# Polarity of points for systems of nonlinear stochastic heat equations in the
critical dimension

Cheuk Yin Lee  Department of Mathematics, National Tsing Hua University, No.
101, Section 2, Kuang-fu Road, Hsinchu 300, Taiwan<EMAIL_ADDRESS>and
Yimin Xiao  Department of Statistics and Probability, Michigan State
University, East Lansing, MI 48824, United States<EMAIL_ADDRESS>
###### Abstract.

Let $u(t,x)=(u_{1}(t,x),\dots,u_{d}(t,x))$ be the solution to the systems of
nonlinear stochastic heat equations

$\begin{split}\frac{\partial}{\partial t}u(t,x)&=\frac{\partial^{2}}{\partial
x^{2}}u(t,x)+\sigma(u(t,x))\dot{W}(t,x),\\\ u(0,x)&=u_{0}(x),\end{split}$

where $t\geq 0$, $x\in\mathbb{R}$,
$\dot{W}(t,x)=(\dot{W}_{1}(t,x),\dots,\dot{W}_{d}(t,x))$ is a vector of $d$
independent space-time white noises, and
$\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ is a matrix-valued function.
We say that a subset $S$ of $\mathbb{R}^{d}$ is polar for $\\{u(t,x),t\geq
0,x\in\mathbb{R}\\}$ if

$\mathbb{P}\\{u(t,x)\in S\text{ for some }t>0\text{ and }x\in\mathbb{R}\\}=0.$

The main result of this paper shows that, in the critical dimension $d=6$, all
points in $\mathbb{R}^{d}$ are polar for $\\{u(t,x),t\geq
0,x\in\mathbb{R}\\}$. This solves an open problem of Dalang, Khoshnevisan and
Nualart (2009, 2013) and Dalang, Mueller and Xiao (2021). We also provide a
sufficient condition for a subset $S$ of $\mathbb{R}^{d}$ to be polar.

###### Key words and phrases:

Hitting probabilities, polarity of points, critical dimension, stochastic heat
equation, Malliavin calculus

###### 2020 Mathematics Subject Classification:

Primary 60H15, 60J45; Secondary 60G60, 60H07

##  1\. Introduction

Hitting probabilities are one of the most studied topics for stochastic
processes. Let $X=\\{X(t),t\in I\\}$ be an ${\mathbb{R}}^{d}$-valued
stochastic process and $A$ be a subset of ${\mathbb{R}}^{d}$. We say that $X$
hits $A$ if

$\displaystyle{\mathbb{P}}\\{X(t)\in A\text{ for some }t\in I\\}>0.$

We say that $A$ is polar for $X$ if ${\mathbb{P}}\\{X(t)\in A\text{ for some
}t\in I\\}=0$.

This paper is motivated by a problem on hitting probabilities of systems of
nonlinear stochastic heat equations in the critical dimension which was left
open in [5, 6, 9]. We consider the solution
$u(t,x)=(u_{1}(t,x),\dots,u_{d}(t,x))$ to the systems of nonlinear stochastic
heat equations

$\displaystyle\begin{split}\frac{\partial}{\partial
t}u(t,x)&=\frac{\partial^{2}}{\partial
x^{2}}u(t,x)+\sigma(u(t,x))\dot{W}(t,x),\\\ u(0,x)&=u_{0}(x),\end{split}$
(1.1)

where $t\geq 0$, $x\in{\mathbb{R}}$,
$\dot{W}(t,x)=(\dot{W}_{1}(t,x),\dots,\dot{W}_{d}(t,x))$ is a vector of $d$
independent space-time white noises, and
$\sigma:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ is a matrix-valued
function. More specific conditions will be given in the next section.

For systems of linear stochastic heat equations, where $\sigma\equiv 1$ and
the solution is a Gaussian random field, hitting probabilities and hitting of
points have been studied by [14, 4, 8]. In particular, it is known that points
are polar if and only if $d\geq 6$.

The results for systems of nonlinear stochastic heat equations are less
complete. Dalang, Khoshnevisan and Nualart [5, 6] proved upper and lower
bounds on hitting probabilities of the form

$\displaystyle\frac{1}{C_{\varepsilon}}\operatorname{Cap}_{d-6+\varepsilon}(A)\leq{\mathbb{P}}\left\\{u(t,x)\in
A\text{ for some }(t,x)\in[1,2]^{2}\right\\}\leq
C_{\varepsilon}\mathscr{H}_{d-6-\varepsilon}(A),$ (1.2)

where $\operatorname{Cap}$ denotes the Bessel–Riesz capacity, $\mathscr{H}$
denotes the Hausdorff measure, and $\varepsilon>0$. When $A=\\{z\\}$, this
result implies that points are non-polar when $d<6$, and are polar when $d>6$,
but the critical case $d=6$ was left open. This problem was unsolved not only
for $\\{u(t,x),(t,x)\in[0,\infty)\times{\mathbb{R}}\\}$ but also for the one-
parameter processes $\\{u(t,x_{0}),t\geq 0\\}$ and
$\\{u(t_{0},x),x\in{\mathbb{R}}\\}$; see [5, 6].

A partial result for the critical case was obtained by Dalang, Mueller and
Xiao [9], who showed that almost every point is polar for $u$ in the critical
dimension $d=6$, where “almost every” is with respect to the Lebesgue measure.
The question of whether every fixed point is polar had remained open.

In this paper, we answer this question and prove that all points are indeed
polar in the critical dimension $d=6$. We also give a sufficient condition for
a compact set $A$ to be polar, which is useful in certain cases even when
(1.2) fails to show whether $A$ is polar or not.

Hitting probabilities are typically more difficult to handle in critical
dimensions. Talagrand [16, 17] established a covering argument to prove that
fractional Brownian fields do not hit points and do not have multiple points
in the respective critical dimensions. Building on Talagrand’s covering
argument, the authors of [8] and [7] proved that points are polar and multiple
points do not exist in critical dimensions for a large class of Gaussian
random fields including the solutions of systems of linear stochastic heat
equations.

In [8], polarity of points for Gaussian random fields is proved by first
establishing that almost every point is polar using Talagrand’s covering
argument. After that, taking conditional expectation of the random field is
required to extend the polarity result from “almost every point” to “every
point”. In particular, for the linear stochastic heat equation, the solution
$u$ is Gaussian,

$\displaystyle
u^{(2)}(t,x)={\mathbb{E}}[u(t,x)|u(t^{\prime},x^{\prime})]\quad\text{and}\quad
u^{(1)}(t,x)=u(t,x)-u^{(2)}(t,x)$ (1.3)

are two independent Gaussian processes and the conditional expectation can be
computed explicitly. In the nonlinear case, $u$ is non-Gaussian and (1.3) no
longer works. Due to this difficulty, Dalang, Mueller and Xiao [9] were only
able to show that almost every point is polar in the critical dimension.

To circumvent this difficulty, we create independence by taking conditional
expectation of the solution with respect to two independent $\sigma$-algebras
$\mathscr{F}^{1}$ and $\mathscr{F}^{2}$. The new idea is to use the formula

$\displaystyle{\mathbb{E}}[X|\sigma(\mathscr{F}^{1}\cup\mathscr{F}^{2})]={\mathbb{E}}[X|\mathscr{F}^{1}]+{\mathbb{E}}[X|\mathscr{F}^{2}]-{\mathbb{E}}[X]$

for a random variable $X\in L^{2}$ (cf. [2, Theorem 3.3.1(j)]) to obtain a
crucial decomposition of the solution:

$\displaystyle u(t,x)=u^{(0)}(t,x)+u^{(1)}(t,x)+u^{(2)}(t,x),$

where $u^{(0)}$ is the (deterministic) solution to the homogeneous heat
equation, and $u^{(1)}\in\mathscr{F}^{1}$ and $u^{(2)}\in\mathscr{F}^{2}$ are
two independent processes with the key property that $u^{(2)}$ is relatively
smooth compared to $u^{(1)}$.

As in the articles [8, 9, 11], we adapt Talagrand’s covering argument [16, 17]
to construct an efficient random cover (which is $\mathscr{F}^{1}$-measurable)
for the level set $u^{-1}(z)\cap I$. To prove that a fixed point
$z\in{\mathbb{R}}^{d}$ is polar in the critical dimension $d=6$, we show that
the “formal” $\phi$-Hausdorff measure of $u^{-1}(z)\cap I$ has finite
expectation, where $\phi(r)=\log\log(1/r)$. The key step is to derive the
conditional probability estimate

$\displaystyle{\mathbb{P}}\\{|u(t,x)-z|\leq r|\mathscr{F}^{1}\\}\leq Cr^{d},$

which, thanks to the independence of $\mathscr{F}^{1}$ and $\mathscr{F}^{2}$,
reduces to proving

$\displaystyle{\mathbb{P}}\\{|u^{(2)}(t,x)-z|\leq r\\}\leq Cr^{d}.$

This is achieved by employing tools from Malliavin calculus to prove, under
suitable assumptions on $\sigma$, that $u^{(2)}(t,x)$ has a uniformly bounded
density function.

##  2\. Assumptions and main results

The initial condition of the stochastic heat equation is given by
$u(0,x)=u_{0}(x)$, where $u_{0}:{\mathbb{R}}\to{\mathbb{R}}$ is a Borel
measurable function. Let
$\dot{W}(t,x)=(\dot{W}_{1}(t,x),\dots,\dot{W}_{d}(t,x))$ be a $d$-dimensional
space-time white noise with i.i.d. components defined on a complete
probability space $(\Omega,\mathscr{F},{\mathbb{P}})$. The natural filtration
is defined by

$\displaystyle\mathscr{F}_{t}=\sigma\\{W_{s}(A):s\in[0,t],A\in\mathscr{B}_{b}({\mathbb{R}})\\}\vee\mathcal{N},\quad
t\geq 0,$

where $\mathscr{B}_{d}({\mathbb{R}})$ is the collection of all bounded Borel
sets in ${\mathbb{R}}$, $W_{s}(A)$ is the martingale measure associated with
the noise $\dot{W}(t,x)$ in the sense of Walsh [18], and $\mathcal{N}$ is the
$\sigma$-algebra generated by all ${\mathbb{P}}$-null sets. Let
$\sigma:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ be a matrix-valued
function. We identify ${\mathbb{R}}^{d\times d}\simeq{\mathbb{R}}^{d^{2}}$ and
let $|z|$ denote the Euclidean norm of $z$. As in [9], we impose the following
assumptions.

######  Assumption 2.1.

(a) The function
$\sigma:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ is globally
Lipschitz continuous, namely, there exists a finite constant $L$ such that for
all $u,v\in{\mathbb{R}}^{d}$,

$\displaystyle|\sigma(u)-\sigma(v)|\leq L|u-v|.$

  1. (b)

There exists a finite constant $\sigma_{1}$ such that

$\displaystyle\sup_{u\in{\mathbb{R}}^{d}}|\sigma(u)|\leq\sigma_{1},$

  2. (c)

The initial condition $u_{0}$ is bounded, i.e.,

$\displaystyle\sup_{x\in{\mathbb{R}}}|u_{0}(x)|<\infty.$

In addition, we assume the nondegeneracy condition that $\sigma$ is uniformly
elliptic and the smoothness condition that $\sigma$ is infinitely
differentiable with bounded derivatives of all orders.

######  Assumption 2.2.

There exists a constant $\sigma_{0}>0$ such that

$\displaystyle|{\xi}^{\top}\sigma(u)|^{2}\geq\sigma_{0}\quad\text{for all
}u\in{\mathbb{R}}^{d},\xi\in{\mathbb{R}}^{d}\text{ with }|\xi|=1.$

######  Assumption 2.3.

For all $1\leq i,j\leq d$, the function $\sigma_{i,j}$ is $C^{\infty}$ with
bounded partial derivatives of all orders, namely, for all $k\geq 1$, there
exists a finite constant $C_{k}$ such that

$\displaystyle\sup_{1\leq i,j\leq
d}\sup_{u\in{\mathbb{R}}^{d}}|D^{k}\sigma_{i,j}(u)|\leq C_{k}.$

Before stating our results, we first recall the rigorous formulation for the
solution of (1.1). We say that
$u=\\{u(t,x),(t,x)\in[0,\infty)\times{\mathbb{R}}\\}$ is a mild solution to
(1.1) if $u$ is an $\mathscr{F}_{t}$-adapted, jointly measurable process such
that for every $(t,x)\in[0,\infty)\times{\mathbb{R}}$,

$\displaystyle
u(t,x)=\int_{\mathbb{R}}G(t,x-y)u_{0}(y)dy+\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy),$
(2.1)

where $G$ is the heat kernel

$G(t,x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{x^{2}}{4t}}$

and last integral in (2.1) is a stochastic integral which is interpreted in
the sense of Walsh [18]. The existence and uniqueness of mild solutions to
(1.1) is well known; see [18, Chapter 3] for the case of $d=1$, and the proof
of which extends directly to $d\geq 1$. The solution $u$ has a continuous
version on $(0,\infty)\times{\mathbb{R}}$ (see [18]) and we will work with
this continuous version.

The following is the main result of this paper. In particular, it shows that
all points are polar in the critical dimension $d=6$, and consequently, points
are polar if and only if $d\geq 6$. This solves an open problem of Dalang,
Khoshnevisan and Nualart [5, 6] and Dalang, Mueller and Xiao [9].

######  Theorem 2.4.

If $d\geq 6$, then every fixed point is polar, i.e., for every
$z\in{\mathbb{R}}^{d}$,

$\mathbb{P}\\{u(t,x)=z\text{ for some }t>0\text{ and }x\in{\mathbb{R}}\\}=0.$

The next result gives a sufficient condition for a compact set
$S\subset{\mathbb{R}}^{d}$ to be polar. It extends Theorem 2.3 of [11] to the
non-Gaussian random field
$u=\\{u(t,x),(t,x)\in[0,\infty)\times{\mathbb{R}}\\}$ and improves
significantly the polarity condition in terms of Hausdorff measure (cf. (1.2)
above) of Dalang, Khoshnevisan and Nualart [5, 6].

######  Theorem 2.5.

Suppose $d\geq 6$. Let $S\subset{\mathbb{R}}^{d}$ be a compact set that
satisfies the following condition: There exist constants $r_{0}>0$, $C>0$,
$\kappa>0$ and a function $f:(0,r_{0}]\to(0,\infty)$ such that

$\displaystyle\lambda_{d}(S^{(r)})\leq f(r)\quad\text{for all }r\in(0,r_{0}],$
(2.2)

where $\lambda_{d}$ denotes the Lebesgue measure on ${\mathbb{R}}^{d}$,

$\displaystyle S^{(r)}=\left\\{x\in{\mathbb{R}}^{d}:\inf_{y\in S}|x-y|\leq
r\right\\}$

denotes the $r$-neighborhood of $S$, and $f$ satisfies

$\displaystyle\frac{f(s)}{f(r)}\leq
C\left(\frac{s}{r}\right)^{\kappa}\quad\text{for }0<r<s\leq r_{0}.$ (2.3)

Suppose that

$\displaystyle\lim_{r\to
0^{+}}\frac{r^{6}}{f(r(\log\log(1/r))^{-1/6})}=\infty.$ (2.4)

Then $S$ is polar for $u$, i.e.,

$\displaystyle{\mathbb{P}}\big{\\{}u(t,x)\in S\text{ for some }t>0\text{ and
}x\in{\mathbb{R}}\big{\\}}=0.$

######  Remark 2.6.

Let $d>6$ and $S$ be an $n$-dimensional interval in ${\mathbb{R}}^{d}$, where
$n=d-6$. Then for any $\varepsilon>0$,
$\operatorname{Cap}_{d-6+\varepsilon}(S)=0$ and
$\mathscr{H}_{d-6-\varepsilon}(S)=\infty$. The hitting probability bounds of
the form (1.2) (see [5, 6]) reduce to

$\displaystyle 0\leq{\mathbb{P}}\\{u(t,x)\in S\text{ for some }t>0\text{ and
}x\in{\mathbb{R}}\\}\leq\infty$

which is not informative for determining the polarity of $S$. Nevertheless,
this problem can be solved using our Theorem 2.5. Indeed, $S$ satisfies
condition (2.2) with $f(r)=r^{6}$ and hence condition (2.4), namely,

$\displaystyle\lim_{r\to
0^{+}}\frac{r^{6}}{f(r(\log\log(1/r))^{-1/6})}=\lim_{r\to
0^{+}}\log\log(1/r)=\infty.$

Therefore, $S$ is polar by Theorem 2.5. In fact, the theorem shows that when
$n\leq d-6$, any such $n$-dimensional interval $S$ is polar.

The arguments that lead to the above results also apply to the processes
$\\{u(t,x^{*}),t\geq 0\\}$ (where $x^{*}\in{\mathbb{R}}$ is fixed) and
$\\{u(t^{*},x),x\in{\mathbb{R}}\\}$ (where $t^{*}>0$ is fixed). In particular,
this solves the open problem on polarity of points for these processes in
their respective critical dimensions (see [5, 6]).

######  Theorem 2.7.

Fix $t^{*}>0$ and $x^{*}\in{\mathbb{R}}$.

  1. (i)

If $d\geq 4$, then for every $z\in{\mathbb{R}}^{d}$,
${\mathbb{P}}\\{u(t,x^{*})=z\text{ for some }t>0\\}=0$.

  2. (ii)

If $d\geq 2$, then for every $z\in{\mathbb{R}}^{d}$,
${\mathbb{P}}\\{u(t^{*},x)=z\text{ for some }x\in{\mathbb{R}}\\}=0$.

The rest of the paper is organized as follows. Sections 3 to 5 provide the
main ingredients for proving the main theorems. In Section 3, we recall some
notations including the stopping times and local decomposition of the solution
introduced in [9], and recall the probability bounds for the oscillations of
the processes. In Section 4, we introduce the crucial decomposition
$u=u^{(0)}+u^{(1)}+u^{(2)}$ of the solution. In Section 5, we use the
Malliavin calculus to prove a key probability estimate for $u^{(2)}$. The
proofs of Theorems 2.4, 2.5 and 2.7 are given in Sections 6, 7 and 8,
respectively.

##  3\. Oscillations of the solution

Consider the following metric on ${\mathbb{R}}^{2}$:

$\displaystyle
d((t,x),(s,y))=\Delta(t-s,x-y):=\max\\{|t-s|^{1/4},|x-y|^{1/2}\\}.$

For any $r>0$, $(t,x)\in{\mathbb{R}}^{2}$, define

$\displaystyle R_{r}(t,x)$
$\displaystyle=\\{(s,y)\in{\mathbb{R}}^{2}:\Delta(t-s,x-y)\leq r\\}$
$\displaystyle=[t-r^{4},t+r^{4}]\times[x-r^{2},x+r^{2}].$

Define the oscillation of an ${\mathbb{R}}^{d}$-valued function $f$ on a
rectangle $R\subset{\mathbb{R}}^{2}$ by

$\operatorname{osc}_{R}(f)=\sup_{(t,x),(s,y)\in R}|f(t,x)-f(s,y)|.$

Fix $t_{0}>0$, $x_{0}\in{\mathbb{R}}$ and $\delta\in(0,1)$. Let
$T_{0}=t_{0}+1$ and $I=[t_{0},T_{0}]\times[x_{0},x_{0}+1]$. We recall the
stopping times and the local decomposition of the solution introduced by
Dalang, Mueller and Xiao [9]. Let

$\displaystyle\tau_{K,1}=\inf\left\\{t\in[t_{0},T_{0}]:\exists\,s\in[t_{0},t),\exists\,x,y\in[x_{0},x_{0}+1],|u(t,x)-u(s,y)|\geq
K\Delta(t-s,x-y)^{1-\delta}\right\\}.$

If the set of $t\in[t_{0},T_{0}]$ above is empty, we understand that
$\tau_{K,1}=T_{0}$. We will also use the same convention below. Let
$\tilde{u}(t,x)=\tilde{u}_{K}(t,x)$ be a continuous version of

$\displaystyle\tilde{u}(t,x)=\int_{{\mathbb{R}}}G(t,x-y)u_{0}(y)dy+\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s\wedge\tau_{K,1},y))W(ds,dy).$

Define

$\displaystyle\tau_{K,2}=\inf\\{t\in[0,T_{0}]:\exists\,x\in{\mathbb{R}},|\tilde{u}(t,x)|\geq
K(1+|x|)\\}.$

Let $v(t,x)$ be a continuous version of

$\displaystyle
v(t,x)=\int_{{\mathbb{R}}}G(t,x-y)u_{0}(y)dy+\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)W(ds,dy).$

Let

$\displaystyle\tau_{K,3}=\inf\\{t\in[0,T_{0}]:\exists\,x\in{\mathbb{R}},|v(t,x)|\geq
K(1+|x|)\\}.$

Fix $\alpha\in(1/2,2/3)$ and $\beta\in(\alpha,2/3)$. For each
$(t_{1},x_{1})\in I$ and $\rho\in(0,t_{0}]$, denote
$t_{1}^{-}=t_{1}^{-}(\rho)=t_{1}-\rho^{4}-\rho^{4(1-\alpha)}$ and
$L_{1}=L_{1}(\rho)=\rho^{2}+\rho^{2(1-\beta)}$. Define the process
$w_{t_{1},x_{1},\rho}(t,x)$ by

$\displaystyle w_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle=\sigma(u(t_{1}^{-}\wedge\tau_{K,1},x_{1}))N^{(0)}(t,x)+N^{(1)}_{t_{1},x_{1},\rho}(t,x)+N^{(2)}_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle\quad-\sigma(u(t_{1}^{-}\wedge\tau_{K,1},x_{1}))\hat{v}^{(1)}_{t_{1},x_{1},\rho}(t,x)+\hat{u}_{t_{1},x_{1},\rho}(t,x),$

where

$\displaystyle N^{(0)}(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)W(ds,dy),$
$\displaystyle N^{(1)}_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle=\int_{t_{1}^{-}}^{t}\int_{x_{1}-L_{1}}^{x_{1}+L_{1}}G(t-s,x-y)[\sigma(u(s\wedge\tau_{K,1},y))-\sigma(u(t_{1}^{-}\wedge\tau_{K,1},x_{1}))]W(ds,dy),$
$\displaystyle N^{(2)}_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle=\int_{t_{1}^{-}}^{t}\int_{[x_{1}-L_{1},x_{1}+L_{1}]^{c}}G(t-s,x-y)[\sigma(u(s\wedge\tau_{K,1},y))-\sigma(u(t_{1}^{-}\wedge\tau_{K,1},x_{1}))]W(ds,dy),$

and

$\displaystyle\hat{u}_{t_{1},x_{1},\rho}(t,x)=\begin{cases}\tilde{u}_{t_{1},x_{1},\rho}(t,x)&\text{if
}t_{1}^{-}\leq\tau_{K,2},\\\ 0&\text{if }t_{1}^{-}>\tau_{K,2},\end{cases}$
$\displaystyle\hat{v}^{(1)}_{t_{1},x_{1},\rho}(t,x)=\begin{cases}v^{(1)}_{t_{1},x_{1},\rho}(t,x)&\text{if
}t_{1}^{-}\leq\tau_{K,3},\\\ 0&\text{if }t_{1}^{-}>\tau_{K,3},\end{cases}$

where

$\displaystyle\tilde{u}_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle=\int_{\mathbb{R}}G(t-t_{1}^{-},x-y)\tilde{u}(t_{1}^{-},y)dy,$
$\displaystyle v^{(1)}_{t_{1},x_{1},\rho}(t,x)$
$\displaystyle=\int_{0}^{t_{1}^{-}}\int_{\mathbb{R}}G(t-s,x-y)W(ds,dy).$

In [9], it is shown that

$\displaystyle\lim_{K\to\infty}{\mathbb{P}}\\{\tau_{K,i}<T_{0}\\}=0,\quad
i=1,2,3.$

Also, on the event $\\{\tau_{K,1}=T_{0}\\}$, $u(t,x)=\tilde{u}(t,x)$ for all
$(t,x)\in[0,T_{0}]\times{\mathbb{R}}$, and on
$\\{\tau_{K,2}\wedge\tau_{K,3}=T_{0}\\}$,
$\tilde{u}(t,x)=w_{t_{1},x_{1},\rho}(t,x)$ for all
$(t,x)\in[t_{1}^{-},\infty)\times{\mathbb{R}}$. Let
$\tau_{K}=\min\\{\tau_{K,1},\tau_{K,2},\tau_{K,3}\\}$. Then

$\displaystyle\lim_{K\to\infty}{\mathbb{P}}\\{\tau_{K}<T_{0}\\}=0.$

Since $\tau_{K}$ is a nondecreasing function of $K$, it follows that

$\displaystyle{\mathbb{P}}\left(\bigcup_{K=1}^{\infty}\\{\tau_{K}=T_{0}\\}\right)=\lim_{K\to\infty}{\mathbb{P}}\\{\tau_{K}=T_{0}\\}=1.$
(3.1)

Moreover,

$\displaystyle\begin{split}\text{on
}\\{\tau_{K}=T_{0}\\},\quad&u(t,x)=\tilde{u}(t,x)\text{ for all }(t,x)\in
I,\text{ and}\\\ &\tilde{u}(t,x)=w_{t_{1},x_{1},\rho}(t,x)\text{ for all
}(t,x)\in[t_{1}^{-},T_{0}]\times{\mathbb{R}}.\end{split}$ (3.2)

Let us recall the probability bounds for the local oscillation of
$w_{t_{1},x_{1},r}$ and global oscillation of $\tilde{u}$, which were proved
in [9].

######  Proposition 3.1.

[9, Theorem 5.2] Under Assumption 2.1, there exist constants $K_{1}<\infty$
and $\rho_{0}\in(0,1/2)$ such that for all $r_{0}\in(0,\rho_{0})$, for all
$(t_{1},x_{1})\in I$, we have

$\displaystyle{\mathbb{P}}\left\\{\exists\,r\in[r_{0}^{2},r_{0}],\operatorname{osc}_{R_{r}(t_{1},x_{1})}(w_{t_{1},x_{1},r})\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}\right\\}\geq
1-2\exp(-(\log(1/r_{0}))^{1/2}).$

Define an “anisotropic dyadic rectangle” of order $p$ as a rectangle of the
form

$\displaystyle
R=[m_{1}2^{-4p},(m_{1}+1)2^{-4p}]\times[m_{2}2^{-2p},(m_{2}+1)2^{-2p}],$

where $m_{1},m_{2}$ are integers. Let $\mathcal{R}_{p}$ denote the collection
of all anisotropic dyadic rectangles $R$ of order $p$ that intersect $I$.

######  Proposition 3.2.

[9, Lemma 7.1] Under Assumption 2.1, there exist positive finite constants
$K_{2}$, $C_{1}$ and $C_{2}$ such that for all $p\geq 1$, we have

$\displaystyle{\mathbb{P}}\left\\{\forall
R\in\mathcal{R}_{2p},\operatorname{osc}_{R}(\tilde{u})\leq
K_{2}2^{-2p}p\right\\}\geq 1-C_{1}\exp(-C_{2}K_{2}^{2}p^{2}).$

##  4\. Decomposition of the solution

Consider the $\sigma$-algebras

$\displaystyle\begin{split}\mathscr{F}^{1}&=\sigma\\{W_{s}(A)-W_{t_{0}/2}(A):s\in[t_{0}/2,\infty),A\in\mathscr{B}_{b}({\mathbb{R}})\\}\vee\mathcal{N},\\\
\mathscr{F}^{2}&=\sigma\\{W_{s}(A):s\in[0,t_{0}/2],A\in\mathscr{B}_{b}({\mathbb{R}})\\}\vee\mathcal{N},\end{split}$
(4.1)

where $\mathscr{B}_{d}({\mathbb{R}})$ is the collection of all bounded Borel
sets in ${\mathbb{R}}$, $W_{s}(A)$ is the martingale measure associated with
the noise $\dot{W}(t,x)$ in the sense of Walsh [18], and $\mathcal{N}$ is the
$\sigma$-algebra generated by all ${\mathbb{P}}$-null sets. Then
$\mathscr{F}^{1}$ and $\mathscr{F}^{2}$ are independent. By [2, Theorem
3.3.1(j)], this implies that for any random variable $X\in
L^{2}(\Omega,\mathscr{F},{\mathbb{P}})$,

$\displaystyle{\mathbb{E}}[X|\sigma(\mathscr{F}^{1}\cup\mathscr{F}^{2})]={\mathbb{E}}[X|\mathscr{F}^{1}]+{\mathbb{E}}[X|\mathscr{F}^{2}]-{\mathbb{E}}[X]\quad\text{a.s.}$
(4.2)

Recall that the mild solution to (1.1) is given by

$\displaystyle
u(t,x)=\int_{\mathbb{R}}G(t,x-y)u_{0}(y)dy+\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy).$

For any $(t,x)\in I$, $u(t,x)$ is
$\sigma(\mathscr{F}^{1}\cup\mathscr{F}^{2})$-measurable. By (4.2), we can
decompose the solution as

$\displaystyle u(t,x)=u^{(0)}(t,x)+u^{(1)}(t,x)+u^{(2)}(t,x),$ (4.3)

where

$\displaystyle u^{(0)}(t,x)$
$\displaystyle=\int_{\mathbb{R}}G(t,x-y)u_{0}(y)dy,$ $\displaystyle
u^{(1)}(t,x)$
$\displaystyle={\mathbb{E}}\left[\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy)\Big{|}\mathscr{F}^{1}\right],$
$\displaystyle u^{(2)}(t,x)$
$\displaystyle={\mathbb{E}}\left[\int_{0}^{t}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy)\Big{|}\mathscr{F}^{2}\right].$

Then, $\\{u^{(1)}(t,x),(t,x)\in I\\}$ and $\\{u^{(2)}(t,x),(t,x)\in I\\}$ are
measurable with respect to $\mathscr{F}^{1}$ and $\mathscr{F}^{2}$
respectively, hence they are two independent processes. Moreover, the
expression for $u^{(2)}(t,x)$ can be simplified. For any fixed $(\tau,x)\in
I$, let

$\displaystyle
M_{t}=\int_{0}^{t}\int_{\mathbb{R}}G(\tau-s,x-y)\sigma(u(s,y))W(ds,dy).$

By the property of the Walsh integral [18, Theorem 2.5], $\\{M_{t},t\geq 0\\}$
is a martingale with respect to the natural filtration $\mathscr{F}_{t}$ (see,
e.g., [3, Proposition 3.1] for the predictability condition). Therefore,

$\displaystyle u^{(2)}(t,x)$
$\displaystyle=\int_{0}^{t_{0}/2}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy).$
(4.4)

An important regularity property is that $u^{(2)}$ is relatively smooth
compared to $u^{(1)}$. The following proposition verifies that $u^{(2)}(t,x)$
is locally almost Lipschitz a.s.

######  Proposition 4.1.

For each $p\geq 2$, there exists a constant $C_{p}<\infty$ such that for all
$(t,x),(t^{\prime},x^{\prime})\in I$,

$\displaystyle\mathbb{E}(|u^{(2)}(t,x)-u^{(2)}(t^{\prime},x^{\prime})|^{p})\leq
C_{p}(|t-t^{\prime}|+|x-x^{\prime}|)^{p}.$ (4.5)

Consequently, for each fixed $\delta\in(0,1)$, $(t,x)\mapsto u^{(2)}(t,x)$ is
a.s. $(1-\delta)$-Hölder continuous on $I$. In particular, there exists an
$\mathscr{F}^{2}$-measurable random variable $Z$ with all moments such that

$\displaystyle{\mathbb{P}}\left\\{|u^{(2)}(t,x)-u^{(2)}(t^{\prime},x^{\prime})|\leq
Z(|t-t^{\prime}|+|x-x^{\prime}|)^{1-\delta}\text{ for all
}(t,x),(t^{\prime},x^{\prime})\in I\right\\}=1.$ (4.6)

###### Proof.

By the Burkholder–Davis–Gundy inequality and Assumption 2.1(b) that $\sigma$
is bounded,

$\displaystyle{\mathbb{E}}(|u^{(2)}(t,x)-u^{(2)}(t^{\prime},x^{\prime})|^{p})$
$\displaystyle\leq
C_{p}\sum_{i=1}^{d}{\mathbb{E}}\left[\left(\sum_{j=1}^{d}\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|(G(t-s,x-y)-G(t^{\prime}-s,x^{\prime}-y))\sigma_{ij}(u(s,y))|^{2}\right)^{p/2}\right]$
$\displaystyle\leq
C_{p}\sigma_{1}^{d}d^{1+p/2}\left(\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|G(t-s,x-y)-G(t^{\prime}-s,x^{\prime}-y)|^{2}\right)^{p/2}.$

By the mean value theorem, there exists $\eta=\eta(t,t^{\prime},s,y)$ taking
value between $t-s$ and $t^{\prime}-s$ such that

$\displaystyle
G(t-s,y)-G(t^{\prime}-s,y)=\frac{y^{2}-2\eta}{8\sqrt{\pi}\,\eta^{5/2}}e^{-\frac{y^{2}}{4\eta}}(t-t^{\prime}).$

In particular, $t_{0}/2\leq\eta\leq t_{0}+1$ whenever
$t,t^{\prime}\in[t_{0},t_{0}+1]$ and $s\in[0,t_{0}/2]$. It follows that

$\displaystyle\begin{split}&\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|G(t-s,x-y)-G(t^{\prime}-s,x-y)|^{2}\\\
&=\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|G(t-s,y)-G(t^{\prime}-s,y)|^{2}\\\
&\leq\frac{1}{64\pi(t_{0}/2)^{5}}|t-t^{\prime}|^{2}\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,\left(y^{2}+2(t_{0}+1)\right)^{2}e^{-\frac{y^{2}}{2(t_{0}+1)}}\\\
&\leq C|t-t^{\prime}|^{2}.\end{split}$ (4.7)

Also, by the mean value theorem, there exists
$\xi=\xi(t^{\prime},s,x,x^{\prime},y)$ taking value between $0$ and
$x-x^{\prime}$ such that

$G(t^{\prime}-s,y+x-x^{\prime})-G(t^{\prime}-s,y)=-\frac{y+\xi}{4\sqrt{\pi}(t^{\prime}-s)^{3/2}}e^{-\frac{(y+\xi)^{2}}{4(t^{\prime}-s)}}(x-x^{\prime}).$

Since $x,x^{\prime}\in[x_{0},x_{0}+1]$, we have $|\xi|\leq 1$. Also, using the
elementary inequality $(y+\xi)^{2}\geq\frac{2}{3}y^{2}-2\xi^{2}$, we get that

$e^{-\frac{(y+\xi)^{2}}{2(t^{\prime}-s)}}\leq
e^{-\frac{y^{2}}{3(t_{0}+1)}}e^{\frac{1}{t_{0}/2}}.$

It follows that

$\displaystyle\begin{split}&\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|G(t^{\prime}-s,x-y)-G(t^{\prime}-s,x^{\prime}-y)|^{2}\\\
&=\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,|G(t^{\prime}-s,y+x-x^{\prime})-G(t^{\prime}-s,y)|^{2}\\\
&\leq\frac{1}{16\pi(t_{0}/2)^{3}}|x-x^{\prime}|^{2}\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,(|y|+1)^{2}e^{-\frac{y^{2}}{3(t_{0}+1)}}e^{\frac{1}{t_{0}/2}}\\\
&\leq C|x-x^{\prime}|^{2}.\end{split}$ (4.8)

Combining (4.7) and (4.8), we get (4.5).

Finally, for any $\delta\in(0,1)$, the a.s. $(1-\delta)$-Hölder continuity of
$(t,x)\mapsto u^{(2)}(t,x)$ follows from (4.5) and Kolmogorov’s continuity
theorem (see, e.g., [10, Theorem 4.3]), and hence (4.6) can be obtained by
taking

$Z=\sup_{\begin{subarray}{c}(t,x),(t^{\prime},x^{\prime})\in I,\\\
(t,x)\neq(t^{\prime},x^{\prime})\end{subarray}}\frac{|u^{(2)}(t,x)-u^{(2)}(t^{\prime},x^{\prime})|}{(|t-t^{\prime}|+|x-x^{\prime}|)^{1-\delta}},$

which is $\mathscr{F}^{2}$-measurable and has finite moments of all orders.
The proof of Proposition 4.1 is complete. ∎

##  5\. Malliavin calculus and probability estimate for $u^{(2)}$

The goal of this section is to prove the following probability estimate for
$u^{(2)}(t,x)$:

$\displaystyle{\mathbb{P}}\\{|u^{(2)}(t,x)-a|\leq r\\}\leq Cr^{d}.$

We will prove this by showing that $u^{(2)}(t,x)$ has a density function
$p_{t,x}(z)$ that is uniformly bounded over $z\in{\mathbb{R}}^{d}$ and
$(t,x)\in I$. This result is given in Proposition 5.7 below and its proof
relies on Malliavin calculus. We first recall some basic elements of Malliavin
calculus following Nualart [15].

Let $\mathcal{H}=L^{2}([0,T]\times{\mathbb{R}},{\mathbb{R}}^{d})$, where
$T=t_{0}+1$, be the Hilbert space of $L^{2}$-integrable functions on
$[0,T]\times{\mathbb{R}}$ with values in ${\mathbb{R}}^{d}$. Let $\mathcal{S}$
denote the set of smooth random variables of the form
$F=f(W(h_{1}),\dots,W(h_{n}))$, where $n\geq 1$,
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a smooth function whose partial
derivatives all have at most polynomial growth, $h_{i}\in\mathcal{H}$, and
$W(h)$ is defined by the Wiener integral

$W(h)=\int_{0}^{T}\int_{\mathbb{R}}h(t,x)\cdot W(dt,dx).$

For $F\in\mathcal{S}$, its derivative is defined as the
${\mathbb{R}}^{d}$-valued stochastic process

$DF=\\{D_{t,x}F=(D_{t,x}^{(1)}F,\dots,D_{t,x}^{(d)}F);\,t\in[0,T],x\in{\mathbb{R}}\\}$

given by

$D_{t,x}F=\sum_{i=1}^{n}\frac{\partial f}{\partial
x_{i}}(W(h_{1}),\dots,W(h_{n}))h_{i}(t,x).$

The $k$-th derivative $D^{k}F$ of $F$ is defined by

$D^{k}_{y}F=\sum_{i_{1},\dots,i_{k}=1}^{n}\frac{\partial}{\partial
x_{i_{1}}}\dots\frac{\partial}{\partial
x_{i_{k}}}f(W(h_{1}),\dots,W(h_{n}))h_{i_{1}}(y_{1})\otimes\dots\otimes
h_{i_{k}}(y_{k}),$

where $y=(y_{1},\dots,y_{k})$ and $y_{i}=(t_{i},x_{k})$, $1\leq i\leq k$, and
$D^{k}F$ is an $\mathcal{H}^{\otimes k}$-valued random variable.

For $p\geq 1$, $k\geq 1$, the space ${\mathbb{D}}^{k,p}$ is defined as the
closure of $\mathcal{S}$ with respect to the norm $\|\cdot\|_{k,p}$ given by

$\|F\|_{k,p}^{p}={\mathbb{E}}(|F|^{p})+\sum_{i=1}^{k}{\mathbb{E}}(\|D^{i}F\|^{p}_{\mathcal{H}^{\otimes
i}}),$

where

$\|D^{i}F\|_{\mathcal{H}^{\otimes
i}}^{2}=\sum_{k_{1},\dots,k_{i}=1}^{d}\int_{0}^{T}ds_{1}\int_{\mathbb{R}}dy_{1}\cdots\int_{0}^{T}ds_{i}\int_{\mathbb{R}}dy_{i}\left(D^{(k_{1})}_{s_{1},y_{1}}\dots
D^{(k_{i})}_{s_{i},y_{i}}F\right)^{2}.$

Define ${\mathbb{D}}^{\infty}=\bigcap_{k\geq 1}\bigcap_{p\geq
1}{\mathbb{D}}^{k,p}$.

The adjoint of the derivative operator $D$ on $L^{2}(\Omega)$ is called the
divergence operator $\delta$, which is an unbounded operator on
$L^{2}(\Omega;\mathcal{H})$ with values in $L^{2}(\Omega)$ and its domain
$\operatorname{Dom}\delta$ is the set of all $u\in L^{2}(\Omega;\mathcal{H})$
such that there exists a constant $c$ such that

$|{\mathbb{E}}(\langle DF,u\rangle_{\mathcal{H}})|\leq c\|F\|_{2}$

for all $F\in{\mathbb{D}}^{1,2}$. If $u\in\operatorname{Dom}\delta$, then
$\delta(u)$ is the element in $L^{2}(\Omega)$ characterized by the relation

${\mathbb{E}}(F\delta(u))={\mathbb{E}}(\langle DF,u\rangle_{\mathcal{H}})$

for all $F\in{\mathbb{D}}^{1,2}$. Each element $u\in\operatorname{Dom}\delta$
is a square integrable ${\mathbb{R}}^{d}$-valued process and $\delta(u)$ is
the Skorohod stochastic integral of $u$. In particular, if $u$ is an adapted
process, then $\delta(u)$ coincides with the Ito-Walsh stochastic integral:

$\delta(u)=\int_{0}^{T}\int_{\mathbb{R}}u(t,x)\cdot W(dt,dx).$

Recall from (4.4) that for $(t,x)\in I$,

$\displaystyle
u^{(2)}(t,x)=\int_{0}^{t_{0}/2}\int_{\mathbb{R}}G(t-s,x-y)\sigma(u(s,y))W(ds,dy).$

The key to proving Proposition 5.7 below is to show that $u^{(2)}(t,x)$ has a
density function $p_{t,x}(z)$ which is uniformly bounded over
$z\in{\mathbb{R}}^{d}$ and $(t,x)\in I$. We will follow the Malliavin calculus
approach of Dalang, Khoshnevisan and Nualart [5, 6], in which this property
was proved for the solutions $u(t,x)$ of the nonlinear stochastic heat
equations with spatial domain $[0,1]$ and ${\mathbb{R}}^{k}$, respectively.
The following result is Proposition 3.4 in [5]:

######  Proposition 5.1.

Let $F\in({\mathbb{D}}^{\infty})^{d}$ and $\gamma_{F}$ be the Malliavin matrix
of $F$ defined by

$\gamma_{F}=(\langle DF_{i},DF_{j}\rangle_{\mathcal{H}})_{1\leq i,j\leq d}.$

Suppose there exist finite numbers $c_{1}(p)>0$ and $c_{2}(k,p)\geq 0$ such
that

  1. (a)

${\mathbb{E}}[(\det\gamma_{F})^{-p}]\leq c_{1}(p)$ for all $p\geq 1$, and

  2. (b)

${\mathbb{E}}(\|D^{k}F_{i}\|^{p}_{\mathcal{H}^{\otimes k}})\leq c_{2}(k,p)$
for all $k\geq 1$, $p\geq 1$, $i=1,\dots,d$.

Then $F$ has a uniformly bounded density, and the bound does not depend on $F$
but only on $c_{1}(p)$ and $c_{2}(k,p)$.

We first establish some lemmas. Then, we verify conditions (a) and (b) in
Proposition 5.1 for the process $\\{u^{(2)}(t,x);(t,x)\in I\\}$ in Lemmas 5.5
and 5.6 below.

######  Lemma 5.2.

For any $t>0$, $x\in{\mathbb{R}}$, $0\leq a\leq b\leq t$ and $0\leq r\leq t$,

$\displaystyle\int_{\mathbb{R}}dy\,G^{2}(t-r,x-y)=\frac{1}{2\sqrt{2\pi}}(t-r)^{-1/2}.$
(5.1)

and

$\displaystyle\int_{a}^{b}ds\int_{\mathbb{R}}dy\,G^{2}(t-s,x-y)=\frac{1}{\sqrt{2\pi}}\left(\sqrt{t-a}-\sqrt{t-b}\right)=\frac{b-a}{\sqrt{2\pi}(\sqrt{t-a}+\sqrt{t-b})}.$
(5.2)

Consequently, for any $t_{0}>0$, there exist positive finite constants $C$ and
$C^{\prime}$ depending on $t_{0}$ such that for all
$(t,x)\in[t_{0},t_{0}+1]\times{\mathbb{R}}$ and $0\leq a\leq b\leq t_{0}/2$,

$\displaystyle
C^{\prime}(b-a)\leq\int_{a}^{b}ds\int_{\mathbb{R}}dy\,G^{2}(t-s,x-y)\leq
C(b-a).$ (5.3)

###### Proof.

The identities (5.1) and (5.2) can be easily shown by direct computation and
the identity $\sqrt{u}-\sqrt{v}=(u-v)/(\sqrt{u}+\sqrt{v})$. The last statement
follows immediately from (5.2). ∎

######  Lemma 5.3.

For any $(t,x)\in[0,T]\times{\mathbb{R}}$, we have
$u(t,x)\in({\mathbb{D}}^{\infty})^{d}$ with the property that for any $k\geq
1$, $p\geq 1$, $T>0$, there exists a finite number $C=C(k,p,T)$ depending on
$k$, $p$ and $T$ such that

$\displaystyle\sup_{1\leq i\leq
d}\sup_{(t,x)\in[0,T]\times{\mathbb{R}}}{\mathbb{E}}\left(\|D^{k}u_{i}(t,x)\|^{p}_{\mathcal{H}^{\otimes
k}}\right)\leq C,$ (5.4)

and there exists a finite number $C^{\prime}=C^{\prime}(p,T)$ such that for
any $0\leq a\leq b\leq t$,

$\displaystyle\sup_{1\leq l\leq d}\sup_{1\leq i\leq
d}\sup_{(t,x)\in[0,T]\times{\mathbb{R}}}{\mathbb{E}}\left[\left(\int_{a}^{b}ds\int_{\mathbb{R}}dy\left(D^{(l)}_{s,y}u_{i}(t,x)\right)^{2}\right)^{p}\,\right]\leq
C^{\prime}(b-a)^{p/2}.$ (5.5)

###### Proof.

When $d=1$, this result follows from Proposition 2.4 and Lemma 2.5 of [12].
The case of $d>1$ can be proved in a similar way, so we omit its proof. ∎

We recall the following Burkholder’s inequality for Hilbert space-valued
martingales [13, E.2. p.212]; see also [5, Lemma 7.6] and [1, (4.18)].

######  Lemma 5.4.

Let $H_{s,y}$ be a predictable
$L^{2}(([0,T]\times{\mathbb{R}})^{m},d\alpha)$-valued process, where $m\geq 1$
and $d\alpha$ denotes the Lebesgue measure. Then, for any $p\geq 1$, there
exists a constant $C_{p}$ such that

$\displaystyle{\mathbb{E}}\left|\int_{{([0,T]\times{\mathbb{R}})}^{m}}\left(\int_{0}^{T}\int_{\mathbb{R}}H_{s,y}(\alpha)W(ds,dy)\right)^{2}d\alpha\right|^{p}\leq
C_{p}\,{\mathbb{E}}\left|\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{{([0,T]\times{\mathbb{R}})}^{m}}H_{s,y}^{2}(\alpha)d\alpha\right)dsdy\right|^{p}.$

######  Lemma 5.5.

For every $(t,x)\in I$, we have $u^{(2)}(t,x)\in(\mathbb{D}^{\infty})^{d}$
with

$\displaystyle\begin{split}&D^{(m_{1})}_{s_{1},y_{1}}\cdots
D^{(m_{k})}_{s_{k},y_{k}}u^{(2)}_{i}(t,x)\\\ &=\sum_{l=1}^{k}{\bf
1}_{[0,t_{0}/2]}(s_{l})G(t-s_{l},x-y_{l})D^{(m_{1})}_{s_{1},y_{1}}\cdots
D^{(m_{l-1})}_{s_{l-1},y_{l-1}}D^{(m_{l+1})}_{s_{l+1},y_{l+1}}\cdots
D^{(m_{k})}_{s_{k},y_{k}}(\sigma_{i,m_{l}}(u(s_{l},y_{l})))\\\
&\quad+\sum_{j=1}^{d}\int_{s_{1}\vee\dots\vee
s_{k}}^{t_{0}/2}\int_{\mathbb{R}}G(t-r,x-z)\prod_{l=1}^{k}D^{(m_{l})}_{s_{l},y_{l}}(\sigma_{ij}(u(r,z)))W_{j}(dr,dz)\end{split}$
(5.6)

if $s_{1}\vee\dots\vee s_{k}\leq t_{0}/2$; and
$D^{(m_{1})}_{s_{1},y_{1}}\cdots D^{(m_{k})}_{s_{k},y_{k}}u^{(2)}_{i}(t,x)=0$
if $s_{1}\vee\dots\vee s_{k}>t_{0}/2$. Moreover, for any $k\geq 1$ and $p\geq
1$, there exists a finite number $C=C(k,p,t_{0})$ depending on $k$, $p$ and
$t_{0}$ such that

$\displaystyle\sup_{1\leq i\leq d}\sup_{(t,x)\in
I}{\mathbb{E}}\left(\|D^{k}u_{i}^{(2)}(t,x)\|_{\mathcal{H}^{\otimes
k}}^{p}\right)\leq C(k,p,t_{0}).$ (5.7)

###### Proof.

The formula (5.6) can be verified by applying the identity
$D_{h}\delta(u)=\delta(D_{h}u)+\langle h,u\rangle_{\mathcal{H}}$ to (4.4)
inductively (see Proposition 1.3.3 and Corollary 1.2.1 of [15]). Finally, we
can use Lemma 5.2, the chain rule, Assumption 2.3 and (5.4) to deduce (5.7)
from (5.6), where the integrals in the last sum can be handled by using
Burkholder’s inequality in Lemma 5.4. ∎

######  Lemma 5.6.

Let $\gamma_{t,x}$ denote the Malliavin matrix of $u^{(2)}(t,x)$ defined by

$\displaystyle\gamma_{t,x}=\left(\langle
Du^{(2)}_{i}(t,x),Du^{(2)}_{j}(t,x)\rangle_{\mathcal{H}}\right)_{1\leq i,j\leq
d}.$

Then, for any $p\geq 1$, there exists a finite number $C$ depending on $p$,
$d$, $\sigma_{0}$ and $t_{0}$ such that

$\displaystyle\sup_{(t,x)\in I}{\mathbb{E}}[(\det\gamma_{t,x})^{-p}]\leq C.$

###### Proof.

It is known that the smallest eigenvalue of any $d\times d$ real symmetric
matrix $A$ is equal to

$\displaystyle\inf_{\xi\in{\mathbb{R}}^{d},|\xi|=1}\xi^{\top}A\xi.$

This implies that

$\displaystyle\det\gamma_{t,x}\geq\left(\inf_{\xi\in{\mathbb{R}}^{d},|\xi|=1}\xi^{\top}\gamma_{t,x}\xi\right)^{d}.$

By the elementary inequality $(a+b)^{2}\geq\frac{2}{3}a^{2}-2b^{2}$ and Lemma
5.5 with $k=1$, for any $\epsilon\in(0,1/2)$, we have

$\displaystyle\xi^{\top}\gamma_{t,x}\xi$
$\displaystyle=\int_{0}^{t_{0}/2}ds\int_{\mathbb{R}}dy\left|\sum_{i=1}^{d}D_{s,y}u^{(2)}_{i}(t,x)\xi_{i}\right|^{2}$
$\displaystyle\geq\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\left|\sum_{i=1}^{d}D_{s,y}u^{(2)}_{i}(t,x)\xi_{i}\right|^{2}$
$\displaystyle\geq I_{1}(\epsilon)-I_{2}(\epsilon),$

where

$\displaystyle I_{1}(\epsilon)$
$\displaystyle=\frac{2}{3}\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\sum_{j=1}^{d}\left(\sum_{i=1}^{d}G(t-s,x-y)\sigma_{ij}(u(s,y))\xi_{i}\right)^{2},$
$\displaystyle I_{2}(\epsilon)$
$\displaystyle=2\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\sum_{j=1}^{d}\left(\sum_{i=1}^{d}\sum_{k=1}^{d}\xi_{i}\int_{s}^{t_{0}/2}\int_{\mathbb{R}}G(t-r,x-z)D^{(k)}_{s,y}(\sigma_{ij}(u(r,z)))W_{k}(dr,dz)\right)^{2}.$

Note that $I_{1}$ and $I_{2}$ also depend on $t$ and $x$ but for for
simplicity, we omit this dependence from the notation. By Assumption 2.2 and
Lemma 5.2,

$\displaystyle\begin{split}\inf_{|\xi|=1}I_{1}(\epsilon)&\geq\frac{2}{3}\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,G^{2}(t-s,x-y)\inf_{|\xi|=1}\sum_{j=1}^{d}\left(\sum_{i=1}^{d}\sigma_{ij}(u(s,y))\xi_{i}\right)^{2}\\\
&\geq\frac{2}{3}\sigma_{0}\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\,G^{2}(t-s,x-y)\\\
&\geq c_{0}\epsilon,\end{split}$ (5.8)

where $c_{0}=(\sigma_{0}^{2}t_{0})/(6\sqrt{2\pi(t_{0}+1)})$. On the other
hand, for any $q\geq 1$, by the Cauchy–Schwarz inequality,

$\displaystyle{\mathbb{E}}\left(\sup_{|\xi|=1}|I_{2}(\epsilon)|^{q}\right)$
$\displaystyle\leq
C\sum_{i,j,k=1}^{d}{\mathbb{E}}\left|\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\left(\int_{s}^{t_{0}/2}\int_{\mathbb{R}}G(t-r,x-z)D_{s,y}^{(k)}(\sigma_{ij}(u(r,z)))W_{k}(dr,dz)\right)^{2}\right|^{q}.$

Then, by Burkholder’s inequality (Lemma 5.4), Assumption 2.3 and Hölder’s
inequality with respect to the measure $G^{2}(t-r,x-z)drdz$, we have

$\displaystyle{\mathbb{E}}\left(\sup_{|\xi|=1}|I_{2}(\epsilon)|^{q}\right)$
$\displaystyle\leq
C\left(\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}dr\int_{\mathbb{R}}dz\,G^{2}(t-r,x-z)\right)^{q-1}$
$\displaystyle\quad\times\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}dr\int_{\mathbb{R}}dz\,G^{2}(t-r,x-z)\sum_{k=1}^{d}{\mathbb{E}}\left(\int_{t_{0}(1/2-\epsilon)}^{t_{0}/2}ds\int_{\mathbb{R}}dy\left(\sum_{l=1}^{d}D^{(k)}_{s,y}u_{l}(r,z)\right)^{2}\right)^{q}.$

This, together with (5.5) and Lemma 5.2, implies that

$\displaystyle\sup_{(t,x)\in
I}{\mathbb{E}}\left(\sup_{|\xi|=1}|I_{2}(\epsilon)|^{q}\right)\leq
C(t_{0}\epsilon)^{3q/2},$ (5.9)

where $C$ is a constant depending on $d$ and $q$. Using the formula
${\mathbb{E}}[Z^{p}]=\int_{0}^{\infty}py^{p-1}\mathbb{P}\\{Z>y\\}dy$ for
$Z\geq 0$ and $p\geq 1$, we get that

$\displaystyle{\mathbb{E}}[(\det\gamma_{t,x})^{-p}]$
$\displaystyle\leq{\mathbb{E}}\left[\left(\inf_{|\xi|=1}\xi^{\top}\gamma_{t,x}\xi\right)^{-pd}\,\right]$
$\displaystyle=\int_{0}^{\infty}pd\,y^{pd-1}\,{\mathbb{P}}\left\\{\left(\inf_{|\xi|=1}\xi^{\top}\gamma_{t,x}\xi\right)^{-1}>y\right\\}\,dy.$

For any $y\geq 1$, take $\epsilon=\epsilon(y)=2/(c_{0}y)$ so that
$c_{0}\epsilon-1/y=1/y$. Then, by (5.8), for any $y\geq 1$,

$\displaystyle{\mathbb{P}}\left\\{\left(\inf_{|\xi|=1}\xi^{\top}\gamma_{t,x}\xi\right)^{-1}>y\right\\}$
$\displaystyle\leq{\mathbb{P}}\left\\{\sup_{|\xi|=1}I_{2}(\epsilon)>\inf_{|\xi|=1}I_{1}(\epsilon)-1/y\right\\}$
$\displaystyle\leq{\mathbb{P}}\left\\{\sup_{|\xi|=1}I_{2}(\epsilon)>1/y\right\\}.$

Take $q=pd$. Then, by Markov’s inequality and (5.9), we have

$\displaystyle\sup_{(t,x)\in I}{\mathbb{E}}[(\det\gamma_{t,x})^{-p}]$
$\displaystyle\leq 1+pd\int_{1}^{\infty}y^{pd-1}y^{q}\sup_{(t,x)\in
I}{\mathbb{E}}\left(\sup_{|\xi|=1}|I_{2}(\epsilon)|^{q}\right)dy$
$\displaystyle\leq
1+Cpd(2t_{0}/c_{0})^{3pd/2}\int_{1}^{\infty}y^{\frac{pd}{2}-1}dy$
$\displaystyle=C_{p,d,\sigma_{0},t_{0}}<\infty.$

This completes the proof of Lemma 5.6. ∎

######  Proposition 5.7.

Under Assumptions 2.2 and 2.3, $u^{(2)}(t,x)$ has a uniformly bounded density
function $p_{t,x}(z)$ such that

$\displaystyle\sup_{(t,x)\in I}\sup_{z\in{\mathbb{R}}^{d}}p_{t,x}(z)<\infty.$
(5.10)

Consequently, there exists a positive finite constant $C$ such that for all
$(t,x)\in I$, $a\in{\mathbb{R}}^{d}$, and all $r>0$,

$\displaystyle{\mathbb{P}}\\{|u^{(2)}(t,x)-a|\leq r\\}\leq Cr^{d}.$ (5.11)

###### Proof.

Lemmas 5.5 and 5.6 above show that there exist finite numbers $c_{1}=c_{1}(p)$
and $c_{2}=c_{2}(k,p)$ such that conditions (a) and (b) in Proposition 4.3 of
[5] (see also Proposition 5.1 above) are satisfied, and both $c_{1}$ and
$c_{2}$ do not depend on $(t,x)\in I=[t_{0},t_{0}+1]\times[x_{0},x_{0}+1]$.
Therefore, by that proposition, we know that $u^{(2)}(t,x)$ has a uniformly
bounded density function $p_{t,x}(z)$ satisfying (5.10). This implies (5.11)
and completes the proof of Proposition 5.7. ∎

##  6\. Proof of Theorem 2.4

We now prove Theorem 2.4. It suffices to prove that, for any fixed
$z\in{\mathbb{R}}^{d}$, $t_{0}>0$ and $x_{0}\in{\mathbb{R}}$,
${\mathbb{P}}(F)=0$, where

$\displaystyle F=\\{u^{-1}(z)\cap I\neq\varnothing\\}\quad\text{and}\quad
I=[t_{0},t_{0}+1]\times[x_{0},x_{0}+1].$

Suppose, towards a contradiction, that ${\mathbb{P}}(F)>0$. Fix
$\delta\in(0,1/2)$. Then, by Proposition 4.1,

$\displaystyle{\mathbb{P}}\left(\bigcup_{K=1}^{\infty}\\{Z\leq K\\}\right)=1.$
(6.1)

Let $E_{K}=\\{\tau_{K}=T_{0}\\}\cap\\{Z\leq K\\}$. It follows from (3.1) and
(6.1) that

$\displaystyle{\mathbb{P}}\left(\bigcup_{K=1}^{\infty}E_{K}\cap F\right)>0.$

Hence, we can find a large enough $K$ such that

$\displaystyle{\mathbb{P}}\\{E_{K}\cap F\\}>0.$ (6.2)

For each $p\geq 1$, consider the random set

$\displaystyle G_{p}=\left\\{(s,y)\in
I:\exists\,r\in[2^{-2p},2^{-p}],\operatorname{osc}_{R_{r}(s,y)}(w_{s,y,r})\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}\right\\}.$

Consider the events

$\displaystyle\Omega_{p,1}$
$\displaystyle=\\{\lambda_{2}(G_{p})\geq\lambda_{2}(I)(1-e^{-\sqrt{p}/4})\\},$
$\displaystyle\Omega_{p,2}$ $\displaystyle=\\{\forall
R\in\mathcal{R}_{2p},\operatorname{osc}_{R}(\tilde{u})\leq K_{2}2^{-2p}p\\},$

where $\lambda_{2}$ denotes the Lebesgue measure on ${\mathbb{R}}^{2}$.
Furthermore, let

$\displaystyle G_{p}^{(1)}=\left\\{(s,y)\in
I:\exists\,r\in[2^{-2p},2^{-p}],\operatorname{osc}_{R_{r}(s,y)}(u^{(1)})\leq
4\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}\right\\}$

and

$\displaystyle\Omega_{p,3}=\\{\lambda_{2}(G_{p}^{(1)})\geq\lambda_{2}(I)(1-e^{-\sqrt{p}/4})\\}.$

Note that on $E_{K}$, $G_{p}\subset G_{p}^{(1)}$ for $p$ large. Indeed, if
$(s,y)\in G_{p}$, there exists $r\in[2^{-2p},2^{-p}]$ such that
$\operatorname{osc}_{R_{r}(s,y)}(w_{s,y,r})\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}$. Then, on $E_{K}$, by (3.2) and
(4.3),

$\displaystyle\operatorname{osc}_{R_{r}(s,y)}(u^{(1)})$
$\displaystyle\leq\operatorname{osc}_{R_{r}(s,y)}(w_{s,y,r})+\operatorname{osc}_{R_{r}(s,y)}(u^{(0)})+\operatorname{osc}_{R_{r}(s,y)}(u^{(2)}).$

Note that $u^{(0)}$ is $C^{1}$ on $I$, so in particular it is Lipschitz on
$I$. Also, by Proposition 4.1, it follows that

$\displaystyle\operatorname{osc}_{R_{r}(s,y)}(u^{(1)})$ $\displaystyle\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}+Cr+K(r^{4}+r^{2})^{1-\delta},$

which is $\leq 4\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}$ for $p$ large since
$2(1-\delta)>1$. Hence, on $E_{K}$, $G_{p}\subset G_{p}^{(1)}$ for $p$ large.
This implies that

$\displaystyle\Omega_{p,1}\cap E_{K}\subset\Omega_{p,3}\cap
E_{K}\quad\text{for }p\text{ large}.$ (6.3)

Let $\Omega_{p}=\Omega_{p,2}\cap\Omega_{p,3}$.

We now construct a random cover for $u^{-1}(z)\cap I$. For any $p\geq 1$ and
$(t,x)\in I$, let $A_{p}(t,x)$ be the unique dyadic rectangle of order $p$ in
$\mathcal{R}_{p}$ that contains $(t,x)$. We say that a dyadic rectangle $A$ of
order $q$ is “good” if

$\displaystyle\operatorname{osc}_{A}(u^{(1)})\leq
8\sigma_{1}K_{1}2^{-q}(\log\log 2^{q})^{-1/6}.$ (6.4)

For each $(s,y)\in G_{p}^{(1)}$, we can find the smallest $q$ with $p\leq
q\leq 2p$ such that for some $r\in[2^{-q-1},2^{-q}]$,

$\displaystyle\operatorname{osc}_{R_{r}(s,y)}(u^{(1)})\leq
4\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}.$

This implies that $A_{q}(s,y)$ is a good dyadic rectangle. We obtain in this
way a family $\mathcal{H}_{p}^{1}=\mathcal{H}_{p}^{1}(\omega)$ of disjoint
dyadic rectangles in $\bigcup_{q=p}^{2p}\mathcal{R}_{q}$ such that

$\displaystyle G^{(1)}_{p}\subset\bigcup_{A\in\mathcal{H}_{p}^{1}}A.$ (6.5)

On the other hand, we consider the family
$\mathcal{H}_{p}^{2}=\mathcal{H}_{p}^{2}(\omega)$ of all dyadic rectangles in
$\mathcal{R}_{2p}$ of order $2p$ that are not contained in any rectangle of
$\mathcal{H}_{p}^{1}$, so that

$\displaystyle\bigcup_{A\in\mathcal{H}_{p}^{2}}A\subset I\setminus
G_{p}^{(1)}.$ (6.6)

Let $\mathcal{H}_{p}=\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}$. Note that
$\mathcal{H}_{p}$ depends only on the random field $\\{u^{(1)}(t,x),(t,x)\in
I\\}$.

For each rectangle $A\in\mathcal{H}_{p}$, label a fixed point
$(t_{A},x_{A})\in A\cap I$. Consider the event

$\displaystyle\Omega_{A}=\\{|u(t_{A},x_{A})-z|\leq r_{A}\\},$

where

$\displaystyle r_{A}=\begin{cases}16\sigma_{1}K_{1}2^{-q}(\log\log
2^{q})^{-1/6}&\text{if }A\in\mathcal{H}_{p}^{1}\cap\mathcal{R}_{q},p\leq q\leq
2p,\\\ K_{2}2^{-2p}p&\text{if }A\in\mathcal{H}_{p}^{2}.\end{cases}$ (6.7)

######  Lemma 6.1.

For $p\geq 1$, let

$\displaystyle\mathcal{F}_{p}=\mathcal{F}_{p}(\omega)=\\{A\in\mathcal{H}_{p}:\Omega_{A}\text{
occurs}\\}.$

Then, for $p$ large enough, on $\Omega_{p}\cap E_{K}$, $\mathcal{F}_{p}$
covers $u^{-1}(z)\cap I$.

###### Proof.

Suppose $\Omega_{p}\cap E_{K}$ occurs and $(t,x)\in u^{-1}(z)\cap I$. By (6.5)
and (6.6), $(t,x)$ lies in some dyadic rectangle
$A\in\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}$. To prove
$A\in\mathcal{F}_{p}$, we need to show that $\Omega_{A}$ occurs.

Case 1: if $A\in\mathcal{H}_{p}^{1}$, then $A=A_{q}(t,x)$ is a good dyadic
rectangle of order $q$, where $p\leq q\leq 2p$, and (6.4) holds. Note that
$z=u(t,x)$. Also, since $E_{K}$ occurs, Proposition 4.1 implies that

$\displaystyle|u(t_{A},x_{A})-z|$ $\displaystyle=|u(t_{A},x_{A})-u(t,x)|$
$\displaystyle\leq|u^{(0)}(t_{A},x_{A})-u^{(0)}(t,x)|+|u^{(1)}(t_{A},x_{A})-u^{(1)}(t,x)|+|u^{(2)}(t_{A},x_{A})-u^{(2)}(t,x)|$
$\displaystyle\leq C(|t_{A}-t|+|x_{A}-x|)+8\sigma_{1}K_{1}2^{-q}(\log\log
2^{q})^{-1/6}+K(|t_{A}-t|+|x_{A}-x|)^{1-\delta}.$

Since $(t,x),(t_{A},x_{A})\in A$ and $\delta\in(0,1/2)$, we have

$\displaystyle(|t_{A}-t|+|x_{A}-x|)^{1-\delta}\leq(2^{-4q}+2^{-2q})^{1-\delta}=o(2^{-q}(\log\log
2^{q})^{-1/6})$

for $p$ large. Hence $|u(t_{A},x_{A})-z|\leq r_{A}$ and $\Omega_{A}$ occurs.

Case 2: if $A\in\mathcal{H}_{p}^{2}$, then $A\in\mathcal{R}_{2p}$. Since
$E_{K}$ occurs, $u=\tilde{u}$ on $A$ by (3.2). Also, since $\Omega_{p,2}$
occurs, we have

$\displaystyle|u(t_{A},x_{A})-z|=|u(t_{A},x_{A})-u(t,x)|\leq
K_{2}2^{-2p}p=r_{A}.$

Hence, $\Omega_{A}$ occurs in both cases and the lemma follows. ∎

######  Lemma 6.2.

Let $\mathscr{F}^{1}$ be the $\sigma$-algebra defined in (4.1). Then, there
exists a positive finite constant $K_{3}$ such that for all rectangles
$A\in\bigcup_{p\geq 1}\mathcal{R}_{p}$,

$\displaystyle\mathbb{P}\\{\Omega_{A}|\mathscr{F}^{1}\\}\leq K_{3}r_{A}^{d}.$
(6.8)

###### Proof.

We can write

$\displaystyle{\mathbb{P}}\\{\Omega_{A}|\mathscr{F}^{1}\\}={\mathbb{P}}\\{|u^{(2)}(t_{A},x_{A})-z^{\prime}|\leq
r_{A}|\mathscr{F}^{1}\\},$

where $z^{\prime}=z-u^{(0)}(t_{A},x_{A})-u^{(1)}(t_{A},x_{A})$. Note that
$z^{\prime}$ and $r_{A}$ are $\mathscr{F}^{1}$-measurable, while $u^{(2)}$,
being $\mathscr{F}^{2}$-measurable, is independent of $\mathscr{F}^{1}$.
Hence, (6.8) follows immediately from Proposition 5.7. ∎

Suppose $d\geq 6$ and consider the random variable

$\displaystyle
V:=\liminf_{p\to\infty}\sum_{A\in\mathcal{F}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p}\cap E_{K}},\quad\text{where
}\phi(r)=r^{6-d}(\log\log(1/r))^{d/6}$ (6.9)

and $D_{A}=2^{-q}$ if $A\in\mathcal{R}_{q}$. We aim to derive a contradiction
by showing that ${\mathbb{E}}[V]<\infty$ but $V=\infty$ with positive
probability. In fact, for each $p\geq 1$,

$\displaystyle{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{F}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p}\cap
E_{K}}\Bigg{]}\leq{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{F}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p,3}}\Bigg{]}={\mathbb{E}}\left[{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{H}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p,3}}{\bf 1}_{\Omega_{A}}\Bigg{|}\mathscr{F}^{1}\Bigg{]}\right].$

Note that $\mathcal{H}_{p}$ and $\Omega_{p,3}$ are
$\mathscr{F}^{1}$-measurable. So, by Lemma 6.2,

$\displaystyle{\mathbb{E}}\left[{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{H}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p,3}}{\bf 1}_{\Omega_{A}}\Bigg{|}\mathscr{F}^{1}\Bigg{]}\right]$
$\displaystyle={\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}}\phi(D_{A}){\mathbb{E}}[{\bf
1}_{\Omega_{A}}|\mathscr{F}^{1}]\Bigg{]}$ $\displaystyle\leq
K_{3}\,{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}}\phi(D_{A})r_{A}^{d}\Bigg{]}.$

Split the sum into two parts where $A\in\mathcal{H}_{p}^{1}$ and
$A\in\mathcal{H}_{p}^{2}$ respectively. For $A\in\mathcal{H}_{p}^{1}$, by the
definition of $\phi$ and $r_{A}$,

$\displaystyle{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{1}}\phi(D_{A})r_{A}^{d}\Bigg{]}$
$\displaystyle\leq(16\sigma_{1}K_{1})^{d}\,{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{H}_{p}^{1}}D_{A}^{6}\Bigg{]}$
$\displaystyle\leq(16\sigma_{1}K_{1})^{d}\lambda_{2}(I).$

The last inequality holds because $D_{A}^{6}=\lambda_{2}(A)$ and the dyadic
rectangles of $\mathcal{H}_{p}^{1}$ are disjoint and all contained in $I$. For
$A\in\mathcal{H}_{p}^{2}$,

$\displaystyle{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{2}}\phi(D_{A})r_{A}^{d}\Bigg{]}$
$\displaystyle\leq K_{2}^{d}\,{\mathbb{E}}\bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{2}}2^{-12p}(\log\log
2^{2p})^{d/6}p^{d}\bigg{]}.$

On $\Omega_{p,3}$, by (6.6), the number of rectangles in $\mathcal{H}_{p}^{2}$
is at most

$\displaystyle Ce^{-\sqrt{p}/4}/(2^{-2p})^{6}$

which implies that

$\displaystyle{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{2}}\phi(D_{A})r_{A}^{d}\Bigg{]}$
$\displaystyle\leq C(\log 2p)^{d/6}p^{d}e^{-\sqrt{p}/4}.$

Therefore, by Fatou’s lemma,

$\displaystyle{\mathbb{E}}[V]\leq\liminf_{p\to\infty}{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{F}_{p}}\phi(D_{A}){\bf
1}_{\Omega_{p}\cap E_{K}}\Bigg{]}\leq
K_{3}(16\sigma_{1}K_{1})^{d}\lambda_{2}(I)<\infty.$ (6.10)

On the other hand, consider the event

$\displaystyle\Omega^{*}=\bigcup_{\ell\geq 1}\bigcap_{p\geq\ell}\Omega_{p}.$
(6.11)

By (6.3), we have

$\displaystyle\Omega^{*}\cap E_{K}\cap F$ $\displaystyle=\bigcup_{\ell\geq
1}\bigcap_{p\geq\ell}(\Omega_{p,2}\cap\Omega_{p,3})\cap E_{K}\cap F$
$\displaystyle\supset\bigcup_{\ell\geq
1}\bigcap_{p\geq\ell}(\Omega_{p,2}\cap\Omega_{p,1})\cap E_{K}\cap F.$

By Proposition 3.1 and Fubini’s theorem, we have
$\sum_{p=1}^{\infty}{\mathbb{P}}(\Omega_{p,1}^{c})<\infty$. Also, by
Proposition 3.2, we have
$\sum_{p=1}^{\infty}{\mathbb{P}}(\Omega_{p,2}^{c})<\infty$. Then, by the
Borel–Cantelli lemma, the event

$\displaystyle\bigcup_{\ell\geq
1}\bigcap_{p\geq\ell}(\Omega_{p,2}\cap\Omega_{p,1})$

has probability 1. This and (6.2) imply that ${\mathbb{P}}\\{\Omega^{*}\cap
E_{K}\cap F\\}>0$. Now, if $\omega\in\Omega^{*}\cap E_{K}\cap F$, then
$u^{-1}(z)\cap I\neq\varnothing$ and Lemma 6.1 implies that
$\mathcal{F}_{p}(\omega)\neq\varnothing$ for all large $p$, and hence
$V(\omega)=\infty$ because $\phi(r)\to\infty$ as $r\to 0$ when $d\geq 6$. This
shows that $V=\infty$ with positive probability, which contradicts (6.10).
Hence ${\mathbb{P}}\\{u^{-1}(z)\cap I\neq\varnothing\\}=0$. This completes the
proof of Theorem 2.4. ∎

##  7\. Proof of Theorem 2.5

We need to prove that, for any fixed $t_{0}>0$ and $x_{0}\in{\mathbb{R}}$,
${\mathbb{P}}(F)=0$, where

$\displaystyle F=\\{u^{-1}(S)\cap I\neq\varnothing\\}\quad\text{and}\quad
I=[t_{0},t_{0}+1]\times[x_{0},x_{0}+1].$

Suppose that ${\mathbb{P}}(F)>0$. As in the proof of Theorem 2.4, we can find
a large enough $K$ such that

$\displaystyle{\mathbb{P}}\\{E_{K}\cap F\\}>0.$ (7.1)

Define the random sets $G_{p}$, $G_{p}^{(1)}$, the events $\Omega_{p,1}$,
$\Omega_{p,2}$, $\Omega_{p,3}$, and the family
$\mathcal{H}_{p}=\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}$ of dyadic
rectangles as in the proof of Theorem 2.4. For each rectangle
$A\in\mathcal{H}_{p}$, consider the event

$\displaystyle\Omega_{A}^{\prime}=\left\\{\inf_{z\in S}|u(t_{A},x_{A})-z|\leq
r_{A}\right\\},$

where $r_{A}$ is as defined in (6.7).

######  Lemma 7.1.

Let

$\displaystyle\mathcal{F}_{p}^{\prime}=\mathcal{F}_{p}^{\prime}(\omega)=\\{A\in\mathcal{H}_{p}:\Omega_{A}^{\prime}\text{
occurs}\\}.$

Then, for $p$ large enough, on $\Omega_{p}\cap E_{K}$, $\mathcal{F}_{p}$
covers $u^{-1}(S)\cap I$.

###### Proof.

Suppose $\Omega_{p}\cap E_{K}$ occurs and $(t,x)\in u^{-1}(S)\cap I$. Then
$u(t,x)=z$ for some $z\in S$ and $(t,x)$ lies in some dyadic cube
$A\in\mathcal{H}_{p}$. Since

$\displaystyle\inf_{z\in S}|u(t_{A},x_{A})-z|\leq|u(t_{A},x_{A})-u(t,x)|,$

the same proof of Lemma 6.1 shows that $|u(t_{A},x_{A})-u(t,x)|\leq r_{A}$,
and hence $\Omega_{A}^{\prime}$ occurs. ∎

######  Lemma 7.2.

Let $\mathscr{F}^{1}$ be the $\sigma$-algebra defined in (4.1). Then, there
exists a positive finite constant $K_{0}$ such that for all rectangles
$A\in\mathcal{R}_{p}$ with $p$ large enough,

$\displaystyle{\mathbb{P}}\\{\Omega_{A}^{\prime}|\mathscr{F}^{1}\\}\leq
K_{0}f(r_{A}).$ (7.2)

###### Proof.

Note that

$\displaystyle{\mathbb{P}}\\{\Omega_{A}^{\prime}|\mathscr{F}^{1}\\}$
$\displaystyle={\mathbb{P}}\left\\{\inf_{z\in S}|u(t_{A},x_{A})-z|\leq
r_{A}\Big{|}\mathscr{F}^{1}\right\\}$
$\displaystyle={\mathbb{P}}\left\\{u^{(2)}(t_{A},x_{A})\in(S-z^{\prime})^{(r_{A})}\Big{|}\mathscr{F}^{1}\right\\},$

where $z^{\prime}=u^{(0)}(t_{A},x_{A})+u^{(1)}(t_{A},x_{A})$. Here,
$z^{\prime}$ and $r_{A}$ are $\mathscr{F}^{1}$-measurable, and $u^{(2)}$ is
independent of $\mathscr{F}^{1}$. By Proposition 5.7, $u^{(2)}(t,x)$ has a
density function $p_{t,x}(z)$ that is uniformly bounded by some constant
$K_{0}<\infty$ for all $(t,x)\in I$. This and condition (2.2) imply that for
all $A\in\mathcal{R}_{p}$ and $p$ large,

$\displaystyle{\mathbb{P}}\\{\Omega_{A}^{\prime}|\mathscr{F}^{1}\\}\leq
K_{0}\lambda_{d}((S-z^{\prime})^{(r_{A})})\leq K_{4}f(r_{A}).$

This completes the proof of the lemma. ∎

Consider the random variable

$\displaystyle
V:=\liminf_{p\to\infty}\sum_{A\in\mathcal{F}_{p}}\phi_{f}(D_{A}){\bf
1}_{\Omega_{p}\cap
E_{K}}\quad\text{where}\quad\phi_{f}(r)=\frac{r^{6}}{f(r(\log\log(1/r))^{-1/6})}$

and $D_{A}=2^{-q}$ if $A\in\mathcal{R}_{q}$. By Lemma 7.2, for $p$ large,

$\displaystyle{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{F}_{p}}\phi_{f}(D_{A}){\bf
1}_{\Omega_{p}\cap E_{K}}\Bigg{]}$ $\displaystyle\leq{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}}\phi_{f}(D_{A}){\mathbb{E}}[{\bf
1}_{\Omega_{A}^{\prime}}|\mathscr{F}^{1}]\Bigg{]}$ $\displaystyle\leq
K_{0}\,{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}}\phi_{f}(D_{A})f(r_{A})\Bigg{]}.$

For $A\in\mathcal{H}_{p}^{1}$,
$r_{A}=16\sigma_{1}K_{1}D_{A}(\log\log(1/D_{A}))^{-1/6}$. By (2.3), for $p$
large,

$\displaystyle{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{1}}\phi_{f}(D_{A})f(r_{A})\Bigg{]}$
$\displaystyle\leq{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{H}_{p}^{1}}\frac{D_{A}^{6}f(16\sigma_{1}K_{1}D_{A}(\log\log(1/D_{A}))^{-1/6})}{f(D_{A}(\log\log(1/D_{A}))^{-1/6})}\Bigg{]}$
$\displaystyle\leq
C(16\sigma_{1}K_{1})^{\kappa}\,{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{H}_{p}^{1}}D_{A}^{6}\Bigg{]}$
$\displaystyle\leq C(16\sigma_{1}K_{1})^{\kappa}\lambda_{2}(I).$

For $A\in\mathcal{H}_{p}^{2}$, the number of rectangles in
$\mathcal{H}_{p}^{2}$ is at most $Ce^{-\sqrt{p}/4}/2^{-12p}$ on
$\Omega_{p,3}$, and by (2.3),

$\displaystyle\phi_{f}(D_{A})f(r_{A})=\frac{2^{-12p}f(K_{2}2^{-2p}p)}{f(2^{-2p}(\log\log
2^{2p})^{-1/6})}\leq C2^{-12p}(\log\log 2^{2p})^{\kappa/6}p^{\kappa}$

for $p$ large. It follows that

$\displaystyle{\mathbb{E}}\Bigg{[}{\bf
1}_{\Omega_{p,3}}\sum_{A\in\mathcal{H}_{p}^{2}}\phi_{f}(D_{A})f(r_{A})\Bigg{]}\leq
C(\log\log 2^{2p})^{\kappa/6}p^{\kappa}e^{-\sqrt{p}/4}.$

Hence, by Fatou’s lemma, we have

$\displaystyle{\mathbb{E}}[V]\leq\liminf_{p\to\infty}{\mathbb{E}}\Bigg{[}\sum_{A\in\mathcal{F}_{p}}\phi_{f}(D_{A}){\bf
1}_{\Omega_{p}\cap E_{K}}\Bigg{]}<\infty.$ (7.3)

On the other hand, we have ${\mathbb{P}}\\{\Omega^{*}\cap E_{K}\cap F\\}>0$,
where $\Omega^{*}$ is the event defined in (6.11). If $\omega\in\Omega^{*}\cap
E_{F}\cap F$, then $u^{-1}(S)\cap I\neq\varnothing$ and Lemma 7.1 implies that
$\mathcal{F}_{p}^{\prime}(\omega)\neq\varnothing$ for all large $p$. Also,
since $\phi_{f}(r)\to\infty$ as $r\to 0$ by condition (2.4), it follows that
$V(\omega)=\infty$. This means that $V=\infty$ with positive probability. This
contradicts (7.3). Hence ${\mathbb{P}}\\{u^{-1}(S)\cap I\neq\varnothing\\}=0$.
The proof of Theorem 2.5 is complete. ∎

##  8\. Proof of Theorem 2.7

The proof follows the arguments that led to Theorem 2.4 above. We only mention
the main steps and the modifications needed. For (i), let
$I_{1}=[t_{0},t_{0}+1]$, $R^{1}_{r}(t)=[t-r^{4},t+r^{4}]$, and consider the
collection $\mathcal{R}^{1}_{p}$ of dyadic intervals of the form

$\displaystyle Q=[m2^{-4p},(m+1)2^{-4p}],\quad m\in{\mathbb{Z}},$

that intersect $I_{1}$. Then, the conclusion of Proposition 3.1 applies to
oscillations on the intervals $R^{1}_{r}(t_{1})\times\\{x^{*}\\}$, that of
Proposition 3.2 applies to oscillations on the dyadic intervals
$R=A\times\\{x^{*}\\}$ where $A\in\mathcal{R}^{1}_{2p}$, and that of
Proposition 4.1 applies to the interval $I_{1}\times\\{x^{*}\\}$. Assume that
${\mathbb{P}}(F_{1})>0$, where
$F=\\{u^{-1}(z)\cap(I_{1}\times\\{x^{*}\\})\neq\varnothing\\}$ and choose $K$
large enough such that ${\mathbb{P}}\\{E_{K}\cap F\\}>0$. We define the random
sets

$\displaystyle G_{p}=\left\\{s\in
I_{1}:\exists\,r\in[2^{-2p},2^{-p}],\operatorname{osc}_{R_{r}^{1}(s)\times\\{x^{*}\\}}(w_{s,x^{*},r})\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}\right\\},$ $\displaystyle
G_{p}^{(1)}=\left\\{s\in
I_{1}:\exists\,r\in[2^{-2p},2^{-p}],\operatorname{osc}_{R_{r}^{1}(s)\times\\{x^{*}\\}}(u^{(1)})\leq
2\sigma_{1}K_{1}r(\log\log(1/r))^{-1/6}\right\\}$

and the events

$\displaystyle\Omega_{p,1}=\\{\lambda_{1}(G_{p})\geq\lambda_{1}(I)(1-e^{-\sqrt{p}/4})\\},$
$\displaystyle\Omega_{p,2}=\\{\forall
Q\in\mathcal{R}^{1}_{2p},\operatorname{osc}_{Q\times\\{x^{*}\\}}(\tilde{u})\leq
K_{2}2^{-2p}p\\},$
$\displaystyle\Omega_{p,3}=\\{\lambda_{1}(G^{(1)}_{p})\geq\lambda_{1}(I_{1})(1-e^{-\sqrt{p}/4})\\},$
$\displaystyle\Omega_{p}=\Omega_{p,2}\cap\Omega_{p,3}.$

We say that a dyadic interval $A\in\mathcal{R}_{q}^{1}$ is “good” if

$\displaystyle\operatorname{osc}_{A\times\\{x^{*}\\}}(u^{(1)})\leq
8\sigma_{1}K_{1}2^{-q}(\log\log 2^{q})^{-1/6}.$

Then, as in the proof of Theorem 2.4, we construct the family
$\mathcal{H}_{p}=\mathcal{H}_{p}^{1}\cup\mathcal{H}_{p}^{2}$, where
$\mathcal{H}_{p}^{1}$ is a family of disjoint good dyadic intervals in
$\bigcup_{q=p}^{2p}\mathcal{R}_{q}^{1}$ that covers $G^{(1)}_{p}$, and
$\mathcal{H}_{p}^{2}$ is a family of dyadic intervals in
$\mathcal{R}_{2p}^{1}$ that are not contained in any intervals of
$\mathcal{H}_{p}^{1}$ but are contained in $I_{1}\setminus G^{(1)}_{p}$.
Define

$\displaystyle\mathcal{F}_{p}=\\{A\in\mathcal{H}_{p}:\Omega_{A}\text{
occurs}\\},$

where

$\displaystyle\Omega_{A}=\\{|u(t_{A},x^{*})-z|\leq r_{A}\\},$

$t_{A}$ is a fixed point in $A\cap I_{1}$ for each $A\in\mathcal{H}_{p}$, and
$r_{A}$ is defined as in (6.7). Then, for $p$ large, $\mathcal{F}_{p}$ covers
$u^{-1}(z)\cap(I_{1}\times\\{x^{*}\\})$ on $\Omega_{p}\cap E_{K}$, and Lemma
6.2 still holds. As in the proof of Theorem 2.4, by considering the random
variable $V$ defined in (6.9) but with $\phi(r)=r^{4-d}(\log\log(1/r))^{d/6}$
and using the properties that $D_{A}^{4}=\lambda_{1}(A)$ for
$A\in\mathcal{H}_{p}$ and that the number of intervals in
$\mathcal{H}_{p}^{2}$ is at most $Ce^{-\sqrt{p}/4}2^{8p}$, we can deduce that
${\mathbb{E}}[V]<\infty$ but $V=\infty$ with positive probability, which is a
contradiction. Hence ${\mathbb{P}}\\{F_{1}\\}=0$ and (i) is proved.

For (ii), let $I_{2}=[x_{0},x_{0}+1]$, $R^{2}_{r}(x)=[x-r^{2},x+r^{2}]$, and
$\mathcal{R}^{2}_{p}$ be the collection of dyadic intervals of the form

$\displaystyle[m2^{-2p},(m+1)2^{-2p}],\quad m\in{\mathbb{Z}},$

that intersect $I_{2}$. We consider the interval $\\{t^{*}\\}\times I_{2}$ and
the dyadic intervals $\\{t^{*}\\}\times A$, where $A\in\mathcal{R}^{2}_{p}$.
Then, the proof of (ii) is similar that of (i). The details are left to the
reader. ∎

Acknowledgements. The authors would like to thank Raluca Balan for helpful
discussions. The first author was supported in part by Taiwan’s National
Science and Technology Council grant NSTC111-2115-M-007-015-MY2. The second
author was supported in part by the NSF grants DMS-1855185 and DMS-2153846.

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