)$ for $Z_{t_{1}}$,

(5.13)
$1=\mathbb{E}\left[\frac{dQ}{dP}\right]=\mathbb{E}\left[\exp\left(Z_{t_{1}}-\frac{1}{2}\langle
Z\rangle_{t_{1}}\right)\right]=\mathbb{E}[\exp\left(2Z_{t_{1}}-2\langle
Z\rangle_{t_{1}}\right)].$

Because $f(s,y)$ is deterministic, we may estimate the Radon-Nikodym
derivative,

$\displaystyle\mathbb{E}\left[\left(\frac{dQ}{dP}\right)^{2}\right]$
$\displaystyle=\mathbb{E}[\exp\left(2Z_{t_{1}}-\langle
Z\rangle_{t_{1}}\right)]=\mathbb{E}[\exp\left(2Z_{t_{1}}-2\langle
Z\rangle_{t_{1}}\right)\cdot\exp(\langle Z\rangle_{t_{1}})]$
$\displaystyle\leq\exp\left(\frac{C}{t_{1}^{\gamma/2}\mathcal{C}_{1}^{2}}\right).$

The last inequality follows from (5.12) and (5.13). The Cauchy-Schwarz
inequality implies

$Q(D_{0})\leq\sqrt{\mathbb{E}\left[\left(\frac{dQ}{dP}\right)^{2}\right]}\cdot\sqrt{P(D_{0})},$

and as a consequence, we get

(5.14)
$\begin{split}P(D_{0})&\geq\exp\left(-\frac{C}{t_{1}^{\gamma/2}\mathcal{C}_{1}^{2}}\right)\exp\left(\frac{4}{\varepsilon^{2}}\ln\left[1-\frac{C_{5}}{\sqrt{c_{0}}}\exp\left(-\frac{C_{6}}{36\mathcal{C}_{2}^{2}c_{0}^{\frac{2-\gamma}{2}}}\right)\right]\right)\\\
&\geq\exp\left(-\frac{C}{t_{1}^{\gamma/2}\mathcal{C}_{1}^{2}}\right).\end{split}$

For the lower bound with a non-deterministic $\sigma(t,x,u)$, we write

$u(t,x)=u_{g}(t,x)+D(t,x)$

where $u_{g}(t,x)$ satisfies the equation

$\partial_{t}u_{g}=\partial^{2}_{x}u_{g}+\sigma(t,x,u_{0}(x))\dot{F}(t,x),$

and

$D(t,x)=\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)[\sigma(s,y,u(s,y))-\sigma(s,y,u_{0}(y))]F(dyds)$

with an initial profile $u_{0}$. Since $u_{g}$ is Gaussian, for an event
defined as

$\widetilde{D}_{0}=\left\\{|u_{g}(t_{1},x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}~{}\text{and}~{}|u_{g}(t,x)|\leq\frac{2}{3}\varepsilon^{2-\gamma}~{}\forall
t\in[0,t_{1}],x\in[-1,1]\right\\},$

we can apply (5.14) to it and get

(5.15)
$P\left(\widetilde{D}_{0}\right)\geq\exp\left(-\frac{C}{t_{1}^{\gamma/2}\mathcal{C}_{1}^{2}}\right).$

Define the stopping time

$\tau=\inf\left\\{t>0:|u(t,x)-u_{0}(x)|>2\varepsilon^{2-\gamma}\text{~{}for
some $x\in[-1,1]$}\right\\},$

and if the set is empty, we set $\tau=\infty$. Clearly we have $\tau>t_{1}$ on
the event $E_{0}$ in (3.4) since
$|u_{0}(x)|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}<\frac{1}{3}\varepsilon^{2-\gamma}$,
and $|u(t,x)|\leq\varepsilon^{2-\gamma}$ for $\forall t\in[0,t_{1}]$ on the
event $E_{0}$. We make another definition

$\widetilde{D}(t,x)=\int_{0}^{t}\int_{-1}^{1}G_{t-s}(x-y)[\sigma(s,y,u(s\wedge\tau,y))-\sigma(s,y,u_{0}(y))]F(dyds),$

and $D(t,x)=\widetilde{D}(t,x)$ for $t\leq t_{1}$ on the event
$\\{\tau>t_{1}\\}$. Moreover, we have

(5.16) $\begin{split}P(E_{0})&\geq
P\left(\widetilde{D}_{0}\bigcap\left\\{\sup_{\begin{subarray}{c}0\leq t\leq
t_{1}\\\
x\in[-1,1]\end{subarray}}|D(t,x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}\right\\}\right)\\\
&=P\left(\left(\widetilde{D}_{0}\bigcap\left\\{\sup_{\begin{subarray}{c}0\leq
t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}|D(t,x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}\right\\}\bigcap\\{\tau>t_{1}\\}\right)\right.\\\
&\hskip
85.35826pt\bigcup\left.\left(\widetilde{D}_{0}\bigcap\left\\{\sup_{\begin{subarray}{c}0\leq
t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}|D(t,x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}\right\\}\bigcap\\{\tau\leq
t_{1}\\}\right)\right).\end{split}$

On the event $\\{\tau>t_{1}\\}$, we have

$\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}|D(t,x)|=\sup_{\begin{subarray}{c}0\leq t\leq
t_{1}\\\ x\in[-1,1]\end{subarray}}|\widetilde{D}(t,x)|,$

and on the event $\widetilde{D}_{0}\cap\\{\tau\leq t_{1}\\}$, we have, for
some $x$,

$|u_{g}(\tau,x)|\leq\frac{2}{3}\varepsilon^{2-\gamma},|u(\tau,x)-u_{0}(x)|>2\varepsilon^{2-\gamma}\quad\text{and}\quad|u_{0}(x)|\leq\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}.$

The above inequalities lead to

$\displaystyle\sup_{\begin{subarray}{c}x\end{subarray}}|D(\tau,x)|$
$\displaystyle=\sup_{\begin{subarray}{c}x\end{subarray}}|u(\tau,x)-u_{g}(\tau,x)|\geq\sup_{\begin{subarray}{c}x\end{subarray}}(|u(\tau,x)|-|u_{g}(\tau,x)|)$
$\displaystyle\geq\sup_{\begin{subarray}{c}x\end{subarray}}|u(\tau,x)|-\frac{2}{3}\varepsilon^{2-\gamma}\geq
2\varepsilon^{2-\gamma}-\frac{1}{3}t_{1}^{\frac{2-\gamma}{4}}-\frac{2}{3}\varepsilon^{2-\gamma}$
$\displaystyle>2\varepsilon^{2-\gamma}-\frac{1}{3}\varepsilon^{2-\gamma}-\frac{2}{3}\varepsilon^{2-\gamma}\geq\varepsilon^{2-\gamma}>\frac{1}{6}\varepsilon^{2-\gamma},$

which implies

$\widetilde{D}_{0}\cap\left\\{\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}|D(t,x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}\right\\}\cap\\{\tau\leq
t_{1}\\}=\phi.$

Combining the above with (5.16) yields

(5.17) $\begin{split}P(E_{0})&\geq
P\left(\widetilde{D}_{0}\cap\left\\{\sup_{\begin{subarray}{c}0\leq t\leq
t_{1}\\\
x\in[-1,1]\end{subarray}}|\widetilde{D}(t,x)|\leq\frac{1}{6}\varepsilon^{2-\gamma}\right\\}\right)\\\
&\geq P\left(\widetilde{D}_{0}\right)-P\left(\sup_{\begin{subarray}{c}0\leq
t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}|\widetilde{D}(t,x)|>\frac{1}{6}\varepsilon^{2-\gamma}\right),\end{split}$

and a union bound gives that

(5.18) $P\left(\sup_{\begin{subarray}{c}0\leq t\leq t_{1}\\\
x\in[-1,1]\end{subarray}}\left|\widetilde{D}(t,x)\right|>\frac{1}{6}\varepsilon^{2-\gamma}\right)\leq\frac{C_{5}}{(1\wedge\sqrt{c_{0}})\varepsilon^{2}}\exp\left(-\frac{C_{6}}{144\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right).$

Consequently, from (5.15), (5.17) and (5.18), we conclude that there exists a
constant $\mathcal{D}_{2}(\gamma)>0$ such that for any
$\mathcal{D}<\mathcal{D}_{2}$,

(5.19)
$\begin{split}P(E_{0})&\geq\exp\left(-C(d,\beta,\mathcal{C}_{1})t_{1}^{-\gamma/2}\right)-C_{5}^{\prime}\exp\left(-\frac{C_{6}^{\prime}}{\mathcal{D}^{2}t_{1}^{1-\gamma/2}}\right)\\\
&\geq\textbf{C}_{6}\exp\left(-\frac{\textbf{C}_{7}}{t_{1}^{\gamma/2}}\right),\end{split}$

which completes the proof of (5.8) as well as Proposition 3.1(b).

##  6\. Acknowledgment

The author would like to thank his advisor Carl Mueller for productive
comments.

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[affiliation=1^†^#]ArnavGoel [affiliation=1^†]MedhaHira
[affiliation=1,2]AnubhaGupta

# Exploring Multilingual Unseen Speaker Emotion Recognition: Leveraging Co-
Attention Cues in Multitask Learning

###### Abstract

Advent of modern deep learning techniques has given rise to advancements in
the field of Speech Emotion Recognition (SER). However, most systems prevalent
in the field fail to generalize to speakers not seen during training. This
study focuses on handling challenges of multilingual SER, specifically on
unseen speakers. We introduce CAMuLeNet, a novel architecture leveraging co-
attention based fusion and multitask learning to address this problem.
Additionally, we benchmark pretrained encoders of Whisper, HuBERT, Wav2Vec2.0,
and WavLM using 10-fold leave-speaker-out cross-validation on five existing
multilingual benchmark datasets: IEMOCAP, RAVDESS, CREMA-D, EmoDB and CaFE
and, release a novel dataset for SER on the Hindi language (BhavVani).
CAMuLeNet shows an average improvement of approximately 8% over all benchmarks
on unseen speakers determined by our cross-validation strategy.

###### keywords:

Speaker emotion recognition, co-attention, multitask learning, new dataset

##  1 Introduction

Figure 1: CAMuLeNet: Co-Attention based Multitask Learning Network

In her seminal work on affective computing, Picard asserts that computers can
achieve genuine intelligence and natural interactivity if we empower them with
the ability to recognize and understand emotions [1]. Besides working with
vision-based cues such as facial expressions and hand movements, humans excel
in discerning emotions even when only auditory information is available [2].
This ability highlights the nuanced and adaptable nature of human emotional
understanding, capable of discerning and processing emotions across a wide
spectrum of speakers and voice types. Building on this premise, it is
imperative to extend the capabilities of Speaker Emotion Recognition (SER)
systems beyond speakers on which they have been trained.

Traditional SER systems rely on features including pitch, energy, MFCCs, and
spectrograms for emotion recognition [3, 4]. With the emergence of deep
learning methods, systems employing CNNs, Bi-Directional RNNs, and LSTMs are
able to learn discernible features [5, 6, 7]. Transformer-based models, a more
recent development, marked a significant advancement with the introduction of
large pre-trained models (PTMs) trained under a self-supervised learning
framework [8]. Recent advancements in weakly-supervised models, such as
Whisper [9], which are trained on extensive corpora, have demonstrated
superior performance on a diverse array of downstream tasks [10, 11, 12].
Attention mechanisms, including cross-attention [13], windowed-attention [14],
and self-attention [15] along with multitask training have been explored to
enhance the performance of SER [16, 17, 18]. Despite these advances, a
critical challenge remains: the inability of modern SER systems to effectively
adapt to unseen scenarios and speakers, resulting in performance that is
inferior to human capabilities [19, 20].

This study contributes to the field by benchmarking various PTM embeddings in
a transfer learning framework, specifically addressing unseen speaker
recognition on five existing benchmark datasets and the 6th newly released
dataset with this work. Although co-attention based fusion mechanisms have
been used previously on the speaker emotion recognition downstream task for
fusing features from multiple modalities [21] and multi-level acoustic
information [22], their use on unseen speaker emotion recognition tasks along
with multitask learning is yet to be thoroughly explored. This study addresses
this gap by proposing an architecture that fuses features from the frequency
domain and PTM embeddings.

Moreover, the variation in emotional expression across languages poses a
distinctive challenge in multilingual SER, which is compounded by the scarcity
of comprehensive datasets. To address this gap, we introduce a novel Hindi SER
dataset, designed to enhance model training and benchmarking in Indian
linguistic contexts. To the best of our knowledge, this is the first open-
source Hindi SER dataset. Extending our efforts, we apply our methodology to
French and German datasets, positioning our work as the first to benchmark
Whisper's encoder in multilingual SER settings. The codes and dataset can be
found on GitHub111https://github.com/arnav10goel/CAMuLeNet.

The key contributions of this work are three-fold:

  1. 1.

We introduce BhavVani, the first-ever Hindi Speech Emotion Recognition dataset
with over 9,000 utterances.

  2. 2.

Our research uniquely benchmarks pre-trained model embeddings across six
datasets in four languages (English, German, French, Hindi) for unseen speaker
recognition, marking the first study of its kind to explore these embeddings,
including Whisper, on this downstream task.

  3. 3.

We introduce CAMuLeNet, a Co-Attention based Multitask Learning Network
architecture that fuses frequency domain features with PTM features in a
multitask framework of emotion and gender recognition, aiming to derive
generalized representations for enhanced speaker emotion recognition.

##  2 Proposed Methodology

The proposed CAMuLeNet architecture described in Figure1, aims to fuse
traditional frequency domain features of the spectrogram and MFCCs, with the
features extracted from a pre-trained Whisper encoder using the co-attention
mechanism described next. We train this architecture through a multitask setup
to improve performance on unseen speakers' emotion recognition.

###  2.1 Extracting Features from the Frequency Domain

To capture frequency and pitch variations in audio clips for generalizing
unseen speakers, we utilize frequency-domain features. The spectrogram and
mel-frequency cepstrum coefficients (MFCCs) are represented as $x_{s}$ and
$x_{m}$, respectively, choosing $40$ MFCCs. Both are used in their two-
dimensional form, capturing time-frequency representation without average
pooling. An audio clip $x$ is preprocessed through padding and filtering for
consistent sequence length, followed by feature extraction via an AlexNet
encoder [23], treating the spectrogram as an image. The latent embeddings from
AlexNet is a one-dimensional feature vector $x^{\prime}_{s}$ of size $4096$.
Concurrently, MFCCs undergo processing through a Bidirectional Gated Recurrent
Unit (Bi-GRU) with two 256-sized hidden layers and a 0.2 dropout, producing a
one-dimensional embedding $x^{\prime}_{m}$ sized (seq_len $\times 512$).

###  2.2 Extracting Features through Transfer Learning

OpenAI's Whisper222https://github.com/openai/whisper, a multilingual encoder-
decoder model, exhibits state-of-the-art performance across various speech-to-
text benchmarks. Trained on a vast and diverse audio dataset, it significantly
improves speech recognition and translation tasks. We hypothesize that
Whisper's encoder produces rich latent representations of audio samples. We
pass the preprocessed audio clip $x$ through the Whisper Encoder (which
converts it into a mel spectrogram for processing), obtaining a 2D latent
representation $x_{w}\in\mathbb{R}^{L\times W}$. We refrain from using pooling
on this 2D representation to maintain the time-frequency information of these
embeddings.

###  2.3 Co-Attention based Fusion

The three embeddings hold vital knowledge representation that we aim to fuse
for improved performance. First, the derived embeddings from the spectrogram
$x^{\prime}_{s}$ and MFCC features $x^{\prime}_{m}$ are passed through a fully
connected (FC) layer with the same output dimension:

$\displaystyle x_{m_{att}}$ $\displaystyle=f_{att_{m}}(x^{\prime}_{m})$ ,
$\displaystyle x_{s_{att}}=f_{att_{s}}(x^{\prime}_{s})$ (1)

where $x_{s_{att}}\in\mathbb{R}^{1\times T}$,
$x_{m_{att}}\in\mathbb{R}^{1\times T}$, and $T(=1024)$ is a hyperparameter.
The two transformed one-dimensional embeddings are concatenated to create a
one-dimensional vector and passed through another FC layer:

$x_{sm_{att}}=f_{att_{ms}}(x_{m_{att}}\oplus x_{s_{att}})$ (2)

where $x_{sm_{att}}\in\mathbb{R}^{1\times L}$. The activated concatenated
features from (2) are sent through a FC layer activated by _ReLU_ and sent
through a _Layer Norm_. This output is multiplied with Whisper features
$x_{w}$ to generate attention-weighted Whisper features

$x_{w_{att}}=f_{att_{ms}}(x_{sm_{att}}\otimes x_{w}),$ (3)

where $x_{w_{att}}\in\mathbb{R}^{1\times W}$. Attention-weighted features are
sent through a 3-layer network with 0.15 dropout, an FC layer with ReLU
activation, and a terminal Layer Norm. These processed features, along with
activated features from (2) sent via a skip-connection, are concatenated to
form the network's final embedding for downstream classification. This co-
attention mechanism attends frequency domain information with frame-level
features from the Whisper encoder.

###  2.4 Multitask Learning

Utilizing multitask learning, our network concurrently trains on emotion and
gender recognition, leveraging the significant influence of speaker gender on
emotion. This approach aims to develop nuanced latent representations that
captures intricate emotion-speaker correlations. We use categorical cross
entropy loss $\mathcal{L}_{cce}$ for multiclass emotion recognition and binary
cross entropy loss $\mathcal{L}_{bce}$ for gender recognition, reflecting the
prevalent binary gender annotation. The combined multitask objective is
defined as

$\mathcal{L}=\alpha\cdot\mathcal{L}_{cce}+\beta\cdot\mathcal{L}_{bce}+\gamma,$
(4)

with $\alpha$ and $\beta$ as tunable weights for each task and $\gamma$ as a
tunable parameter providing stability while training. Experimentally, setting
$\alpha$ three times higher than $\beta$ effectively balances training across
tasks, attributed to the class count ratio and the relative simplicity of
emotion recognition, thus imposing a higher penalty for better learning from
the latter.

##  3 Datasets

###  3.1 Benchmark Datasets

This study utilizes five widely-used multilingual datasets (English, German,
French) to assess pre-trained model embeddings and our methodology for unseen
speaker emotion recognition. They are summarised next.

The CREMA-D dataset [24], featuring 7,442 clips from 91 voice actors (48
males, 43 females), offers recordings of 12 sentences expressed in 6 emotions
in English. The IEMOCAP dataset [25] includes 10,039 clips from 10 actors (5
males, 5 females) in both scripted and spontaneous conversations, annotated
with 4 emotion classes. The RAVDESS [26] dataset comprises 1,440 recordings
from 24 actors (12 males, 12 females), each articulating two sentences across
8 emotional states. Each recording features one speaker expressing a single
emotion, with equal distribution of male and female voices across the dataset.
Among these, CREMA-D stands out with the largest variety of unique speakers
from various ethnicity and is the largest corpus of speech samples among the
existing datasets.

We utilize EmoDB [27] and CaFE [28] datasets for German and French languages,
respectively. EmoDB includes 535 recordings from 10 actors (5 males, 5
females), featuring 10 sentences in _7 emotional states_. CaFE provides a
collection of 936 clips from 12 actors (6 males, 6 females), each expressing 6
sentences across _7 emotions_.

###  3.2 Our Novel Hindi SER Dataset: BhavVani

In response to the under-representation of Indic languages in Speech Emotion
Recognition (SER) and their absence in multilingual benchmarks such as SERAB
[29], we release BhavVani, a novel dataset tailored for Hindi SER. This
initiative aims to enrich the SER research landscape by incorporating the
linguistic and cultural diversity of Indian languages, leveraging their
morphological richness and unique emotional expressiveness.

####  3.2.1 Data Collection

The BhavVani dataset comprises approximately _13 hours of audio_ across 8734
utterances, with an average clip length of 5.08 seconds. The audio clips are
curated from the popular Indian sitcom "Sarabhai vs Sarabhai", sourced from
prior work that used the show's text for sarcasm detection tasks [30]. To our
knowledge, BhavVani is the first open-source Hindi dataset tailored for
speaker emotion recognition. The dataset will be released to the community for
further work. Each utterance is a single-speaker dialogue, annotated into one
of seven categories: Neutral, Surprise, Enjoyment, Disgust and Anger, Fear,
Sadness based off Ekman's seven basic human emotions [31].

####  3.2.2 Data Annotation and Validation

The BhavVani dataset was annotated by 18 native Hindi speakers, who received
preliminary training to ensure annotation consistency and accuracy. Each
annotator evaluated approximately 750 audio clips, identifying the speaker’s
gender, name, and perceived emotion while excluding clips with excessive
background noise or multiple speakers. To uphold the integrity of the dataset,
a rigorous validation procedure was followed involving three independent
annotators reviewing each annotation. Annotations that were agreed by all
three annotators, were marked as confirmed, while those with disagreements
were re-evaluated, if necessary, and an alternative emotion was suggested.
This meticulous annotation and validation method, evidenced by a Fleiss Kappa
score of 0.637, highlights substantial annotator agreement, ensuring the
reliability and quality of the dataset.

####  3.2.3 Dataset Statistics

The pie-charts in Figure 2 provide insights of the thoroughly annotated and
curated dataset that had no missing labels. The dataset is balanced with male
and female speakers. The gender has been annotated for all speakers, while the
speaker name has been annotated primarily for the main characters. None of the
other characters had an individual occurrence greater than 41, hence they were
positioned under the umbrella of `Others'.

Figure 2: BhavVani Dataset Statistics

##  4 Experiments

###  4.1 Baseline Pre-Trained Models Employed

We employed encoders from pre-trained models (PTMs) such as Wav2Vec2.0 [32],
WavLM [33], HuBERT [34], and Whisper [9] as baselines via transfer learning.
Initially trained for ASR tasks, Wav2Vec2.0, WavLM, and HuBERT are
transformer-based models with 95M parameters each trained through self-
supervised learning on English datasets. We fine-tuned these models for
multilingual SER using CTC loss. Whisper, distinguished as an encoder-decoder
model, is trained with weak supervision on a diverse 680000-hour multilingual
dataset, with our focus on its Base (74M) and Medium (769M) variants. We adapt
Whisper models for SER by converting their 2D time-frequency outputs into 1D
embeddings via average pool. Chosen for their excellence in speech processing,
their potential with unseen speakers is yet to be fully explored.

Table 1: Comparison of Baseline Methods and CAMuLeNet across various datasets

PTM | CREMA-D | IEMOCAP | RAVDESS | EmoDB | CaFE | BhavVani
---|---|---|---|---|---|---
WA ($\uparrow$) |  WF1 ($\uparrow$) |  WA ($\uparrow$) |  WF1 ($\uparrow$) |  WA ($\uparrow$) |  WF1 ($\uparrow$) |  WA ($\uparrow$) |  WF1 ($\uparrow$) |  WA ($\uparrow$) |  WF1 ($\uparrow$) |  WA ($\uparrow$) |  WF1 ($\uparrow$)
Wav2Vec2.0 |  0.451 |  0.409 |  0.427 |  0.408 |  0.396 |  0.377 |  0.469 |  0.455 |  0.417 |  0.372 |  0.251 |  0.317
HuBERT |  0.529 |  0.499 |  0.461 |  0.458 |  0.587 |  0.546 |  0.486 |  0.479 |  0.424 |  0.397 |  0.253 |  0.332
WavLM |  0.587 |  0.547 |  0.512 |  0.475 |  0.654 |  0.612 |  0.806 |  0.781 |  0.444 |  0.402 |  0.303 |  0.305
Whisper-Base |  0.603 |  0.591 |  0.573 |  0.588 |  0.593 |  0.587 |  0.743 |  0.732 |  0.452 |  0.441 |  0.261 |  0.265
Whisper-Medium |  0.666 |  0.649 |  0.648 |  0.667 |  0.713 |  0.735 |  0.831 |  0.799 |  0.623 |  0.551 |  0.412 |  0.439
Architecture | Experiments on Co-Attention and Multitask Learning
CAMuLeNet (Ours) |  0.762 |  0.768 |  0.734 |  0.728 |  0.823 |  0.826 |  0.862 |  0.847 |  0.709 |  0.691 |  0.453 |  0.441
| Ablation Study
Ours (w/o Multitask) |  0.719 |  0.721 |  0.703 |  0.697 |  0.782 |  0.784 |  0.853 |  0.844 |  0.672 |  0.683 |  0.431 |  0.429
Ours (w/o Co-Att & Multitask) |  0.671 |  0.653 |  0.659 |  0.662 |  0.721 |  0.728 |  0.817 |  0.793 |  0.605 |  0.593 |  0.407 |  0.398

Note: WA stands for weighted accuracy and WF1 stands for weighted F1 score.
These measures account for class imbalance.

###  4.2 Baseline Experiment Setup

In the baseline experiment, we provided 1D-embedding obtained in Section 4.1
as input to a CNN based feature extractor comprising of a 1D convolutional
layer, batch normalization, ReLU activation, dropout (0.3), and max pool
followed by flattening and two fully connected (FC) layers for classification.
We utilized cross-entropy loss and Adam optimizer with a learning rate of
$10^{-4}$ as higher rates led to model overshoot, rapid loss increase, and
early underfitting within a few epochs. We trained the models on a NVIDIA
A5000 GPU. Training was capped at 20 epochs with early stopping based on
validation loss to prevent overfitting. For evaluating performance on unseen
speakers, we followed 10-fold leave-speaker-out cross validation, wherein each
dataset was segmented into 10 folds with each fold containing unique speakers.
Thus, CREMA-D had around 9 new speakers at the time of validation that the
model had not seen at the training time.

###  4.3 CAMuLeNet Training Setup

We extracted MFCC and spectrogram features from the pre-processed audio
waveform. The spectrogram is derived by applying a Hamming window based short-
time Fourier transform, with a window length of 40, a hop length of 10,
setting the size of the FFT window to 800. The calculation of MFCCs involved
generating 40 MFCC values with a hop length of 160, ensuring that these
coefficients were compatible with the Hidden Markov Model Toolkit (HTK). The
training of our model architecture was conducted using a NVIDIA A5000 GPU
using batches of 64, leveraging the Adam optimizer with a learning rate of
$5\times 10^{-5}$. Dropout was maintained at 0.15 throughout the network. In
the context of multitask learning, we determined experimentally that the model
achieved optimal stability across different datasets when the weighting
factors were set to $\alpha=0.4$, $\beta=0.1$ and $\gamma=0.2$. These values,
however, may require tuning to accommodate variations in dataset
characteristics and training configurations. Remaining conditions are as
described in Section-4.2.

##  5 Results and Discussion

This Section details our comparative analysis of baseline and proposed methods
from 4.2, quantified by Weighted Accuracy (WA) and Weighted F1 score (WF1)
across six datasets, presented in Table 1, complemented by an ablation study
discussed later. We report mean metric values averaged over ten folds.

###  5.1 Analysis of Baseline Transfer Learning Results

As per Table-1, Whisper-Medium shows a huge performance jump over self-
supervised based PTMs and its base architecture. The increase is noteworthy on
the IEMOCAP (13% over WavLM), CaFE (18% over Whisper-Base) and BhavVani (14%
over WavLM). The increase is higher on French language dataset due to Whisper
being trained on a lot of audio chunks from the French language. Objective
performance, however, remains sub-optimal on our Hindi SER dataset indicating
the need for interventions to create more robust models and resources for
Indic languages.

###  5.2 Analysis of Results on CAMuLeNet

The baseline analysis indicated Whisper-Medium as an optimal pre-trained
embedding for our co-attention fusion method. As detailed in Section 4.3, we
established a multi-task training framework and observed improved performance
across all benchmarks. The introduction of the parameter $\gamma$ into our
loss function contributed to stabilized training, evidenced by a consistent
decrease in loss values. Remarkably, the employment of CAMuLeNet yielded a
substantial enhancement in accuracy, particularly on the English-language
CREMA-D benchmark, _with an increase of 10%_ , and _an 11% improvement on the
RAVDESS benchmark_. Results on CREMA-D are worth noting due to the presence of
speakers from various ethnicity uttering emotions at different levels of
valence. Hence, Figure-3 contrasts the t-SNE visualizations between Whisper-
Medium and CAMuLeNet, with the latter exhibiting pronounced inter-class
separation and reduced intra-class variance, which underscores the efficacy of
our model in delineating clearer classification boundaries.

Furthermore, the results on multilingual benchmarks, such as CaFE, mirrored
the performance gains observed in baseline experiments, underscoring the
generalizability of our approach. The multi-task training paradigm notably
amplified model's performance on gender recognition tasks, achieving over 95%
accuracy on all benchmarks. These findings confirm the effectiveness of our
multi-task and co-attention strategy to improve the model's performance on
unseen speakers from various linguistic backgrounds.

###  5.3 Ablation Study

Our proposed methods utilise features from the frequency domain and latent
representations generated by a Whisper encoder. Table 1 additionally shows our
ablation study by removing the multi-task training setup (1) and then
additionally removing co-attention to fuse the features (2). We observed that
removing multitask training and replacing it with a single-task training setup
reduces performance by an average of approximately 4% over CAMuLeNet.
Replacing co-attention based feature fusion with normal concatenation reduced
performance across all benchmark datasets below the best-performing baseline
methods by an average of around 10% over CAMuLeNet. This could be attributed
largely to performing concatenation without accounting for the underlying
correlations, emphasising the importance of using co-attention based fusion
methods.

a) t-SNE for Whisper Med

b) t-SNE for CAMuLeNet

Figure 3: t-SNE visualisation of Feature Distribution. (a) Distribution of
final extracted features from Whisper-Medium on the entire CREMA-D dataset.
(b) Distribution of final extracted features from CAMuLeNet before the final
classification layer on the entire CREMA-D dataset.

##  6 Limitations and Conclusion

This study exposes a critical limitation in SER for low-resource languages
through our novel BhavVani benchmark, where gains were modest and overall
performance was subdued, emphasizing the challenges in unseen speaker
recognition. Future work will prioritize these languages and investigate
alternative fusion mechanisms for robust generalizations. Our contributions
include the introduction of BhavVani dataset, comprehensive benchmark of
pretrained embeddings across six datasets using a rigorous 10-fold leave-
speaker-out cross-validation strategy, and the novel CAMuLeNet architecture,
which synergizes frequency domain features with PTM Whisper embeddings through
co-attention based feature fusion and multitask training. These efforts
spotlight the issue of emotion recognition of unseen speakers for advancing
research in this field.

##  7 Acknowledgments

We acknowledge the contributions of: Adamya, Aditya, Akshat, Arnav, Arush,
Divyanshi, Kartikay, Kuleen, Manaswi, Manya, Rishitej, Rushil, Sahaj, Tanish,
Tanishq and Yash from IIIT Delhi in annotating the BhavVani dataset. This work
is supported by the Infosys Foundation via Infosys Centre for AI (CAI), IIIT
Delhi.

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