# Local times of anisotropic Gaussian random fields and stochastic heat
equation

Cheuk Yin Lee and Yimin Xiao

###### Abstract.

We study the local times of a large class of Gaussian random fields satisfying
strong local nondeterminism with respect to an anisotropic metric. We
establish moment estimates and Hölder conditions for the local times of the
Gaussian random fields. Our key estimates rely on geometric properties of
Voronoi partitions with respect to an anisotropic metric and the use of
Besicovitch’s covering theorem. As a consequence, we deduce sample path
properties of the Gaussian random fields that are related to Chung’s law of
the iterated logarithm and modulus of non-differentiability. Moreover, we
apply our results to systems of stochastic heat equations with additive
Gaussian noise and determine the exact Hausdorff measure function with respect
to the parabolic metric for the level sets of the solutions.

###### Key words and phrases:

Local times, Gaussian random fields, anisotropy, stochastic heat equation,
strong local nondeterminism, Hausdorff measure, Voronoi partition,
Besicovitch’s covering theorem

###### 2010 Mathematics Subject Classification:

60G15, 60G60, 60G17

##  1\. Introduction

The objective of this paper is to study the regularities of local times and
related sample path properties of a large class of Gaussian random fields
satisfying strong local nondeterminism with respect to an anisotropic metric.
The solution of systems of stochastic heat equations with additive Gaussian
noise is an important example of these Gaussian random fields.

We consider a mean zero, continuous Gaussian random field
$X=\\{X(t),t\in{\mathbb{R}}^{N}\\}$ with values in ${\mathbb{R}}^{d}$ defined
by

$X(t)=(X_{1}(t),\dots,X_{d}(t)),$ (1.1)

where $X_{1},\dots,X_{d}$ are i.i.d. copies of a real-valued Gaussian field
$Y=\\{Y(t),t\in{\mathbb{R}}^{N}\\}$.

Throughout this paper, let $T$ denote a closed (or open) bounded interval
(rectangle) in ${\mathbb{R}}^{N}$. We say that $X$ satisfies condition (A) on
$T$ if there exists a constant vector $H=(H_{1},\dots,H_{N})\in(0,1)^{N}$ such
that the following conditions (A1) and (A2) hold.

  * (A1)

There exist constants $0<C_{1}\leq C_{2}<\infty$ such that

$C_{1}\rho^{2}(t,s)\leq{\mathbb{E}}[(Y(t)-Y(s))^{2}]\leq C_{2}\rho^{2}(t,s)$

for all $t,s\in T$, where $\rho$ is the anisotropic metric defined by

$\displaystyle\rho(t,s)=\sum_{j=1}^{N}|t_{j}-s_{j}|^{H_{j}}.$ (1.2)

  * (A2)

There exists a constant $C_{3}>0$ such that for all integers $n\geq 1$ and all
$t,t^{1},\dots,t^{n}\in T$,

$\mathrm{Var}(Y(t)|Y(t^{1}),\dots,Y(t^{n}))\geq C_{3}\min_{0\leq k\leq
n}\rho^{2}(t,t^{k}),$

where $t^{0}=0$.

Condition (A2) is called the strong local nondeterminism in metric $\rho$. The
concept of local nondeterminism (LND) was first introduced by Berman [4] to
study the existence and joint continuity of the local times of Gaussian
processes. Berman’s LND property was extended by Pitt [29] to Gaussian random
fields and by Cuzick [6] to local $\phi$-nondeterminism for an arbitrary
positive function $\phi$. Moreover, strong LND and other forms of LND were
developed and they are useful for studying various sample path properties of
Gaussian random fields (see, e.g., [36, 37] and references therein for more
information).

It was proved by Pitt [29] that the multiparameter fractional Brownian motion
with Hurst index $H\in(0,1)$ satisfies strong LND with $\rho(t,s)=|t-s|^{H}$.
The Hölder conditions for the local times and the Hausdorff measure of the
level sets of Gaussian fields with stationary increments and strong LND were
studied by Xiao [35]. Similar results were obtained by Ayache et al. [3] for
fractional Brownian sheets, and by Wu and Xiao [34] and Lee [21, 22] for other
anisotropic Gaussian fields satisfying the property of sectorial local
nondeterminism. This latter property is weaker than the above property (A2)
and was first proved by Khoshnevisan and Xiao [19] for the Brownian sheet, by
Wu and Xiao [33] for fractional Brownian sheets, and Lee and Xiao [24] for the
solution of stochastic wave equations with Gaussian noise which is white in
time and colored in space.

Conditions (A1) and (A2) above were considered in Xiao [38] to study various
properties of sample paths and local times. In [38], moment estimates for the
local times on Euclidean balls were proved, but those estimates were not sharp
enough to obtain exact regularity results and sharp sample path properties. In
the current paper, we consider the local times of $X$ on anisotropic balls and
strengthen the results by proving sharp moment estimates as well as local and
global Hölder conditions for the local times. In the case of anisotropic
balls, the powers appearing in our bounds are different from those in [38, 34,
21, 22].

In [38, 26], sufficient conditions for Gaussian random fields with stationary
increments to satisfy condition (A) were obtained in terms of their spectral
densities or spectral measures. One of our motivations is to provide a more
general framework for studying Gaussian random fields without stationary
increments. We remark that the solution $u=\\{u(t,x),t\geq
0,x\in{\mathbb{R}}^{N}\\}$ of a system of stochastic heat equations with
additive Gaussian noise does not have stationary increments in the time-space
variable $(t,x)$. For another example of Gaussian random fields without
stationary increments that satisfies (A), see Lee and Xiao [23]. In Section 5,
we show that the solution of a system of stochastic heat equations with
additive Gaussian noise satisfies condition (A) as well as the conditions in
Dalang et al. [10]. Hence our results are applicable to $u=\\{u(t,x),t\geq
0,x\in{\mathbb{R}}^{N}\\}$. Furthermore, we obtain a precise result regrading
the Hausdorff measure function (in the metric $\rho$) for the level sets of
$u=\\{u(t,x),t\geq 0,x\in{\mathbb{R}}^{N}\\}$.

The novelty of the current paper is that we employ two geometric tools to
obtain our key estimates. First, for any given points $t^{1},\dots,t^{n}\in
S\subset T$, in order to estimate integrals of the form

$\displaystyle\int_{S}\left[\min_{1\leq l\leq
n}\rho(t,t^{l})\right]^{-\beta}dt,$ (1.3)

we consider the Voronoi partition

$\displaystyle\Gamma_{l}=\left\\{t\in
S:\tilde{\rho}(t,t^{l})\leq\tilde{\rho}(t,t^{j})\text{ for all
}j\in\\{1,\dots,n\\}\right\\},\quad l=1,\dots,n,$

of $S$ generated by these fixed points, where $\tilde{\rho}(t,s)=\max_{1\leq
j\leq N}|t_{j}-s_{j}|^{H_{j}}$ is an anisotropic metric which is equivalent to
$\rho$. We study the geometric properties of the Voronoi partition and prove
that each $\Gamma_{l}$ (which contains $t^{l}$) satisfies an anisotropic star
shape property at the point $t^{l}$ (see Lemma 2.2). Our integral estimate for
(1.3) hinges on this crucial geometric property of the Voronoi partition. We
refer to, e.g., [20, 2] for general theory, properties, and applications of
Voronoi partitions and diagrams.

Secondly, as in [35], finding moment estimates for the increments of the local
times relies on estimates for the integrals

$\displaystyle\int_{S^{n}}\prod_{j=1}^{n}\left[\min_{0\leq l\leq
j-1}\rho(t^{j},t^{l})\right]^{-d}\prod_{j=1}^{n}\left[\min_{0\leq l\leq
n,l\neq j}\rho(t^{j},t^{l})\right]^{-\gamma}dt^{1}\cdots dt^{n}.$

We point out that there was a gap in the proof of Lemma 2.5 in [35], namely,
the second inequality in (2.20) on page 140 in [35] may not hold in general if
$N\geq 2$. The missing part that is needed in order to complete the proof is
the following: For arbitrary $n$ given points $t^{0},t^{1},\dots,t^{n}$ and
$0\leq i\leq n$, we need a universal bound $K=K_{N}$ depending only on $N$
(but not on $n$, $i$ or the points) for the quantity

$\displaystyle\\#\left\\{j\in\\{1,\dots,n\\}:\tilde{\rho}(t^{j},t^{i})\leq\tilde{\rho}(t^{j},t^{l})\text{
for all }l\in\\{0,1,\dots,n\\}\setminus\\{j\\}\right\\}.$

This universal bound is established in Lemma 2.6. Our idea is to relate this
problem to covering points with balls in metric $\tilde{\rho}$ and then apply
a version of Besicovitch’s covering theorem adapted to comparable intervals
(see Lemma 2.5). This allows us to fill the gap in the proof of [35] and give
a complete proof for the moment estimates for the increments of local times.

The rest of the paper is organized as follows. In Section 2, first we
establish some auxiliary lemmas including Lemmas 2.2 and 2.6 about the
anisotropic star shape property for the Voronoi partitions with respect to an
anisotropic metric and a universal bound involving Besicovitch’s covering
theorem, respectively. Then, we use these tools to derive sharp moment
estimates for the local times in Proposition 2.4 and moment estimates for the
increments of the local times in Proposition 2.8. In Section 3, we use the
moment estimates in Section 2 to prove Theorem 3.2 which concerns local and
global Hölder conditions for the local times, and deduce Theorem 3.3 which
provides lower bounds for Chung’s law of the iterated logarithm and the
modulus of non-differentiability for the Gaussian random field. Then in
Section 4, we discuss the Hausdorff dimension and Hausdorff measure of the
level sets. Finally in Section 5, we apply the results to the solution of a
system of stochastic heat equations and determine the exact Hausdorff measure
function of the level sets with respect to the parabolic metric.

Throughout this paper, we let $C$ denote a constant whose value may be
different in each appearance, and $C_{1},C_{2},K_{1},K_{2},\dots$ denote
specific constants. We let $\lambda_{N}$ denote the Lebesgue measure on
${\mathbb{R}}^{N}$, and for any Borel set $S\subset{\mathbb{R}}^{N}$, let
$\mathscr{B}(S)$ denote the $\sigma$-algebra of Borel subsets of $S$. For
$a\in{\mathbb{R}}^{N}$ and $r>0$, let
$B_{\rho}(a,r)=\\{t\in{\mathbb{R}}^{N}:\rho(t,a)\leq r\\}$ be the closed ball
centered at $a$ with radius $r$ in the metric $\rho$. We denote a finite
sequence of points in ${\mathbb{R}}^{N}$ by $t^{1},t^{2},\dots,t^{n}$, and for
a given $k\in\\{1,\dots,n\\}$, the coordinates of the point $t^{k}$ are
written as $t^{k}=(t^{k}_{1},t^{k}_{2},\dots,t^{k}_{N})$.

##  2\. Moment Estimates for the Local Times

In this section, we study the joint continuity of local times for Gaussian
random fields satisfying condition (A). Let us recall the definition and
properties of local times. Let $X(t)$ be a vector field on ${\mathbb{R}}^{N}$
with values in ${\mathbb{R}}^{d}$ and $S\in\mathscr{B}({\mathbb{R}}^{N})$. The
occupation measure of $X$ on $S$ is the Borel measure on ${\mathbb{R}}^{d}$
defined by

$\mu_{S}(B)=\lambda_{N}\\{t\in S:X(t)\in B\\},\quad
B\in\mathscr{B}({\mathbb{R}}^{d}).$

We say that $X$ has a _local time_ on $S$ if the occupation measure $\mu_{S}$
is absolutely continuous with respect to the Lebesgue measure $\lambda_{d}$.
In this case, the local time of $X$ is defined as the Radon–Nikodym
derivative:

$L(x,S)=\frac{d\mu_{S}}{d\lambda_{d}}(x).$

Note that if $X$ has a local time on $S$, then it also has a local time on any
Borel set $A\subset S$.

By Theorem 6.3 of Geman and Horowitz [12], when the local time exists on $S$,
one can choose a version of the local time, still denoted by $L(x,S)$, which
is a kernel in the sense that

  1. (i)

$L(\cdot,A)$ is $\mathscr{B}({\mathbb{R}}^{d})$-measurable for each fixed
$A\in\mathscr{B}(S)$; and

  2. (ii)

$L(x,\cdot)$ is a Borel measure on $\mathscr{B}(S)$ for each fixed
$x\in{\mathbb{R}}^{d}$.

Moreover, by Theorem 6.4 of [12], $L(x,S)$ satisfies the following _occupation
density formula_ : for any nonnegative Borel function $f$ on
${\mathbb{R}}^{d}$,

$\int_{S}f(X(t))\,dt=\int_{{\mathbb{R}}^{d}}f(x)L(x,S)\,dx.$ (2.1)

Let $T=\prod_{j=1}^{N}[\tau_{j},\tau_{j}+h_{j}]$ be a compact interval, where
$h_{j}>0$ for $j=1,\dots,N$. We say that the local time is _jointly
continuous_ on $T$ if we can find a version of the local time such that a.s.
$L\big{(}x,\prod_{j=1}^{N}[\tau_{j},\tau_{j}+s_{j}]\big{)}$ is jointly
continuous in all variables $(x,s)$ in
${\mathbb{R}}^{d}\times\prod_{j=1}^{N}[0,h_{j}]$. Throughout this paper, we
will always use a jointly continuous version of the local time whenever it
exists. When a local time is jointly continuous, it can be uniquely extended
to a kernel and if, in addition, $X$ is continuous, $L(x,\cdot)$ defines a
Borel measure supported on the level set $X^{-1}(x)\cap T=\\{t\in
T:X(t)=x\\}$; see [12, p.12, Remark (c)] or [1, Theorem 8.6.1].

Let $X$ be a Gaussian random field defined by (1.1). If condition (A1) is
satisfied on $T$, then by Theorem 8.1 of Xiao [38], $X$ has a local time on
$T$ with $L(\cdot,T)\in L^{2}(\lambda_{d}\times{\mathbb{P}})$ if any only if
$d<\sum_{j=1}^{N}(1/H_{j})$. Moreover, when the latter condition holds, the
local time has the following representation in
$L^{2}(\lambda_{d}\times{\mathbb{P}})$: for any $S\in\mathscr{B}(T)$,

$L(x,S)=(2\pi)^{-d}\int_{{\mathbb{R}}^{d}}du\,e^{-i\langle
u,x\rangle}\int_{S}dt\,e^{i\langle u,X(t)\rangle}.$ (2.2)

It follows that for any integer $n\geq 1$ and any $x\in{\mathbb{R}}^{d}$,

${\mathbb{E}}[L(x,S)^{n}]=(2\pi)^{-nd}\int_{{\mathbb{R}}^{nd}}d\bar{u}\int_{S^{n}}d\bar{t}\,e^{-i\sum_{j=1}^{n}\langle
u^{j},x\rangle}\,{\mathbb{E}}\left[e^{i\sum_{j=1}^{n}\langle
u^{j},X(t^{j})\rangle}\right]$ (2.3)

and for any even integer $n\geq 2$ and $x,y\in{\mathbb{R}}^{d}$,

$\displaystyle\begin{aligned} &{\mathbb{E}}[(L(x,S)-L(y,S))^{n}]\\\
&=(2\pi)^{-nd}\int_{{\mathbb{R}}^{nd}}d\bar{u}\int_{S^{n}}d\bar{t}\prod_{j=1}^{n}\left[e^{-i\langle
u^{j},x\rangle}-e^{-i\langle
u^{j},y\rangle}\right]{\mathbb{E}}\left[e^{i\sum_{l=1}^{n}\langle
u^{l},X(t^{l})\rangle}\right],\end{aligned}$ (2.4)

where $\bar{u}=(u^{1},\dots,u^{n})$ and $\bar{t}=(t^{1},\dots,t^{n})$. See,
e.g., [12, §25].

In fact, under condition (A), the condition $d<\sum_{j=1}^{N}(1/H_{j})$
implies not only the existence of local times but also their joint continuity.
This has been proved in [38, Theorem 8.2]:

######  Theorem 2.1.

Suppose $X$ satisfies condition (A) on $T$ and $d<\sum_{j=1}^{N}(1/H_{j})$.
Then $X$ has a jointly continuous local time on $T$.

The proof in [38] is based on moment estimates of the local times on the
Euclidean balls and a multiparameter version of Kolmogorov’s continuity
theorem. However, those moment estimates are not sharp enough to deduce sharp
Hölder conditions of the local times and the exact Hausdorff measure of the
level sets.

The goal of this section is to prove sharp moment estimates for the local
times on anisotropic balls. These estimates are provided in Propositions 2.4
and 2.8 below. We extend the method of Xiao [35, Lemma 2.5]. To facilitate
some of our arguments, let us define the metric

$\displaystyle\tilde{\rho}(t,s)=\max_{1\leq j\leq N}|t_{j}-s_{j}|^{H_{j}}$
(2.5)

which is equivalent to the metric $\rho$ defined in (1.2). Indeed, we have

$\tilde{\rho}(t,s)\leq\rho(t,s)\leq N\tilde{\rho}(t,s)\quad\text{for all
}t,s\in{\mathbb{R}}^{N}.$ (2.6)

Let us begin with the following lemma, which is concerned with the geometric
properties of the Voronoi partition generated by $m$ given points with respect
to the anisotropic metric $\tilde{\rho}$.

######  Lemma 2.2.

Fix $m$ distinct points $t^{1},\dots,t^{m}\in{\mathbb{R}}^{N}$. For
$l=1,\dots,m$, define

$\displaystyle\Gamma_{l}=\left\\{t\in{\mathbb{R}}^{N}:\tilde{\rho}(t,t^{l})=\min_{1\leq
k\leq m}\tilde{\rho}(t,t^{k})\right\\}.$ (2.7)

Then the following properties hold:

  1. (i)

${\mathbb{R}}^{N}=\bigcup_{l=1}^{m}\Gamma_{l}$ and
$\lambda_{N}(\Gamma_{l}\cap\Gamma_{l^{\prime}})=0$ whenever $l\neq
l^{\prime}$.

  2. (ii)

$\Gamma_{l}$ satisfies the following anisotropic star shape property at the
point $t^{l}$:

$\displaystyle t\in\Gamma_{l}\quad\text{implies}\quad
t^{l}+\varepsilon^{E}(t-t^{l})\in\Gamma_{l}\text{ for all
}\varepsilon\in(0,1),$ (2.8)

where $E$ is the diagonal matrix $\operatorname{diag}(1/H_{1},\dots,1/H_{N})$.

###### Proof.

(i). It is clear that the union of all $\Gamma_{l}$’s is ${\mathbb{R}}^{N}$.
For $l\neq l^{\prime}$, we have

$\displaystyle\lambda_{N}(\Gamma_{l}\cap\Gamma_{l^{\prime}})$
$\displaystyle\leq\lambda_{N}\left\\{t\in{\mathbb{R}}^{N}:\tilde{\rho}(t,t^{l})=\tilde{\rho}(t,t^{l^{\prime}})\right\\}$
$\displaystyle\leq\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{N}\left\\{t\in{\mathbb{R}}^{N}:|t_{i}-t^{l}_{i}|^{H_{i}}=|t_{j}-t^{l^{\prime}}_{j}|^{H_{j}}\right\\}=0.$

(ii). It suffices to prove the property (2.8) for $l=1$. Also, since

$\displaystyle\Gamma_{1}=\bigcap_{k=2}^{m}\left\\{t\in{\mathbb{R}}^{N}:\tilde{\rho}(t,t^{1})\leq\tilde{\rho}(t,t^{k})\right\\},$

we can further reduce the proof to the case $l=1$ and $m=2$. With this
reduction in mind, we assume $t\in\Gamma_{1}$, i.e.,

$\displaystyle\tilde{\rho}(t,t^{1})\leq\tilde{\rho}(t,t^{2}),$ (2.9)

and aim to show that for all $\varepsilon\in(0,1)$, the point
$s=s(\varepsilon):=t^{1}+\varepsilon^{E}(t-t^{1})$ is in $\Gamma_{1}$, i.e.,

$\displaystyle\tilde{\rho}(s,t^{1})\leq\tilde{\rho}(s,t^{2}).$ (2.10)

The points $t$, $t^{1}$ and $t^{2}$ may have different configurations. Since
only their relative positions are relevant to us, we may assume without loss
of generality that $t^{2}$ is in the positive orthant relative to $t$, namely,
$t^{2}_{j}\geq t_{j}$ for all $j\in\\{1,\dots,N\\}$; see Figure 1. By the
definition of the metric $\tilde{\rho}$ in (2.5), we have
$\tilde{\rho}(t,t^{2})=|t_{j}-t^{2}_{j}|^{H_{j}}$ for some
$j\in\\{1,\dots,N\\}$. For simplicity, we assume that $j=1$, i.e.,

$\displaystyle\tilde{\rho}(t,t^{2})=|t_{1}-t^{2}_{1}|^{H_{1}}=(t^{2}_{1}-t_{1})^{H_{1}}$
(2.11)

since the proof below also works for the other cases $j\neq 1$ in the exact
same way. Note that

$\displaystyle\tilde{\rho}(s,t^{1})=\max_{1\leq j\leq
N}|t^{1}_{j}+\varepsilon^{1/H_{j}}(t_{j}-t^{1}_{j})-t^{1}_{j}|^{H_{j}}=\varepsilon\tilde{\rho}(t,t^{1})$
(2.12)

and

$\displaystyle\begin{split}\tilde{\rho}(s,t^{2})\geq|s_{1}+t^{2}_{1}|^{H_{1}}&=|t^{1}_{1}+\varepsilon^{1/H_{1}}(t_{1}-t^{1}_{1})-t^{2}_{1}|^{H_{1}}\\\
&=|t_{1}-t^{2}_{1}+(1-\varepsilon^{1/H_{1}})(t^{1}_{1}-t_{1})|^{H_{1}}\\\
&=|t^{2}_{1}-t_{1}|^{H_{1}}\left|1-(1-\varepsilon^{1/H_{1}})\frac{t^{1}_{1}-t_{1}}{t^{2}_{1}-t_{1}}\right|^{H_{1}}.\end{split}$
(2.13)

In order to show (2.10), we consider two cases: (1) $t_{1}^{1}\geq t_{1}$, and
(2) $t_{1}^{1}<t_{1}$; see Figure 1.

$t$$t^{2}$$t^{1}$

$t$$t^{2}$$t^{1}$

Figure 1. Two cases of configuration: (1) $t_{1}^{1}\geq t_{1}$ (left), and
(2) $t^{1}_{1}<t_{1}$ (right)

Case (1): $t_{1}^{1}\geq t_{1}$. By (2.9) and (2.11),
$(t^{1}_{1}-t_{1})^{H_{1}}\leq\tilde{\rho}(t,t^{1})\leq\tilde{\rho}(t,t^{2})=(t^{2}_{1}-t_{1})^{H_{1}}$.
It follows that

$\displaystyle 0\leq\frac{t^{1}_{1}-t_{1}}{t^{2}_{1}-t_{1}}\leq
1\quad\text{and hence}\quad\varepsilon^{1/H_{1}}\leq
1-(1-\varepsilon^{1/H_{1}})\frac{t^{1}_{1}-t_{1}}{t^{2}_{1}-t_{1}}\leq 1.$

This together with (2.13), (2.11), (2.9) and (2.12) implies that

$\displaystyle\tilde{\rho}(s,t^{2})\geq\varepsilon|t^{2}_{1}-t_{1}|^{H_{1}}=\varepsilon\tilde{\rho}(t,t^{2})\geq\varepsilon\tilde{\rho}(t,t^{1})=\tilde{\rho}(s,t^{1}).$

Case (2): $t^{1}_{1}<t_{1}$. In this case, we have $t^{2}_{1}-t_{1}\geq 0$. It
follows that

$\displaystyle\frac{t^{1}_{1}-t_{1}}{t^{2}_{1}-t_{1}}<0\quad\text{and
hence}\quad
1-(1-\varepsilon^{1/H_{1}})\frac{t^{1}_{1}-t_{1}}{t^{2}_{1}-t_{1}}>1\geq\varepsilon^{1/H_{1}}.$

Again, this together with (2.13), (2.11), (2.9) and (2.12) implies that
$\tilde{\rho}(s,t^{2})\geq\tilde{\rho}(s,t^{1})$. This proves (2.10) in both
cases and finishes the proof of (2.8). ∎

The next lemma is an integral estimate which will be useful later in the
moment estimates of the local times.

######  Lemma 2.3.

Let $T\subset{\mathbb{R}}^{N}$ be any compact interval. Suppose that
$0<d\leq\beta_{0}<Q$, where $Q=\sum_{j=1}^{N}(1/H_{j})$. Then there is a
finite constant $C=C(N,H,Q,\beta_{0})$ such that for all intervals $S$ in $T$,
for all $\beta\in[d,\beta_{0}]$, all integers $m\geq 1$, and all distinct
points $t^{1},\dots,t^{m}\in S$, we have

$\int_{S}\left[\min_{0\leq k\leq m-1}\rho(t,t^{k})\right]^{-\beta}dt\leq
Cm^{\beta/Q}\lambda_{N}(S)^{1-\beta/Q}.$ (2.14)

In particular, for all $a\in{\mathbb{R}}^{N}$, $0<r<1$ and all distinct
$t^{1},\dots,t^{m}\in B_{\rho}(a,r)\subset T$, we have

$\int_{B_{\rho}(a,r)}\left[\min_{0\leq k\leq
m-1}\rho(t,t^{k})\right]^{-\beta}dt\leq Cm^{\beta/Q}r^{Q-\beta}.$ (2.15)

###### Proof.

Let $I$ denote the integral on the left-hand side of (2.14). Since
$\lambda_{N}(S)=\lambda_{N}(\overline{S})$ for all intervals $S\subset T$, we
may assume that $S$ is a closed interval. We consider the Voronoi partition
$\\{\Gamma_{l}\\}_{l=0}^{m-1}$ of $S$ generated by the points
$t^{0},t^{1},\dots,t^{m-1}$ with respect to the metric $\tilde{\rho}$ defined
in (2.5), i.e.,

$\Gamma_{l}=\left\\{t\in S:\tilde{\rho}(t,t^{l})=\min_{0\leq k\leq
m-1}\tilde{\rho}(t,t^{k})\right\\},\quad l=0,1,\dots,m-1.$

By (2.6) and $S=\bigcup_{l=0}^{m-1}\Gamma_{l}$ (see Lemma 2.2), we have

$\displaystyle I$ $\displaystyle\leq\int_{S}\left[\min_{0\leq k\leq
m-1}\tilde{\rho}(t,t^{k})\right]^{-\beta}dt$
$\displaystyle\leq\sum_{l=0}^{m-1}\int_{\Gamma_{l}}[\tilde{\rho}(t,t^{l})]^{-\beta}dt\leq
N^{\beta}\sum_{l=0}^{m-1}\int_{{\mathbb{R}}^{N}}\Bigg{(}\sum_{j=1}^{N}|t_{j}-t^{l}_{j}|^{H_{j}}\Bigg{)}^{-\beta}{\bf
1}_{\Gamma_{l}}(t)\,dt.$

Fix $l\in\\{0,1,\dots,m-1\\}$. Let us consider the following anisotropic
spherical coordinates $\Gamma_{l}$, namely, we let
$t=t^{l}+h^{E}\Psi(\theta)$, where $E$ is the diagonal matrix
$\operatorname{diag}(1/H_{1},\dots,1/H_{N})$ and

$\displaystyle\Psi(\theta)=(\Psi_{1}(\theta),\dots,\Psi_{N}(\theta))^{T}:A\to{\mathbb{R}}^{N},\quad\theta\in
A:=[0,2\pi]\times[0,\pi]^{N-2},$

is defined by

$\displaystyle\begin{cases}\Psi_{1}(\theta)=[\cos(\theta_{1})]^{2/H_{1}},\\\
\Psi_{2}(\theta)=[\sin(\theta_{1})\cos(\theta_{2})]^{2/H_{2}},\\\
\quad\vdots\\\
\Psi_{N-1}(\theta)=[\sin(\theta_{1})\dots\sin(\theta_{N-2})\cos(\theta_{N-1})]^{2/H_{N-1}},\\\
\Psi_{N}(\theta)=[\sin(\theta_{1})\dots\sin(\theta_{N-2})\sin(\theta_{N-1})]^{2/H_{N}}.\end{cases}$

Here, $[x]^{p}:=x|x|^{p-1}$ for any $x\in{\mathbb{R}}$. The Jacobian $J_{l}$
for this transformation is well-defined and $|\det
J_{l}|=h^{Q-1}\varphi(\theta)$, where $\varphi$ is a bounded nonnegative
function. Under this change of coordinates, we have

$\displaystyle\int_{{\mathbb{R}}^{N}}\Bigg{(}\sum_{j=1}^{N}|t_{j}-t^{l}_{j}|^{H_{j}}\Bigg{)}^{-\beta}{\bf
1}_{\Gamma_{l}}(t)\,dt=\int_{A}d\theta\,\varphi(\theta)\int_{0}^{\infty}h^{Q-1-\beta}\,{\bf
1}_{\Gamma_{l}}(t^{l}+h^{E}\Psi(\theta))\,dh.$

The anisotropic star shape property of $\Gamma_{l}$ from Lemma 2.2 shows that
for any fixed $\theta\in A$,

$\displaystyle t^{l}+h^{E}\Psi(\theta)\in\Gamma_{l}\quad\text{implies}\quad
t^{l}+(\varepsilon h)^{E}\Psi(\theta)\in\Gamma_{l}\text{ for all
}\varepsilon\in(0,1).$

Here, we have also used the fact that $S$ is a closed interval. It follows
that there exists $h_{l}(\theta)>0$ such that ${\bf
1}_{\Gamma_{l}}(t^{l}+h^{E}\Psi(\theta))={\bf 1}_{(0,h_{l}(\theta)]}(h)$ and
hence

$\displaystyle I$ $\displaystyle\leq
N^{\beta}\sum_{l=0}^{m-1}\int_{A}d\theta\,\varphi(\theta)\int_{0}^{h_{l}(\theta)}h^{Q-1-\beta}dh=\frac{N^{\beta}}{Q-\beta}\sum_{l=0}^{m-1}\int_{A}[h_{l}(\theta)]^{Q-\beta}\varphi(\theta)\,d\theta.$

By the same variables change, the Lebesgue measure of $\Gamma_{l}$ can be
computed as follows:

$\displaystyle\begin{aligned}
\lambda_{N}(\Gamma_{l})=\int_{A}d\theta\,\varphi(\theta)\int_{0}^{h_{l}(\theta)}h^{Q-1}dh=\frac{c_{N,H}}{Q}\int_{A}[h_{l}(\theta)]^{Q}\sigma(d\theta),\end{aligned}$
(2.16)

where the constant $c_{N,H}=\int_{A}\varphi(\theta)d\theta$ depends only on
$N$ and $H$, and $\sigma(d\theta)=c_{N,H}^{-1}\varphi(\theta)d\theta$ is a
probability measure on $A$. Since $0<\beta<Q$, the function $x\mapsto
x^{1-\beta/Q}$ is concave on the interval $[0,\infty)$. Then, we can apply
Jensen’s inequality for the probability measure $\sigma$ and the relation
(2.16) to obtain

$\displaystyle I$
$\displaystyle\leq\frac{c_{N,H}N^{\beta}}{Q-\beta}\sum_{l=0}^{m-1}\int_{A}\left\\{[h_{l}(\theta)]^{Q}\right\\}^{1-\beta/Q}\sigma(d\theta)$
$\displaystyle\leq\frac{c_{N,H}N^{\beta_{0}}}{Q-\beta_{0}}\sum_{l=0}^{m-1}\left\\{\int_{A}[h_{l}(\theta)]^{Q}\sigma(d\theta)\right\\}^{1-\beta/Q}$
$\displaystyle=\frac{c_{N,H}N^{\beta_{0}}}{Q-\beta_{0}}\sum_{l=0}^{m-1}\left\\{\frac{Q}{c_{N,H}}\lambda(\Gamma_{l})\right\\}^{1-\beta/Q}$
$\displaystyle=C\sum_{l=0}^{m-1}{\lambda(\Gamma_{l})}^{1-\beta/Q},$

where $C$ is a constant that depends only on $N,H,Q$ and $\beta_{0}$, and the
last equality is valid thanks to part (i) of Lemma 2.2. Then, applying
Jensen’s inequality for the uniform probability measure on
$\\{0,1,\dots,m-1\\}$, we get that

$\displaystyle I$ $\displaystyle\leq
Cm\left\\{\frac{1}{m}\sum_{l=0}^{m-1}\lambda(\Gamma_{l})\right\\}^{1-\beta/Q}=Cm^{\beta/Q}\lambda(S)^{1-\beta/Q}.$

This proves (2.14). Finally, observe from (2.6) that $B_{\rho}(a,r)\subset
B_{\tilde{\rho}}(a,r)$ and the latter is a closed interval. Hence, (2.15)
follows from (2.14) by taking $S=B_{\tilde{\rho}}(a,r)$ and using the fact
that $\lambda_{N}(B_{\tilde{\rho}}(a,r))=2^{N}r^{Q}$. ∎

For any Gaussian vector $(Z_{1},\dots,Z_{n})$, let
$\mathrm{Cov}(Z_{1},\dots,Z_{n})$ denote its covariance matrix. Recall that
the determinant of this matrix can be evaluated by using the following formula
[17, Corollary A.2]:

$\det\mathrm{Cov}(Z_{1},\dots,Z_{n})=\mathrm{Var}(Z_{1})\prod_{m=1}^{n}\mathrm{Var}(Z_{m}|Z_{1},\dots,Z_{m-1}).$
(2.17)

The following proposition provides moment estimates for the local time.

######  Proposition 2.4.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$, where
$Q=\sum_{j=1}^{N}(1/H_{j})$. Then there exists a finite constant $C$ such that
for all intervals $S$ in $T$, for all $x\in{\mathbb{R}}^{d}$ and all integers
$n\geq 1$, we have

${\mathbb{E}}[L(x,S)^{n}]\leq C^{n}(n!)^{d/Q}\lambda_{N}(S)^{n(1-d/Q)}.$

In particular, for all $a\in T$ and $r\in(0,1)$ with $B_{\rho}(a,r)\subset T$,
we have

${\mathbb{E}}[L(x,B_{\rho}(a,r))^{n}]\leq C^{n}(n!)^{d/Q}r^{n(Q-d)}.$

###### Proof.

By (2.3), we have

$\displaystyle{\mathbb{E}}[L(x,S)^{n}]$
$\displaystyle=(2\pi)^{-nd}\int_{{\mathbb{R}}^{nd}}d\bar{u}\int_{S^{n}}d\bar{t}\,e^{-i\sum_{j=1}^{n}\langle
u^{j},x\rangle}\,{\mathbb{E}}\left[e^{i\sum_{j=1}^{n}\langle
u^{j},X(t^{j})\rangle}\right],$

where $\bar{u}=(u^{1},\dots,u^{n})$ and $\bar{t}=(t^{1},\dots,t^{n})$. Since
$X_{1},\dots,X_{d}$ are i.i.d. copies of $Y$, we have

$\displaystyle{\mathbb{E}}[L(x,S)^{n}]$
$\displaystyle\leq(2\pi)^{-nd}\int_{S^{n}}d\bar{t}\prod_{k=1}^{d}\int_{{\mathbb{R}}^{n}}d\bar{u}_{k}\,e^{-\frac{1}{2}\mathrm{Var}(\sum_{j=1}^{n}u^{j}_{k}Y(t^{j}))}$
$\displaystyle=(2\pi)^{-nd/2}\int_{S^{n}}\left[\det\mathrm{Cov}(Y(t^{1}),\dots,Y(t^{n}))\right]^{-d/2}d\bar{t},$

where $\bar{u}_{k}=(u^{1}_{k},\dots,u^{n}_{k})$. By (2.17),

$\det\mathrm{Cov}(Y(t^{1}),\dots,Y(t^{n}))=\mathrm{Var}(Y(t^{1}))\prod_{m=2}^{n}\mathrm{Var}(Y(t^{m})|Y(t^{1}),\dots,Y(t^{m-1})).$

It follows from condition (A2) that

${\mathbb{E}}[L(x,S)^{n}]\leq
C(2\pi)^{-nd/2}\int_{S^{n}}\prod_{m=1}^{n}\left[\min_{0\leq k\leq
m-1}\rho(t^{m},t^{k})\right]^{-d}d\bar{t}.$ (2.18)

If we integrate (2.18) in the order of $dt^{n},dt^{n-1},\dots,dt^{1}$, and
apply Lemma 2.3 (with $\beta=d$) repeatedly, we deduce that

$\displaystyle{\mathbb{E}}[L(x,S)^{n}]\leq
C^{n}(n!)^{d/Q}\lambda_{N}(S)^{n(1-d/Q)}.$

This completes the proof of the proposition. ∎

Next, we aim to prove moment estimates for the increments of the local time in
both time and space variables. We are mostly following the approach of Lemma
2.5 in Xiao [35], but let us point out that there is an error in the proof of
(2.9) of that lemma: we observe that the second inequality in (2.20) of the
proof may not be true if $N\geq 2$, because there may be multiple $j$’s such
that $\min\\{\xi(|t_{\pi(j)}-t_{i}|)^{\gamma}:i=0\textup{ or
}i\neq\pi(j)\\}=\xi(|t_{\pi(j)}-t_{\pi(1)}|)^{\gamma}$. [When $N=1$ and $\pi$
is the permutation such that $t_{\pi(1)}\leq t_{\pi(2)}\leq\ldots\leq
t_{\pi(n)}$, then (2.20) in [35] holds. When $N\geq 2$, the definition of the
permutation $\pi$ on page 139 is not sufficient for the second inequality in
(2.20) to hold.] Nevertheless, with the help of Lemma 2.6 provided below, we
are able to give a complete proof for the moment estimates in Proposition 2.8
below and therefore Lemma 2.5 in [35] can be corrected in a similar way.

The proof of Lemma 2.6 below is based on Besicovitch’s covering theorem. The
original theorem is stated for balls under the Euclidean metric (cf., e.g.,
[27, p.30]). In order to apply it in our anisotropic setting, we will use the
following more general version of the covering theorem for comparable
intervals in ${\mathbb{R}}^{N}$.

######  Lemma 2.5.

[14, Theorem 1.1 and Remark 5] There exists a positive integer $M=M(N)$
depending only on $N$ with the following property. For any bounded subset $A$
of ${\mathbb{R}}^{N}$ and any family $\mathscr{B}=\\{Q(x):x\in A\\}$ of closed
intervals such that $Q(x)$ is centered at $x$ for every $x\in A$ and, for
every two pints $x_{1}$ and $x_{2}$ of $A$, $Q(x_{1})$ and $Q(x_{2})$ can be
translated to be concentric such that one is contained in the other, there
exists a sequence $\\{Q_{i}\\}$ in $\mathscr{B}$ such that:

  1. $(i)$

$A\subset\bigcup_{i}Q_{i}$;

  2. $(ii)$

the intervals of $\\{Q_{i}\\}$ can be distributed in $M$ families of disjoint
intervals.

We use the Besicovitch covering theorem above to prove the next lemma.

######  Lemma 2.6.

There exists a positive integer $K=K(N)$ depending only on $N$ such that for
any integer $n\geq 1$ and any distinct points
$s^{0},s^{1},\dots,s^{n}\in{\mathbb{R}}^{N}$, the cardinality of the set of
all $j\in\\{1,\dots,n\\}$ such that

$\tilde{\rho}(s^{j},s^{0})=\min\\{\tilde{\rho}(s^{j},s^{i}):0\leq i\leq
n,i\neq j\\}$ (2.19)

is at most $K$, where $\tilde{\rho}$ is the equivalent metric defined in
(2.5).

###### Proof.

Without loss of generality, we may assume that (2.19) is satisfied for
$j=1,\dots,k$. Note that for $s\in{\mathbb{R}}^{N}$ and $r>0$, the ball
$B_{\tilde{\rho}}(s,r):=\\{t\in{\mathbb{R}}^{N}:\tilde{\rho}(t,s)\leq r\\}$
under the metric $\tilde{\rho}$ is the closed interval (hypercube) centered at
$s$ with side lengths $2r^{1/H_{1}},\dots,2r^{1/H_{N}}$. We will make use of
Besicovitch’s covering theorem (Lemma 2.5) to show that $k\leq K$ for some
positive integer $K=K(N)$ that depends only on the dimension $N$.

To this end, let

$\delta_{0}=\min\left\\{\frac{\tilde{\rho}(s^{i},s^{0})}{\tilde{\rho}(s^{j},s^{0})}:i,j\in\\{1,\dots,k\\}\right\\}.$

Note that $0<\delta_{0}\leq 1$. Choose a small $0<\varepsilon_{0}<1$ such that
$(1-\varepsilon_{0})^{1/H_{p}}(1+\delta_{0}^{1/H_{p}})\geq 1$ for all
$p\in\\{1,\dots,N\\}$. Let
$\varepsilon=\varepsilon_{0}\min\\{\tilde{\rho}(s^{i},s^{0}):1\leq i\leq
k\\}$. Let $A=\\{s^{1},\dots,s^{k}\\}$ and consider the family
$\mathscr{B}=\\{B_{\tilde{\rho}}(s^{1},r_{1}),\dots,B_{\tilde{\rho}}(s^{k},r_{k})\\}$
of intervals, where $r_{i}=\tilde{\rho}(s^{i},s^{0})-\varepsilon$. By
Besicovitch’s covering theorem (Lemma 2.5), we can find sub-families
$\mathscr{B}_{1},\dots,\mathscr{B}_{M}\subset\mathscr{B}$, which we denote by
$\mathscr{B}_{i}=\\{B_{\tilde{\rho}}(s^{i,1},r_{i,1}),\dots,B_{\tilde{\rho}}(s^{i,J(i)},r_{i,J(i)})\\}$,
such that

$A=\\{s^{1},\dots,s^{k}\\}\subset\bigcup_{i=1}^{M}\bigcup_{j=1}^{J(i)}B_{\tilde{\rho}}(s^{i,j},r_{i,j})$

and for each sub-family $\mathscr{B}_{i}$, the intervals in $\mathscr{B}_{i}$
are pairwise disjoint, where $M=M(N)$ is a positive integer depending only on
$N$. For each $1\leq j\leq k$, by the assumption (2.19), if $\ell\neq j$, then
$\tilde{\rho}(s^{j},s^{\ell})\geq\tilde{\rho}(s^{j},s^{0})>r_{j}$. In other
words, for each $1\leq j\leq k$, the interval $B_{\tilde{\rho}}(s^{j},r_{j})$
does not contain any other $s^{\ell}$ with $\ell\neq j$. This means that at
least $k$ intervals are needed to cover the set $A$, and hence $k\leq
J(1)+\dots+J(M)$.

Let us fix $i$ and estimate the cardinality $J(i)$ of the family
$\mathscr{B}_{i}$. Consider the family
$\mathscr{B}_{i}^{*}=\\{B_{\tilde{\rho}}(s^{i,1},r^{*}_{i,1}),\dots,B_{\tilde{\rho}}(s^{i,J(i)},r^{*}_{i,J(i)})\\}$,
where $r^{*}_{i,j}=\tilde{\rho}(s^{i,j},s^{0})$. Since the intervals in
$\mathscr{B}_{i}$ are pairwise disjoint, this means that for any pair
$s^{i,\ell}\neq s^{i,j}$ we can find some $p\in\\{1,\dots,N\\}$ such that

$|s^{i,\ell}_{p}-s^{i,j}_{p}|>r_{i,\ell}^{1/H_{p}}+r_{i,j}^{1/H_{p}}.$

Then, by the definitions of $r_{i}$, $\varepsilon$, and $\delta_{0}$, and by
the choice of $\varepsilon_{0}$,

$\displaystyle|s^{i,\ell}_{p}-s^{i,j}_{p}|$
$\displaystyle>(\tilde{\rho}(s^{i,\ell},s^{0})-\varepsilon)^{1/H_{p}}+(\tilde{\rho}(s^{i,j},s^{0})-\varepsilon)^{1/H_{p}}$
$\displaystyle\geq(1-\varepsilon_{0})^{1/H_{p}}\left(\tilde{\rho}(s^{i,\ell},s^{0})^{1/H_{p}}+\tilde{\rho}(s^{i,j},s^{0})^{1/H_{p}}\right)$
$\displaystyle\geq(1-\varepsilon_{0})^{1/H_{p}}(\delta_{0}^{1/H_{p}}+1)\tilde{\rho}(s^{i,j},s^{0})^{1/H_{p}}$
$\displaystyle\geq\tilde{\rho}(s^{i,j},s^{0})^{1/H_{p}}$
$\displaystyle=(r_{i,j}^{*})^{1/H_{p}}.$

It follows that $\tilde{\rho}(s^{i,\ell},s^{i,j})>r_{i,j}^{*}$, which means
that the interval $B(s^{i,j},r^{*}_{i,j})$ does not contain any other
$s^{i,\ell}$ with $\ell\neq j$. On the other hand, every interval in
$\mathscr{B}^{*}_{i}$ contains the point $s^{0}$, so these intervals are not
pairwise disjoint. Then another application of Besicovitch’s covering theorem
(Lemma 2.5) to the set $\\{s^{i,1},\dots,s^{i,J(i)}\\}$ and the family
$\mathscr{B}_{i}^{*}$ implies that $J(i)\leq M$. Hence, we conclude that
$k\leq M^{2}$, and the proof is finished by taking $K=M^{2}$. ∎

Recall the following lemma in [7, Lemma 2].

######  Lemma 2.7.

Let $Y_{1},\dots,Y_{n}$ be mean zero Gaussian random variables that are
linearly independent and assume that $\int_{\mathbb{R}}g(v)e^{-\varepsilon
v^{2}}dv<\infty$ for all $\varepsilon>0$. Then

$\int_{{\mathbb{R}}^{n}}g(v_{1})\exp\bigg{[}-\frac{1}{2}\mathrm{Var}\bigg{(}\sum_{l=1}^{n}v_{l}Y_{l}\bigg{)}\bigg{]}dv_{1}\dots
dv_{n}=\frac{(2\pi)^{(n-1)/2}}{\det\mathrm{Cov}(Y_{1},\dots,Y_{n})^{1/2}}\int_{\mathbb{R}}g(v/\sigma_{1})e^{-v^{2}/2}dv,$

where $\sigma_{1}^{2}=\mathrm{Var}(Y_{1}|Y_{2},\dots,Y_{n})$.

Recall also the equivalent metric $\tilde{\rho}$ defined in (2.5). Note that
condition (A2) implies

$\mathrm{Var}(Y(t)|Y(t^{1}),\dots,Y(t^{n}))\geq C_{3}\min_{0\leq l\leq
n}\tilde{\rho}^{2}(t,t^{l})$ (2.20)

for all $n\geq 1$ and all $t,t^{1},\dots,t^{n}\in T$.

Now, we derive moment estimates for the increments of the local time.

######  Proposition 2.8.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q=\sum_{j=1}^{N}(1/H_{j})$.
Then there exist positive finite constants $C=C(T,N,d,H)$ and $K=K(N)>1$ such
that for all $\gamma\in(0,1)$ small enough, for all intervals $S\subseteq T$,
$x,y\in{\mathbb{R}}^{d}$, and all even integers $n\geq 2$, we have

${\mathbb{E}}[(L(x,S)-L(y,S))^{n}]\leq
C^{n}|x-y|^{n\gamma}(n!)^{d/Q+K\gamma/Q}\lambda_{N}(S)^{n(1-(d+\gamma)/Q)}.$

In particular, for all $a\in T$, $0<r<1$ with $B_{\rho}(a,r)\subset T$, we
have

${\mathbb{E}}[(L(x,B_{\rho}(a,r))-L(y,B_{\rho}(a,r)))^{n}]\leq
C^{n}|x-y|^{n\gamma}(n!)^{d/Q+K\gamma/Q}r^{n(Q-d-\gamma)}.$

###### Proof.

By (LABEL:Eq:local_time_incre_mom), for any even integer $n\geq 2$ and
$x,y\in{\mathbb{R}}^{d}$,

$\displaystyle\begin{aligned} &{\mathbb{E}}[(L(x,S)-L(y,S))^{n}]\\\
&=(2\pi)^{-nd}\int_{S^{n}}d\bar{t}\int_{{\mathbb{R}}^{nd}}d\bar{u}\prod_{j=1}^{n}\left[e^{-i\langle
u^{j},x\rangle}-e^{-i\langle
u^{j},y\rangle}\right]{\mathbb{E}}\left[e^{i\sum_{l=1}^{n}\langle
u^{l},X(t^{l})\rangle}\right],\end{aligned}$ (2.21)

where $\bar{u}=(u^{1},\dots,u^{n})\in{\mathbb{R}}^{nd}$ and
$\bar{t}=(t^{1},\dots,t^{n})\in S^{n}$. Note that

${\mathbb{E}}\left[e^{i\sum_{l=1}^{n}\langle
u^{l},X(t^{l})\rangle}\right]=\exp\Bigg{[}{-\frac{1}{2}\mathrm{Var}\Bigg{(}\sum_{l=1}^{n}\sum_{k=1}^{d}u_{k}^{l}X_{k}(t^{l})\Bigg{)}}\Bigg{]}$
(2.22)

for all $u^{1},\dots,u^{n}\in{\mathbb{R}}^{d}$ and $t^{1},\dots,t^{n}\in S$.
For any $0<\gamma<1$, we have $|e^{iu}-1|\leq 2^{1-\gamma}|u|^{\gamma}$ and
$|u+v|^{\gamma}\leq|u|^{\gamma}+|v|^{\gamma}$ for all $u,v\in{\mathbb{R}}$. It
follows that

$\prod_{j=1}^{n}\left|e^{-i\langle u^{j},x\rangle}-e^{-i\langle
u^{j},y\rangle}\right|\leq
2^{(1-\gamma)n}|x-y|^{n\gamma}\sum_{\bar{k}}\prod_{j=1}^{n}{\big{|}u_{k_{j}}^{j}\big{|}}^{\gamma}$
(2.23)

for all $u^{1},\dots,u^{n},x,y\in{\mathbb{R}}^{d}$, where the summation runs
over all $\bar{k}=(k_{1},\dots,k_{n})\in\\{1,\dots,d\\}^{n}$. Then (2.21),
(2.22) and (2.23) imply that

$\displaystyle{\mathbb{E}}[(L(x,S)-L(y,S))^{n}]\leq(2\pi)^{-nd}\,2^{n}|x-y|^{n\gamma}\sum_{\bar{k}}\int_{S^{n}}J(\bar{t},\bar{k})\,d\bar{t},$
(2.24)

where

$J(\bar{t},\bar{k})=\int_{{\mathbb{R}}^{nd}}\Bigg{(}\prod_{j=1}^{n}{\big{|}u_{k_{j}}^{j}\big{|}}^{\gamma}\Bigg{)}\exp\Bigg{[}{-\frac{1}{2}\mathrm{Var}\Bigg{(}\sum_{l=1}^{n}\sum_{k=1}^{d}u_{k}^{l}X_{k}(t^{l})\Bigg{)}}\Bigg{]}d\bar{u}$

for $\bar{t}\in S^{n}$ and $\bar{k}\in\\{1,\dots,d\\}^{n}$. We can assume that
$(t^{1},\dots,t^{n})\in S^{n}$ are distinct points because those which are not
distinct form a set of Lebesgue measure 0. By the generalized Hölder
inequality,

$\displaystyle J(\bar{t},\bar{k})$
$\displaystyle\leq\prod_{j=1}^{n}\Bigg{\\{}\int_{{\mathbb{R}}^{nd}}{\big{|}u_{k_{j}}^{j}\big{|}}^{n\gamma}\exp\Bigg{[}{-\frac{1}{2}\mathrm{Var}\Bigg{(}\sum_{l=1}^{n}\sum_{k=1}^{d}u_{k}^{l}X_{k}(t^{l})\Bigg{)}}\Bigg{]}d\bar{u}\Bigg{\\}}^{1/n}.$

Fix $\bar{t}$, $\bar{k}$, and $j$. By condition (A2), the random variables
$\\{X_{k}(t^{l}):1\leq l\leq n,1\leq k\leq d\\}$ are linearly independent.
Then by Lemma 2.7 and the fact that $X_{1},\dots,X_{d}$ are i.i.d. copies of
$Y$,

$\displaystyle\int_{{\mathbb{R}}^{nd}}{\big{|}u_{k_{j}}^{j}\big{|}}^{n\gamma}\exp\Bigg{[}{-\frac{1}{2}\mathrm{Var}\Bigg{(}\sum_{l=1}^{n}\sum_{k=1}^{d}u_{k}^{l}X_{k}(t^{l})\Bigg{)}}\Bigg{]}d\bar{u}$
$\displaystyle=\frac{(2\pi)^{(nd-1)/2}}{\det\mathrm{Cov}(Y(t^{1}),\dots,Y(t^{n}))^{d/2}}\int_{\mathbb{R}}{\left|\frac{v}{\sigma_{j}}\right|}^{n\gamma}e^{-v^{2}/2}dv,$

where

$\displaystyle\sigma_{j}^{2}=\mathrm{Var}\left(X_{k_{j}}(t^{j})\,\big{|}\,X_{k}(t^{l}):(k,l)\neq(k_{j},j)\right)=\mathrm{Var}\left(Y(t^{j})\,\big{|}\,Y(t^{l}):l\neq
j\right).$

By Jensen’s inequality and Gaussian moment estimates,

$\int_{\mathbb{R}}|v|^{n\gamma}e^{-v^{2}/2}dv\leq\sqrt{2\pi}\left(\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}|v|^{n}e^{-v^{2}/2}dv\right)^{\gamma}=\sqrt{2\pi}\left((n-1)!!\right)^{\gamma}\leq\sqrt{2\pi}(n!)^{\gamma}.$

It follows that

$J(\bar{t},\bar{k})\leq\frac{C^{n}(n!)^{\gamma}}{\det\mathrm{Cov}(Y(t^{1}),\dots,Y(t^{n}))^{d/2}}\prod_{j=1}^{n}\frac{1}{\sigma_{j}^{\gamma}}.$
(2.25)

By the covariance formula (2.17) and condition (A2), or (2.20),

$\displaystyle\int_{S^{n}}J(\bar{t},\bar{k})\,d\bar{t}\leq
C^{n}(n!)^{\gamma}\int_{S^{n}}\prod_{j=1}^{n}\left[\min_{0\leq l\leq
j-1}\tilde{\rho}(t^{j},t^{l})\right]^{-d}\prod_{j=1}^{n}\left[\min_{0\leq
l\leq n,l\neq j}\tilde{\rho}(t^{j},t^{l})\right]^{-\gamma}d\bar{t}.$

To estimate the integral over $S^{n}$, we consider, for any
$j\in\\{2,\dots,n\\}$ and any given distinct points
$t^{0},t^{1},\dots,t^{j-1}$, the Voronoi partition
$\\{\Gamma_{j,i}\\}_{i=0}^{j-1}$ of $S$ generated by the points
$t^{0},t^{1},\dots,t^{j-1}$ with respect to the anisotropic metric
$\tilde{\rho}$, namely,

$\displaystyle\Gamma_{j,i}=\left\\{t^{j}\in
S:\tilde{\rho}(t^{j},t^{i})=\min_{0\leq l\leq
j-1}\tilde{\rho}(t^{j},t^{l})\right\\},\quad 0\leq i\leq j-1.$

Then

$\displaystyle\int_{S^{n}}\prod_{j=1}^{n}\left[\min_{0\leq l\leq
j-1}\tilde{\rho}(t^{j},t^{l})\right]^{-d}\prod_{j=1}^{n}\left[\min_{0\leq
l\leq n,l\neq j}\tilde{\rho}(t^{j},t^{l})\right]^{-\gamma}d\bar{t}$
$\displaystyle=\sum_{i_{2}=0}^{1}\dots\sum_{i_{n}=0}^{n-1}\int_{S}\int_{\Gamma_{2,i_{2}}}\dots\int_{\Gamma_{n,i_{n}}}d\bar{t}\,\prod_{j=1}^{n}\tilde{\rho}(t^{j},t^{i_{j}})^{-d}\prod_{j=1}^{n}\left[\min_{0\leq
l\leq n,l\neq j}\tilde{\rho}(t^{j},t^{l})\right]^{-\gamma},$

where $i_{1}=0$.

Let $(t^{1},t^{2},\dots,t^{n})\in
S\times\Gamma_{2,i_{2}}\times\cdots\times\Gamma_{n,i_{n}}$. For each $1\leq
j\leq n$, let $\alpha_{n}(j)$ be the largest index in
$\\{0,1,\dots,n\\}\setminus\\{j\\}$ such that

$\tilde{\rho}(t^{j},t^{\alpha_{n}(j)})=\min_{0\leq l\leq n,l\neq
j}\tilde{\rho}(t^{j},t^{l}).$

Let $m_{n}$ be the number of $j$’s in $\\{1,\dots,n\\}$ such that
$\alpha_{n}(j)=n$. By Lemma 2.6, we have $0\leq m_{n}\leq K$, where $K$ is a
universal bound depending only on $N$. Then

$\displaystyle\prod_{j=1}^{n}\min_{0\leq l\leq n,l\neq
j}\tilde{\rho}(t^{j},t^{l})$ $\displaystyle=\prod_{\begin{subarray}{c}1\leq
j<n\\\
\alpha_{n}(j)=n\end{subarray}}\tilde{\rho}(t^{j},t^{n})\prod_{\begin{subarray}{c}1\leq
j\leq n\\\ \alpha_{n}(j)\neq n\end{subarray}}\min_{0\leq l\leq n,l\neq
j}\tilde{\rho}(t^{j},t^{l})$
$\displaystyle\geq\left[\tilde{\rho}(t^{n},t^{i_{n}})\right]^{m_{n}}\prod_{\begin{subarray}{c}1\leq
j\leq n\\\ \alpha_{n}(j)\neq n\end{subarray}}\min_{0\leq l\leq n,l\neq
j}\tilde{\rho}(t^{j},t^{l}),$

where in the last inequality, we have used the fact that
$t^{n}\in\Gamma_{n,i_{n}}$. Next, for $1\leq j\leq n$ with $\alpha_{n}(j)\neq
n$, we consider the largest index $\alpha_{n-1}(j)$ in
$\\{0,1,\dots,n\\}\setminus\\{j\\}$ such that

$\tilde{\rho}(t^{j},t^{\alpha_{n-1}(j)})=\min_{0\leq l\leq n,l\neq
j}\tilde{\rho}(t^{j},t^{l})$

and the number $m_{n-1}$ of $j$’s in $\\{1,\dots,n\\}$ such that
$\alpha_{n}(j)\neq n$ and $\alpha_{n-1}(j)=n-1$. Inductively, we can find
integers $m_{n-1},\dots,m_{2},m_{1}$, with $0\leq m_{j}\leq K$ for all $j$,
such that

$m_{1}+\dots+m_{n}=n\quad\text{and}\quad\prod_{j=1}^{n}\min_{0\leq l\leq
n,l\neq
j}\tilde{\rho}(t^{j},t^{l})\geq\prod_{j=1}^{n}\left[\tilde{\rho}(t^{j},t^{i_{j}})\right]^{m_{j}}.$

It follows that

$\displaystyle\int_{S^{n}}\prod_{j=1}^{n}\left[\min_{0\leq l\leq
j-1}\tilde{\rho}(t^{j},t^{l})\right]^{-d}\prod_{j=1}^{n}\left[\min_{0\leq
l\leq n,l\neq j}\tilde{\rho}(t^{j},t^{l})\right]^{-\gamma}d\bar{t}$
$\displaystyle\leq\sum_{(m_{1},\dots,m_{n})\in
M}\,\sum_{i_{2}=0}^{1}\dots\sum_{i_{n}=0}^{n-1}\int_{S}\int_{\Gamma_{2,i_{2}}}\dots\int_{\Gamma_{n,i_{n}}}d\bar{t}\,\prod_{j=1}^{n}\left[\tilde{\rho}(t^{j},t^{i_{j}})\right]^{-d-m_{j}\gamma},$

where

$M=\big{\\{}(m_{1},\dots,m_{n})\in\\{0,1,\dots,K\\}^{n}:m_{1}+\dots+m_{n}=n\big{\\}}.$

The cardinality of $M$ is bounded by $(K+1)^{n}$. Fix any $\beta_{0}\in(d,Q)$.
Then for all $\gamma\in(0,1)$ small enough, we have

$d<d+K\gamma<\beta_{0}<Q.$

If we fix $(t^{1},t^{2},\dots,t^{n-1})\in
S\times\Gamma_{2,i_{2}}\times\cdots\times\Gamma_{n-1,i_{n-1}}$, then by (2.6)
and Lemma 2.3,

$\displaystyle\sum_{i_{n}=0}^{n-1}\int_{\Gamma_{n,i_{n}}}\left[\rho(t^{n},t^{i_{n}})\right]^{-d-m_{n}\gamma}\,dt^{n}\leq
Cn^{d/Q+m_{n}\gamma/Q}\lambda_{N}(S)^{1-d/Q-m_{n}\gamma/Q}.$

Inductively, we integrate in the order of $dt^{n-1},\dots,dt^{1}$ to deduce
that

$\displaystyle\int_{S^{n}}\prod_{j=1}^{n}\Big{[}\min_{0\leq l\leq
j-1}\tilde{\rho}(t^{j},t^{l})\Big{]}^{-d}\prod_{j=1}^{n}\Big{[}\min_{0\leq
l\leq n,l\neq j}\tilde{\rho}(t^{j},t^{l})\Big{]}^{-\gamma}d\bar{t}$
$\displaystyle\leq C^{n}\sum_{(m_{1},\dots,m_{n})\in
M}\,\prod_{j=1}^{n}\Big{[}j\,^{d/Q+K\gamma/Q}\Big{]}\lambda_{N}(S)^{n(1-d/Q)-(m_{1}+\dots+m_{n})\gamma/Q}$
$\displaystyle\leq
C^{n}(K+1)^{n}(n!)^{d/Q+K\gamma/Q}\lambda_{N}(S)^{n(1-(d+\gamma)/Q)}.$

Therefore,

$\int_{S^{n}}J(\bar{t},\bar{k})\,d\bar{t}\leq
C^{n}(n!)^{d/Q+K\gamma/Q}\lambda_{N}(S)^{n(1-(d+\gamma)/Q)}.$ (2.26)

Note that this bound does not depend on $\bar{k}$. Combining (2.24) and
(2.26), we have

${\mathbb{E}}[(L(x,S)-L(y,S))^{n}]\leq
C^{n}|x-y|^{n\gamma}(n!)^{d/Q+K\gamma/Q}\lambda_{N}(S)^{n(1-(d+\gamma)/Q)}.$

This completes the proof of Proposition 2.8. ∎

We end this section with some lemmas, which will be useful later.

######  Lemma 2.9.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$, where
$Q=\sum_{j=1}^{N}(1/H_{j})$. For any $b>0$, there exists a finite constant $c$
such that the following hold.

  1. (i)

For all $a\in T$ and $0<r<1$ with $D:=B_{\rho}(a,r)\subset T$, for all
$x\in{\mathbb{R}}^{d}$ and $u>1$,

${\mathbb{P}}\left\\{L(x,D)\geq c\,r^{Q-d}u^{d/Q}\right\\}\leq\exp(-bu).$
(2.27)

  2. (ii)

For all $\gamma\in(0,1)$ small enough, for all $a\in T$ and $0<r<1$ with
$D:=B_{\rho}(a,r)\subset T$, for all $x,y\in{\mathbb{R}}^{d}$ with $|x-y|\leq
1$, for all $u>1$,

$\displaystyle\begin{aligned} {\mathbb{P}}\left\\{|L(x,D)-L(y,D)|\geq
c\,|x-y|^{\gamma}r^{Q-d-\gamma}u^{d/Q+K\gamma/Q}\right\\}\leq\exp(-bu),\end{aligned}$
(2.28)

where $K=K(N)>1$ is a finite constant.

###### Proof.

Let $A>0$ be a constant. By Chebyshev’s inequality and Proposition 2.4,

$\displaystyle{\mathbb{P}}\left\\{L(x,D)\geq Ar^{Q-d}n^{d/Q}\right\\}$
$\displaystyle\leq\frac{{\mathbb{E}}[L(x,D)^{n}]}{(Ar^{Q-d}n^{d/Q})^{n}}$
$\displaystyle\leq\left(\frac{C}{A}\right)^{n}\left(\frac{n!}{n^{n}}\right)^{d/Q}.$

By Stirling’s formula, given $b>0$, we can choose $A$ large enough such that
for all $n\geq 1$,

${\mathbb{P}}\left\\{L(x,D)\geq Ar^{Q-d}n^{d/Q}\right\\}\leq\exp(-2bn).$

This implies (2.27). Similarly, (2.28) can be proved by using Proposition 2.8.
∎

######  Lemma 2.10.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$. Then, there exists a
positive finite constant $C$ such that the following statements hold.

  1. (i)

For all $a\in T$ and $0<r<1$ with $D:=B_{\rho}(a,r)\subset T$, for all
$x\in{\mathbb{R}}^{d}$, for all integers $n\geq 1$,

${\mathbb{E}}[L(x+X(a),D)^{n}]\leq C^{n}(n!)^{d/Q}r^{n(Q-d)}.$ (2.29)

  2. (ii)

For all $\gamma\in(0,1)$ small enough, for all $a\in T$ and $0<r<1$ with
$D:=B_{\rho}(a,r)\subset T$, for all $x,y\in{\mathbb{R}}^{d}$ with $|x-y|\leq
1$, for all even integers $n\geq 2$,

$\displaystyle\begin{aligned} {\mathbb{E}}[(L(x+X(a),D)-L(y&+X(a),D))^{n}]\\\
&\leq C^{n}|x-y|^{n\gamma}(n!)^{d/Q+K\gamma/Q}r^{n(Q-d-\gamma)},\end{aligned}$
(2.30)

where $K=K(N)>1$ is a finite constant.

###### Proof.

For any fixed $a\in T$, consider the Gaussian random field
$\tilde{X}(t)=X(t)-X(a)$. If $X$ has a local time $L(x,S)$ on $S$, then
$\tilde{X}$ also has a local time $\tilde{L}(x,S)$ on $S$, and it follows from
(2.1) that $\tilde{L}(x,S)=L(x+X(a),S)$. Moreover, $\tilde{X}$ satisfies
Condition (A1) and a slightly different version of condition (A2) with the
inequality

$\mathrm{Var}(\tilde{Y}(t)|\tilde{Y}(t^{1}),\dots,\tilde{Y}(t^{n}))\geq
C_{3}\min\\{\rho^{2}(t,t^{k}):t\in\\{a,t^{0},t^{1},\dots,t^{n}\\}\\}.$

With little modification for Lemma 2.3, the proofs of Proposition 2.4 and 2.8
can be carried over to the Gaussian field $\tilde{X}$ to obtain (2.29) and
(2.30). ∎

######  Lemma 2.11.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$, where
$Q=\sum_{j=1}^{N}(1/H_{j})$. For any $b>0$, there exists a finite constant $c$
such that the following hold.

  1. (i)

For all $a\in T$ and $0<r<1$ with $D:=B_{\rho}(a,r)\subset T$, for all
$x\in{\mathbb{R}}^{d}$ and $u>1$,

${\mathbb{P}}\left\\{L(x+X(a),D)\geq c\,r^{Q-d}u^{d/Q}\right\\}\leq\exp(-bu).$
(2.31)

  2. (ii)

For all $\gamma\in(0,1)$ small enough, for all $a\in T$ and $0<r<1$ with
$D:=B_{\rho}(a,r)\subset T$, for all $x,y\in{\mathbb{R}}^{d}$ with $|x-y|\leq
1$, for all $u>1$,

$\displaystyle\begin{aligned}
{\mathbb{P}}\Big{\\{}|L(x+X(a)&,D)-L(y+X(a),D)|\\\ &\geq
c\,|x-y|^{\gamma}r^{Q-d-\gamma}u^{d/Q+K\gamma/Q}\Big{\\}}\leq\exp(-bu),\end{aligned}$
(2.32)

where $K=K(N)>1$ is a finite constant.

###### Proof.

As in Lemma 2.9, the proof is based on Lemma 2.10 and Chebyshev’s inequality.
∎

##  3\. Hölder conditions for the Local Times

In this section, we study local and global Hölder conditions in the set
variable for the local times, and discuss related sample path properties
including Chung’s law of the iterated logarithm and the modulus of non-
differentiability.

From Lemma 2.9 and the Borel–Cantelli lemma, one can easily derive the
following law of the iterated logarithm for the local time $L(x,\cdot)$: there
exists a finite constant $C$ such that for any $x\in{\mathbb{R}}^{d}$ and
$t\in T$,

$\limsup_{r\to 0}\frac{L(x,B_{\rho}(t,r))}{\varphi(r)}\leq C\quad a.s.,$ (3.1)

where $\varphi(r)=r^{Q-d}(\log\log(1/r))^{d/Q}$. It follows from Fubini’s
theorem that with probability one, (3.1) holds for $\lambda_{N}$-almost every
$t\in T$.

Next, we are going to prove a stronger version of this result, which will be
useful later in determining the exact Hausdorff measure of the level sets of
$X$.

Recall from Theorem 2.1 that if $X$ satisfies condition (A) on $T$ and
$d<\sum_{j=1}^{N}(1/H_{j})$, then $X$ has a jointly continuous local time on
$T$. From now on, we always use the jointly continuous version for the local
time whenever it exists, and still denote it by $L(x,\cdot)$.

######  Theorem 3.1.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q=\sum_{j=1}^{N}(1/H_{j})$.
Then there exists a finite constant $C$ such that for any
$x\in{\mathbb{R}}^{d}$, with probability 1,

$\limsup_{r\to 0}\frac{L(x,B_{\rho}(t,r))}{\varphi(r)}\leq C$ (3.2)

for $L(x,\cdot)$-almost every $t\in T$, where
$\varphi(r)=r^{Q-d}(\log\log(1/r))^{d/Q}$.

###### Proof.

The proof is similar to that of Proposition 4.1 in [35] and Theorem 8.10 in
[38]. For any $x\in{\mathbb{R}}^{d}$ and any integer $k\geq 1$, consider the
random measure $L_{k}(x,\cdot)$ on Borel subsets $B$ of $T$ defined by

$\displaystyle\begin{aligned}
L_{k}(x,B)&=\int_{B}\left(\frac{k}{2\pi}\right)^{d/2}\exp\left({-\frac{k|X(t)-x|^{2}}{2}}\right)\,dt\\\
&=\int_{B}\int_{{\mathbb{R}}^{d}}\frac{1}{(2\pi)^{d}}\exp\left(-\frac{|u|^{2}}{2k}+i\langle
u,X(t)-x\rangle\right)du\,dt.\end{aligned}$ (3.3)

By the occupation density formula (2.1) and the continuity of $y\mapsto
L(y,B)$ for all rectangles $B$ in $T$, one can verify that a.s. for all $B$,
$L_{k}(x,B)\to L(x,B)$ as $k\to\infty$. It follows that a.s., $L_{k}(x,\cdot)$
converges weakly to $L(x,\cdot)$.

For each $m\geq 1$, define $f_{m}(t)=L(x,B_{\rho}(t,2^{-m}))$. By Propositions
2.4 and 2.8, and the multiparameter version of Kolmogorov’s continuity theorem
[18], $f_{m}(t)$ is a.s. bounded and continuous on $T$. Then by the a.s. weak
convergence of $L_{k}(x,\cdot)$, for all $m,n\geq 1$,

$\int_{T}\,[f_{m}(t)]^{n}L(x,dt)=\lim_{k\to\infty}\int_{T}\,[f_{m}(t)]^{n}L_{k}(x,dt)\quad\text{a.s.}$

Hence, by the dominated convergence theorem, (3.3) and (2.3), we have

$\displaystyle\quad{\mathbb{E}}\int_{T}[L(x,B_{\rho}(t,2^{-m}))]^{n}L(x,dt)$
$\displaystyle=\frac{1}{(2\pi)^{d}}\lim_{k\to\infty}{\mathbb{E}}\int_{T}dt\int_{{\mathbb{R}}^{d}}du\,\exp\left(-\frac{|u|^{2}}{2k}+i\langle
u,X(t)-x\rangle\right)\left[L(x,B(t,2^{-m}))\right]^{n}$
$\displaystyle=\frac{1}{(2\pi)^{d}}\int_{T}dt\int_{{\mathbb{R}}^{d}}du\,{\mathbb{E}}\big{[}e^{i\langle
u,X(t)-x\rangle}L(x,B(t,2^{-m}))^{n}\big{]}$
$\displaystyle=\frac{1}{(2\pi)^{(n+1)d}}\int_{T}\int_{{B_{\rho}(t,2^{-m})}^{n}}d\bar{s}\int_{{\mathbb{R}}^{(n+1)d}}d\bar{u}\,e^{-i\sum_{\ell=1}^{n+1}\langle
x,u^{\ell}\rangle}{\mathbb{E}}\left(e^{i\sum_{\ell=1}^{n+1}\langle
u^{\ell},X(s^{\ell})\rangle}\right),$

where $\bar{u}=(u^{1},\dots,u^{n+1})\in{\mathbb{R}}^{(n+1)d}$,
$\bar{s}=(t,s^{2},\dots,s^{n+1})$ and $s^{1}=t$. Similar to the proof of
Proposition 2.4, we can deduce that

$\displaystyle\begin{aligned}
&\quad{\mathbb{E}}\int_{T}[L(x,B_{\rho}(t,2^{-m}))]^{n}L(x,dt)\\\ &\leq
C^{n}\int_{T\times{B_{\rho}(t,2^{-m})}^{n}}\frac{d\bar{s}}{[\det\mathrm{Cov}(Y(t),Y(s^{2}),\dots,Y(s^{n+1}))]^{d/2}}\\\
&\leq C^{n}(n!)^{d/Q}2^{-nm(Q-d)}.\end{aligned}$ (3.4)

Let $A>0$ be a constant whose value will be determined. Consider the random
set

$B_{m}=\\{t\in T:L(x,B_{\rho}(t,2^{-m}))>A\varphi(2^{-m})\\}.$

Consider the random measure $\mu$ on $T$ defined by $\mu(B)=L(x,B)$ for any
$B\in\mathscr{B}(T)$. Take $n=\lfloor\log m\rfloor$, the integer part of $\log
m$. Then by (LABEL:Eq:E_integral_f) and Stirling’s formula,

$\displaystyle{\mathbb{E}}\,\mu(B_{m})$
$\displaystyle\leq\frac{{\mathbb{E}}\int_{T}[L(x,B_{\rho}(t,2^{-m}))]^{n}L(x,dt)}{[A\varphi(2^{-m})]^{n}}$
$\displaystyle\leq\frac{C^{n}(n!)^{d/Q}2^{-nm(Q-d)}}{A^{n}2^{-nm(Q-d)}(\log
m)^{nd/Q}}\leq m^{-2}$

provided $A>0$ is chosen large enough. This implies that

${\mathbb{E}}\sum_{m=1}^{\infty}\mu(B_{m})<\infty.$

By the Borel–Cantelli lemma, with probability 1, for $\mu$-a.e. $t\in T$, we
have

$\limsup_{m\to\infty}\frac{L(x,B_{\rho}(t,2^{-m}))}{\varphi(2^{-m})}\leq A.$
(3.5)

For any $r>0$ small enough, there exists an integer $m$ such that $2^{-m}\leq
r<2^{-m+1}$ and (3.5) can be applied. Since $\varphi(r)$ is increasing near
$r=0$, we can use a monotonicity argument to obtain (3.2). ∎

Next, we study the local and global Hölder conditions for $L^{*}$, the
supremum of the local time defined by

$L^{*}(B)=\sup_{x\in{\mathbb{R}}^{d}}L(x,B),\quad
B\in\mathscr{B}({\mathbb{R}}^{N}).$

######  Theorem 3.2.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$, where
$Q=\sum_{j=1}^{N}(1/H_{j})$. Then there exist finite constants $C$ and
$C^{\prime}$ such that for any $t\in T$,

$\limsup_{r\to 0}\frac{L^{*}(B_{\rho}(t,r))}{\varphi(r)}\leq
C\quad\text{a.s.}$ (3.6)

and

$\limsup_{r\to 0}\sup_{t\in T}\frac{L^{*}(B_{\rho}(t,r))}{\Phi(r)}\leq
C^{\prime}\quad\text{a.s.}$ (3.7)

where $\varphi(r)=r^{Q-d}(\log\log(1/r))^{d/Q}$ and
$\Phi(r)=r^{Q-d}(\log(1/r))^{d/Q}$.

###### Proof.

As in [11, 35], the proof of Theorem 3.2 is based on Lemma 2.11 and a chaining
argument. We will give a sketch of the proof with necessary modifications.

In order to prove (3.6), it suffices to show that for any $a\in T$,

$\limsup_{n\to\infty}\frac{L^{*}(B_{n})}{\varphi(2^{-n})}\leq
C\quad\textup{a.s.}$ (3.8)

where $B_{n}=B_{\rho}(a,2^{-n})$. We divide the proof of (3.8) into four
steps.

(1). By Lemma 2.1 of Talagrand [30], there exist positive constants $c_{1}$
and $c_{2}$ such that for any $r\in(0,1)$ and $u>c_{1}r$,

${\mathbb{P}}\left\\{\sup_{t\in B_{\rho}(a,r)}|X(t)-X(a)|\geq
u\right\\}\leq\exp\left(-c_{2}(u/r)^{2}\right).$

Taking $u_{n}=2^{-n}\sqrt{2c_{2}^{-1}\log n}$, we have

${\mathbb{P}}\left\\{\sup_{t\in B_{n}}|X(t)-X(a)|\geq u_{n}\right\\}\leq
n^{-2}.$

It follows from the Borel–Cantelli lemma that almost surely, for all $n$
large,

$\sup_{t\in B_{n}}|X(t)-X(a)|\leq u_{n}.$ (3.9)

(2). Let $\theta_{n}=2^{-n}(\log\log 2^{n})^{-K}$, where $K>1$ is the constant
in (2.31). Define

$G_{n}=\left\\{x\in{\mathbb{R}}^{d}:|x|\leq u_{n}\textup{ with
}x=\theta_{n}p\textup{ for some }p\in{\mathbb{Z}}^{d}\right\\}.$

Then, when $n$ is large enough, the cardinality of $G_{n}$ satisfies

$\operatorname{card}G_{n}\leq C(\log n)^{(K+1)d}.$

It follows from (2.31) of Lemma 2.11 (with $b=2$) that we can find a finite
constant $c$ such that for all $n$ large,

${\mathbb{P}}\left\\{\max_{x\in G_{n}}L(x+X(a),B_{n})\geq
c\,\varphi(2^{-n})\right\\}\leq C(\log n)^{(K+1)d}\,n^{-2}.$

By the Borel–Cantelli lemma, almost surely, for all $n$ large,

$\max_{x\in G_{n}}L(x+X(a),B_{n})\leq c\,\varphi(2^{-n}).$ (3.10)

(3). For integers $n,k\geq 1$ and $x\in G_{n}$, define

$F(n,k,x)=\bigg{\\{}y\in{\mathbb{R}}^{d}:y=x+\theta_{n}\sum_{j=1}^{k}\varepsilon_{j}2^{-j},\varepsilon_{j}\in\\{0,1\\}^{d}\textup{
for }1\leq j\leq k\bigg{\\}}.$

A pair of points $y_{1},y_{2}\in F(n,k,x)$ is said to be linked if
$y_{2}-y_{1}=\theta_{n}\varepsilon 2^{-k}$ for some
$\varepsilon\in\\{0,1\\}^{d}$. Consider the event $F_{n}$ defined by

$\displaystyle F_{n}=\bigcup_{x\in G_{n}}\bigcup_{k\geq
1}\bigcup_{y_{1},y_{2}}\Big{\\{}|L$
$\displaystyle(y_{1}+X(a),B_{n})-L(y_{2}+X(a),B_{n})|$ $\displaystyle\geq
c\,2^{-n(Q-d-\gamma)}|y_{1}-y_{2}|^{\gamma}(k\log n)^{d/Q+K\gamma/Q}\Big{\\}}$

where $\bigcup_{y_{1},y_{2}}$ denotes the union over all linked pairs
$y_{1},y_{2}\in F(n,k,x)$, the constant $c$ is given by Lemma 2.11 with $b=2$
and a small $\gamma\in(0,1)$ is chosen so that (2.32) holds. Note that there
are at most $2^{kd}3^{d}$ linked pairs in $F(n,k,x)$. It follows that for $n$
large,

$\displaystyle\begin{aligned} {\mathbb{P}}(F_{n})&\leq C(\log
n)^{(K+1)d}\sum_{k=1}^{\infty}2^{kd}\exp(-2k\log n)\\\ &=C(\log
n)^{(K+1)d}\frac{2^{d}n^{-2}}{1-2^{d}n^{-2}}.\end{aligned}$ (3.11)

Since $\sum_{n=1}^{\infty}{\mathbb{P}}(F_{n})<\infty$, it follows from the
Borel–Cantelli lemma that a.s. $F_{n}$ occurs only finitely many times.

(4). For $y\in{\mathbb{R}}^{d}$ with $|y|\leq u_{n}$, $n\geq 1$, we can
represent $y$ in the form $y=\lim_{k\to\infty}y_{k}$ with

$y_{k}=x+\theta_{n}\sum_{j=1}^{k}\varepsilon_{j}2^{-j},$ (3.12)

where $y_{0}=x\in G_{n}$ and $\varepsilon_{j}\in\\{0,1\\}^{d}$ for
$j=1,\dots,k$. Since the local time $L$ is jointly continuous, by (3.12) and
the triangle inequality, we see that on the event $F_{n}^{c}$,

$\displaystyle\begin{aligned} &|L(y+X(a),B_{n})-L(x+X(a),B_{n})|\\\
&\leq\sum_{k=1}^{\infty}|L(y_{k}+X(a),B_{n})-L(y_{k-1}+X(a),B_{n})|\\\
&\leq\sum_{k=1}^{\infty}c\,2^{-n(Q-d-\gamma)}|y_{k}-y_{k-1}|^{\gamma}(k\log
n)^{d/Q+K\gamma/Q}\\\ &\leq C\varphi(2^{-n}).\end{aligned}$ (3.13)

Combining (3.10) and (LABEL:LT:tri_ineq), we get that a.s. for all $n$ large,

$\sup_{|y|\leq u_{n}}L(y+X(a),B_{n})\leq C\varphi(2^{-n}).$ (3.14)

Since $L^{*}(B_{n})=\sup\\{L(x,B_{n}):x\in\overline{X(B_{n})}\\}$, we can
deduce (3.8) from (3.14) and (3.9). This proves (3.6).

The proof of (3.7) is similar. It suffices to prove that

$\limsup_{n\to\infty}\sup_{B\in\mathscr{B}_{n}}\frac{L^{*}(B)}{\varphi(2^{-n})}\leq
C,$ (3.15)

where, for each $n\geq 1$, $\mathscr{B}_{n}$ is a covering of $T$ consisting
of disjoint anisotropic cubes of side lengths
$2^{-n/H_{1}},\dots,2^{-n/H_{N}}$. Note that
$\operatorname{card}\mathscr{B}_{n}\leq C2^{nQ}$. Define

$G_{n}=\left\\{x\in{\mathbb{R}}^{d}:|x|\leq n\textup{ with
}x=\theta_{n}p\textup{ for some }p\in{\mathbb{Z}}^{d}\right\\}.$

Then we can use Lemma 2.9 to find a constant $c$ such that a.s. for all $n$
large,

$\max_{B\in\mathscr{B}_{n}}\max_{x\in G_{n}}L(x,B)\leq c\,\Phi(2^{-n}).$
(3.16)

Define $F(n,k,x)$ as before and

$\displaystyle F_{n}=\bigcup_{B\in\mathscr{B}_{n}}\bigcup_{x\in
G_{n}}\bigcup_{k\geq 1}\bigcup_{y_{1},y_{2}}\Big{\\{}|L$
$\displaystyle(y_{1},B)-L(y_{2},B)|$ $\displaystyle\geq
c\,2^{-n(Q-d-\gamma)}|y_{1}-y_{2}|^{\gamma}(k\log
2^{n})^{d/Q+K\gamma/Q}\Big{\\}}.$

As in (3.11), we can use (2.28) to show that a.s. $F_{n}$ occurs only finitely
many times. Since $X(t)$ is continuous, there exists $n_{0}=n_{0}(\omega)$
such that $\sup_{t\in T}|X(t)|\leq n_{0}$ a.s. If $|y|\leq n$, then by the
chaining argument as in (3.12) and (LABEL:LT:tri_ineq), we can prove that on
$F_{n}^{c}$,

$|L(y,B)-L(x,B)|\leq C\Phi(2^{-n})$

for some $x\in G_{n}$. This and (3.16) imply that a.s. for all $n$ large,

$\sup_{B\in\mathscr{B}_{n}}\sup_{|y|\leq n}L(y,B)\leq
C\Phi(2^{-n})\quad\textup{a.s.}$ (3.17)

Since $L(y,T)=0$ for $|y|>n_{0}$, (3.15) follows from (3.17). This completes
the proof. ∎

As pointed out by Berman [4] (see also Ehm [11]), the Hölder conditions of the
local times are closely related to the degree of oscillations of the sample
paths of $X(t)$. As a consequence of Theorem 3.2 and the inequality (3.20)
below, we obtain lower bounds for Chung’s law of iterated logarithm and the
modulus of non-differentiability for $X(t)$.

######  Theorem 3.3.

Suppose $X$ satisfies condition (A) on $T$ and let
$Q=\sum_{j=1}^{N}(1/H_{j})$. Then there exist positive constants $C$ and
$C^{\prime}$ such that for any $t\in T$,

$\liminf_{r\to 0}\sup_{s\in
B_{\rho}(t,r)}\frac{|X(s)-X(t)|}{r(\log\log(1/r))^{-1/Q}}\geq
C\quad\text{a.s.}$ (3.18)

and

$\liminf_{r\to 0}\inf_{t\in T}\sup_{s\in
B_{\rho}(t,r)}\frac{|X(s)-X(t)|}{r(\log(1/r))^{-1/Q}}\geq
C^{\prime}\quad\text{a.s.}$ (3.19)

In particular, the sample paths of $X$ are a.s. nowhere differentiable in $T$.

###### Proof.

It is enough to consider the case $d=1$. Let $I$ denote the smallest closed
interval containing the range of $X$ on $B_{\rho}(t,r)$. It follows from the
occupation density formula (2.1) that

$\displaystyle\begin{aligned}
\lambda_{N}(B_{\rho}(t,r))&=\int_{I}L(x,B_{\rho}(t,r))\,dx\\\ &\leq
L^{*}(B_{\rho}(t,r))\times\sup_{s,s^{\prime}\in
B_{\rho}(t,r)}|X(s)-X(s^{\prime})|.\end{aligned}$ (3.20)

Since $\lambda_{N}(B_{\rho}(t,r))=Cr^{Q}$, (3.18) follows from (3.20) and
(3.6). Similarly, (3.19) follows from (3.20) and (3.7). ∎

We end this section with the following remark on the optimality of the
inequalities in (3.20), (3.7), (3.18), and (3.19).

######  Remark 3.4.

As pointed out in Ehm [11], if the left-hand side of (3.18) [or (3.19), resp.]
is also bounded above by a finite constant a.s., then (3.20) implies that
(3.6) [or (3.7), resp.] is also bounded below by a positive constant a.s. and
hence the Hölder conditions for the local time will be optimal. Indeed, if
$X$, in addition to satisfying Condition (A), has stationary increments or
satisfies Assumption 2.1 in [10], then (3.18) is a.s. equal to some positive
finite constant, see [25] and [23], respectively. On the other hand, Wang, Su
and Xiao [32] determined the exact modulus of non-differentiability for a
class of Gaussian random fields with stationary and isotropic increments.
Hence (3.7) is also optimal for these Gaussian random fields.

##  4\. The Exact Hausdorff Measure of Level Sets

Let us consider the class $\mathscr{C}$ of functions
$\varphi:[0,\delta_{0}]\to{\mathbb{R}}_{+}$ such that $\varphi$ is
nondecreasing, continuous, $\varphi(0)=0$, and satisfies the doubling
condition, i.e. there exists a finite constant $c_{0}>0$ such that

$\frac{\varphi(2s)}{\varphi(s)}\leq c_{0}$ (4.1)

for all $s\in(0,\delta_{0}/2)$.

Let $\varphi\in\mathscr{C}$ and let $\rho$ be a metric on ${\mathbb{R}}^{N}$.
For any Borel set $A$ in ${\mathbb{R}}^{N}$, the _Hausdorff measure_ of $A$
with respect to the function $\varphi$, in metric $\rho$ is defined by

$\displaystyle\mathcal{H}^{\varphi}_{\rho}(A)=\lim_{\varepsilon\to
0}\inf\Bigg{\\{}\sum_{n=1}^{\infty}\varphi(2r_{n}):A\subseteq\bigcup_{n=1}^{\infty}B_{\rho}(t^{n},r_{n})\text{
where }t^{n}\in{\mathbb{R}}^{N}\text{ and }r_{n}\leq\varepsilon\text{ for all
}n\Bigg{\\}}.$

We use the notation $\mathcal{H}^{\varphi}(A)$ if $\rho$ is the Euclidean
metric.

When $\varphi(s)=s^{\alpha}$, where $\alpha>0$ is a real number,
$\mathcal{H}^{\alpha}_{\rho}(A)=\mathcal{H}^{\varphi}_{\rho}(A)$ is called the
$\alpha$-dimensional Hausdorff measure of $A$ in metric $\rho$, and the
Hausdorff dimension of $A$ in $\rho$ is defined as

$\dim_{H}^{\rho}(A)=\inf\\{\alpha>0:\mathcal{H}^{\alpha}_{\rho}(A)=0\\}.$

Hausdorff dimension in metric $\rho$ is useful in studying the fractal
properties of anisotropic Gaussian random fields; see Wu and Xiao [33] and
Xiao [38].

Suppose $X$ satisfies condition (A) on a compact interval
$T\subset{\mathbb{R}}^{N}$. Let $Q=\sum_{j=1}^{N}(1/H_{j})$ and

$\rho(t,s)=\sum_{j=1}^{N}|t_{j}-s_{j}|^{H_{j}}.$

By Theorem 7.1 of Xiao [38], for any $x\in{\mathbb{R}}^{d}$, if $Q<d$, then
$X^{-1}(x)\cap T=\varnothing$ a.s.; if $d<Q$, then with positive probability,
the Hausdorff dimension of $X^{-1}(x)\cap T$ in the Euclidean metric is
(assuming that $0<H_{1}\leq H_{2}\leq\dots\leq H_{N}<1$)

$\displaystyle\begin{aligned} \dim_{H}(X^{-1}(x)\cap T)&=\min_{1\leq k\leq
N}\bigg{\\{}\sum_{j=1}^{k}\frac{H_{k}}{H_{j}}+N-k-H_{k}d\bigg{\\}}\\\
&=\sum_{j=1}^{\tau}\frac{H_{\tau}}{H_{j}}+N-\tau-H_{\tau}d,\end{aligned}$
(4.2)

where $\tau$ is the unique integer between $1$ and $N$ such that
$\sum_{j=1}^{\tau-1}(1/H_{j})\leq d<\sum_{j=1}^{\tau}(1/H_{j})$. More
generally, Biermé, Lacaux and Xiao [5] determined the Hausdorff dimension of
the inverse image $X^{-1}(F)$ for Borel sets $F$ in ${\mathbb{R}}^{N}$.

The Hausdorff dimension of the level set may be different when the underlying
metric is not the Euclidean metric. Theorem 4.2 of Wu and Xiao [34] shows that
if $d<Q$, then almost surely, the Hausdorff dimension of $X^{-1}(x)\cap T$ in
the metric $\rho$ is

$\dim_{H}^{\rho}(X^{-1}(x)\cap T)=Q-d$

for all $x\in{\mathbb{R}}^{d}$ such that $L(x,T)>0$.

For the special case where $H_{1}=\dots=H_{N}=H$, we have
$\dim_{H}(X^{-1}(x)\cap T)=N-Hd$ and $\dim_{H}^{\rho}(X^{-1}(x)\cap
T)=\frac{N}{H}-d$. In this case, since $\rho(t,s)\asymp|t-s|^{H}$, it is easy
to see that for any function $\varphi\in\mathscr{C}$, there exist positive
finite constants $C_{1}$ and $C_{2}$ such that

$C_{1}\mathcal{H}^{\psi}(A)\leq\mathcal{H}^{\varphi}_{\rho}(A)\leq
C_{2}\mathcal{H}^{\psi}(A)$ (4.3)

for all $A\in\mathscr{B}({\mathbb{R}}^{N})$, where $\psi$ is defined by
$\psi(r)=\varphi(r^{H})$.

The Hausdorff measure with respect to a suitable function provides a way to
measure the size of the level sets, especially when the level sets have
trivial Lebesgue measure or $\alpha$-dimensional Hausdorff measure. It would
be interesting to determine the exact Hausdorff measure function (or gauge
function) of the level set, that is, to find a function $\varphi$ such that

$0<\mathcal{H}^{\varphi}_{\rho}(X^{-1}(x)\cap T)<\infty$

almost surely or with positive probability.

Recall that the $\rho$-upper $\varphi$-density of a finite Borel measure $\mu$
on ${\mathbb{R}}^{N}$ at the point $t\in{\mathbb{R}}^{N}$ is defined by

$\overline{D}^{\varphi,\rho}_{\mu}(t):=\limsup_{r\to
0}\frac{\mu(B_{\rho}(t,r))}{\varphi(r)}.$

There exists a positive constant $c\geq 1$ depending only on $c_{0}$ in (4.1)
such that

$c^{-1}\mathcal{H}^{\varphi}_{\rho}(E)\inf_{t\in
E}\overline{D}^{\varphi,\rho}_{\mu}(t)\leq\mu(E)\leq
c\,\mathcal{H}^{\varphi}_{\rho}(E)\sup_{t\in
E}\overline{D}^{\varphi,\rho}_{\mu}(t)$ (4.4)

for any finite Borel measure $\mu$ on ${\mathbb{R}}^{N}$ and any Borel set $E$
in ${\mathbb{R}}^{N}$ (see Theorem 4.1 of [34]).

The following is a partial result giving a lower bound for the Hausdorff
measure. As shown by Theorem 5.3 below, it is possible to show
$\mathcal{H}^{\varphi}_{\rho}(X^{-1}(x)\cap T)<\infty$ under certain extra
conditions.

######  Theorem 4.1.

Suppose $X$ satisfies condition (A) on $T$ and $d<Q$, where
$Q=\sum_{j=1}^{N}(1/H_{j})$. Let $\varphi(r)=r^{Q-d}(\log\log(1/r))^{d/Q}$.
Then there is a constant $C>0$ such that for any $x\in{\mathbb{R}}^{d}$,

$CL(x,T)\leq\mathcal{H}^{\varphi}_{\rho}(X^{-1}(x)\cap T)\quad\text{a.s.}$

In particular, if $H_{1}=\dots=H_{N}=H$, then

$CL(x,T)\leq\mathcal{H}^{\psi}(X^{-1}(x)\cap T)\quad\text{a.s.}$

where $\psi(r)=r^{N-Hd}(\log\log(1/r))^{Hd/N}$.

###### Proof.

Take $\mu=L(x,\cdot\cap T)$, which is a.s. a finite Borel measure on
${\mathbb{R}}^{N}$ whose support is $X^{-1}(x)\cap T$. By Theorem 3.1, there
exists a finite constant $C$ such that

$\sup_{t\in E}\overline{D}^{\varphi,\rho}_{\mu}(t)\leq C\quad\text{a.s.}$

Then we can use the upper bound of (4.4) with $E=X^{-1}(x)\cap T$ to obtain
the result. The special case where $H_{1}=\dots=H_{N}=H$ follows from (4.3) ∎

In view of Theorem 4.1, a natural question is whether $X^{-1}(x)=\varnothing$
a.s. when $d\geq Q$. As shown by Dalanget al. [10, Theorem 2.6], this is
indeed the case if, in addition to satisfying conditions of Theorem 4.1, $X$
also satisfies Assumptions 2.1 and 2.4 in [10].

##  5\. Systems of Stochastic Heat Equations

As an example, we consider the following system of stochastic heat equations:

$\displaystyle\begin{cases}\displaystyle\frac{\partial}{\partial
t}u_{j}(t,x)=\Delta u_{j}(t,x)+\dot{W}_{j}(t,x),&t\geq
0,x\in{\mathbb{R}}^{N},\\\ u_{j}(0,x)=0&j=1,\dots,d,\end{cases}$ (5.1)

where $\dot{W}=(\dot{W}_{1},\dots,\dot{W}_{d})$ is a $d$-dimensional Gaussian
noise. In this section, we will apply our main results to study the local
times and level sets of the solution $u$ of (5.1). At the end of this section,
we will also discuss a more general system (5.18) where each component of the
solution may depend on all the $\dot{W}_{j}$’s ($j=1,\dots,d$).

We assume that $\dot{W}_{1},\dots,\dot{W}_{d}$ are i.i.d. and
$\dot{W}_{j}(t,x)$ is either (i) white in time and colored in space with
covariance

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

for $N\geq 1$ and $0<\beta<2\wedge N$, or (ii) the space-time white noise for
$N=1$ (take $\beta=1$ in this case).

The solution of (5.1) is the Gaussian random field $u=\\{u(t,x):t\geq
0,x\in{\mathbb{R}}^{N}\\}$ with i.i.d. components $u_{1},\dots,u_{d}$, given
by

$u_{j}(t,x)=\int_{0}^{t}\int_{{\mathbb{R}}^{N}}G(t-s,x-y)W_{j}(ds\,dy),$

where $G$ is the fundamental solution of the heat equation:

$G(t,x)=\frac{1}{(4\pi t)^{N/2}}\exp\Big{(}-\frac{|x|^{2}}{4t}\Big{)}{\bf
1}_{\\{t>0\\}}.$

Recall that for any $0<a<b<\infty$, there exist positive finite constants
$C_{1},C_{2}$ such that

$C_{1}\rho((t,x),(s,y))\leq({\mathbb{E}}|u(t,x)-u(s,y)|^{2})^{1/2}\leq
C_{2}\rho((t,x),(s,y))$ (5.2)

for all $(t,x),(s,y)\in[a,b]\times[-b,b]^{N}$, where

$\displaystyle\rho((t,x),(s,y))=|t-s|^{\frac{2-\beta}{4}}+|x-y|^{\frac{2-\beta}{2}}.$
(5.3)

See [9], Lemma 4.2. Hence $u$ satisfies (A1) on any compact interval in
$(0,\infty)\times{\mathbb{R}}^{N}$.

We are going to prove that $u$ also satisfies condition (A2), the strong local
nondeterminism in variables $(t,x)$ jointly with respect to the metric $\rho$.
In fact, using the idea of string process in Mueller and Tribe [28], it is
shown in Herrell et al. [15] that $u$ admits the decomposition

$u(t,x)=U(t,x)-Y(t,x),$ (5.4)

where $U(t,x)$ has stationary increments and satisfies strong LND in metric
$\rho$, whereas $Y(t,x)$ is a.s. continuously differentiable in
$(t,x)\in[0,\infty)\times{\mathbb{R}}^{N}$. We mention that, by applying the
decomposition (5.4) and the stationarity of the increments of $U(t,x)$,
Herrell et al. [15] proved the regularity properties such as the exact uniform
and local moduli of continuity and Chung’s law of the iterated logarithm for
$u(t,x)$. Lee and Xiao [23] showed that the regularity properties such as
those studied in [15] can be established under the more general framework of
Dalang et al [10] for Gaussian random fields whose increments may not be
stationary.

In Proposition 5.1 below, we will prove directly that $u$ itself satisfies the
strong LND property in (A2).

Let $n\geq 1$,
$(t^{1},x^{1}),\dots,(t^{n},x^{n})\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{N}$
and $a_{1},\dots,a_{n}\in{\mathbb{R}}$. Let

$g(s,y)=\sum_{j=1}^{n}a_{j}G(t^{j}-s,x^{j}-y){\bf 1}_{[0,t^{j}]}.$

Then by Plancherel’s theorem, we have

$\displaystyle\begin{aligned}
{\mathbb{E}}\Bigg{[}\bigg{(}\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}=C\int_{{\mathbb{R}}}d\tau\int_{{\mathbb{R}}^{N}}d\xi\,|\mathscr{F}g(\tau,\xi)|^{2}\,|\xi|^{\beta-N}.\end{aligned}$
(5.5)

In the above, $\mathscr{F}g$ denotes the Fourier transform of $g$, that is,

$\mathscr{F}g(\tau,\xi)=\int_{{\mathbb{R}}}\int_{{\mathbb{R}}^{N}}e^{-i\tau
s-i\langle\xi,y\rangle}g(t,x)\,ds\,dy.$

One can directly verify that

$\mathscr{F}(G(t-\cdot,x-\cdot){\bf
1}_{[0,t]})(\tau,\xi)=e^{-i\langle\xi,x\rangle}\frac{e^{-i\tau
t}-e^{-t|\xi|^{2}}}{|\xi|^{2}-i\tau}.$ (5.6)

######  Proposition 5.1.

For any $0<a<b<\infty$, there exists a constant $C>0$ such that for all
integers $n\geq 1$ and all
$(t,x),(t^{1},x^{1}),\dots,(t^{n},x^{n})\in[a,b]\times[-b,b]^{N}$,

$\mathrm{Var}\left(u_{1}(t,x)|u_{1}(t^{1},x^{1}),\dots,u_{1}(t^{n},x^{n})\right)\geq
C\min_{1\leq i\leq n}\rho((t,x),(t^{i},x^{i}))^{2},$ (5.7)

where $\rho$ is the metric in (5.3).

###### Proof.

Since $u$ is Gaussian, the conditional variance in (5.7) is the squared
$L^{2}$-distance of $u_{1}(t,x)$ from the linear subspace of
$L^{2}({\mathbb{P}})$ generated by
$u_{1}(t^{1},x^{1}),\dots,u_{1}(t^{n},x^{n})$, that is,

$\mathrm{Var}\left(u_{1}(t,x)|u_{1}(t^{1},x^{1}),\dots,u_{1}(t^{n},x^{n})\right)=\inf_{a_{1},\dots,a_{n}\in{\mathbb{R}}}{\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}.$

Therefore, it suffices to show that there exists a positive constant $C$ such
that

${\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}\geq
Cr^{2-\beta},$

for any $n\geq 1$, any
$(t,x),(t^{1},x^{1}),\dots,(t^{n},x^{n})\in[a,a^{\prime}]\times[-b,b]^{N}$,
and any $a_{1},\dots,a_{n}\in{\mathbb{R}}$, where

$r=\min_{1\leq j\leq n}(|t-t^{j}|^{1/2}\vee|x-x^{j}|).$

From (5.5) and (5.6), we have

$\displaystyle\quad{\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}$
(5.8)
$\displaystyle=C\int_{{\mathbb{R}}}d\tau\int_{{\mathbb{R}}^{N}}d\xi\,\bigg{|}e^{-i\langle\xi,x\rangle}(e^{-i\tau
t}-e^{-t|\xi|^{2}})-\sum_{j=1}^{n}a_{j}e^{-i\langle\xi,x^{j}\rangle}(e^{-i\tau
t^{j}}-e^{-t^{j}|\xi|^{2}})\bigg{|}^{2}\frac{|\xi|^{\beta-N}}{|\xi|^{4}+|\tau|^{2}}.$

Let $M$ be such that $|t-s|^{1/2}\vee|x-y|\leq M$ for all
$(t,x),(s,y)\in[a,a^{\prime}]\times[-b,b]^{N}$. Let $h=\min\\{a/M^{2},1\\}$.
Let $\varphi:{\mathbb{R}}\to{\mathbb{R}}$ and
$\psi:{\mathbb{R}}^{N}\to{\mathbb{R}}$ be nonnegative smooth test functions
that vanish outside the interval $(-h,h)$ and the unit ball respectively and
satisfy $\varphi(0)=\psi(0)=1$. Let
$\varphi_{r}(\tau)=r^{-2}\varphi(r^{-2}\tau)$ and
$\phi_{r}(\xi)=r^{-N}\psi(r^{-1}\xi)$. Consider the integral

$\displaystyle
I:=\int_{{\mathbb{R}}}d\tau\int_{{\mathbb{R}}^{N}}d\xi\bigg{[}e^{-i\langle\xi,x\rangle}(e^{-i\tau
t}-e^{-t|\xi|^{2}})-\sum_{j=1}^{n}a_{j}e^{-i\langle\xi,x^{j}\rangle}(e^{-i\tau
t^{j}}-e^{-t^{j}|\xi|^{2}})\bigg{]}$ $\displaystyle\times
e^{i\langle\xi,x\rangle}e^{i\tau
t}\widehat{\varphi}_{r}(\tau)\widehat{\psi}_{r}(\xi).$

By inverse Fourier transform, we have

$\displaystyle I=(2\pi)^{1+N}\bigg{[}\varphi_{r}(0)\psi_{r}(0)$
$\displaystyle-\varphi_{r}(t)(p_{t}\ast\psi_{r})(0)$
$\displaystyle-\sum_{j=1}^{n}a_{j}\Big{(}\varphi_{r}(t-t^{j})\psi_{r}(x-x^{j})-\varphi_{r}(t)(p_{t^{j}}\ast\psi_{r})(x-x^{j})\Big{)}\bigg{]},$

where $p_{t}(x)=G(t,x)$ is the heat kernel. By the definition of $r$,
$|t-t^{j}|\geq r^{2}$ or $|x-x^{j}|\geq r$ for every $j$, thus
$\varphi_{r}(t-t^{j})\psi_{r}(x-x^{j})=0$. Moreover, since $t/r^{2}\geq
a/M^{2}\geq h$, we have $\varphi_{r}(t)=0$ and hence

$I=(2\pi)^{1+N}r^{-2-N}.$ (5.9)

On the other hand, by the Cauchy–Schwarz inequality and (5.8),

$\displaystyle I^{2}\leq
C\,{\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}\int_{{\mathbb{R}}}\int_{{\mathbb{R}}^{N}}\big{|}\widehat{\varphi}_{r}(\tau)\widehat{\psi}_{r}(\xi)\big{|}^{2}\big{(}|\xi|^{4}+|\tau|^{2}\big{)}|\xi|^{N-\beta}d\tau\,d\xi.$

Note that $\widehat{\varphi}_{r}(\tau)=\widehat{\varphi}(r^{2}\tau)$ and
$\widehat{\psi}_{r}(\xi)=\widehat{\psi}(r\xi)$. Then by scaling,

$\displaystyle\int_{{\mathbb{R}}}\int_{{\mathbb{R}}^{N}}\big{|}\widehat{\varphi}_{r}(\tau)\widehat{\psi}_{r}(\xi)\big{|}^{2}\big{(}|\xi|^{4}+|\tau|^{2}\big{)}|\xi|^{N-\beta}d\tau\,d\xi$
$\displaystyle=r^{-6+\beta-2N}\int_{{\mathbb{R}}}\int_{{\mathbb{R}}^{N}}\big{|}\widehat{\varphi}(\tau)\widehat{\psi}(\xi)\big{|}^{2}\big{(}|\xi|^{4}+|\tau|^{2}\big{)}|\xi|^{N-\beta}d\tau\,d\xi.$

The last integral is finite since $\widehat{\varphi}$ and $\widehat{\psi}$ are
rapidly decreasing functions. It follows that

$I^{2}\leq
C_{0}r^{-6+\beta-2N}\,{\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}$
(5.10)

for some finite constant $C_{0}$. Combining (5.9) and (5.10), we get that

${\mathbb{E}}\Bigg{[}\bigg{(}u_{1}(t,x)-\sum_{j=1}^{n}a_{j}u_{1}(t^{j},x^{j})\bigg{)}^{2}\Bigg{]}\geq(2\pi)^{2+2N}C_{0}^{-1}r^{2-\beta}.$

The proof is complete. ∎

We have shown that $u$ satisfies condition (A) on any compact interval $T$ in
$(0,\infty)\times{\mathbb{R}}^{N}$. Therefore, the following result is a
direct consequence of Theorems 2.1, 3.1, and 3.2.

######  Corollary 5.2.

Suppose $d<Q:=\frac{2(2+N)}{2-\beta}$ and $T$ is any compact interval in
$(0,\infty)\times{\mathbb{R}}^{N}$. Then $u(t,x)$ has a jointly continuous
local time $L(z,T)$ on $T$ satisfying the Hölder conditions (3.2), (3.6) and
(3.7).

Finally, we consider the level sets $u^{-1}(z)\cap T=\\{(t,x)\in
T:u(t,x)=z\\}$, where $z\in{\mathbb{R}}^{d}$. Recall from Section 4 that
$u^{-1}(z)\cap T=\varnothing$ a.s. if $Q<d$. The same is true when $Q=d$,
which was proved by Dalang, Mueller and Xiao [10].

If $d<Q$, we are able to obtain a precise result for the level sets of $u$.
The following theorem determines the exact Hausdorff measure function for the
level sets.

######  Theorem 5.3.

Suppose $d<Q:=\frac{2(2+N)}{2-\beta}$. Let $T$ be a compact interval in
$(0,\infty)\times{\mathbb{R}}^{N}$ and
$\varphi(r)=r^{Q-d}(\log\log(1/r))^{d/Q}$. Then there exists a constant $C>0$
such that for any $z\in{\mathbb{R}}^{d}$,

$CL(z,T)\leq\mathcal{H}_{\rho}^{\varphi}(u^{-1}(z)\cap
T)<\infty\quad\text{a.s.}.$ (5.11)

######  Remark 5.4.

We conjecture that (5.11) can be strengthened to: There exist positive finite
constants $C_{1}$ and $C_{2}$ such that

$C_{1}L(z,T)\leq\mathcal{H}_{\rho}^{\varphi}(u^{-1}(z)\cap T)\leq
C_{2}L(x,T)\quad\text{a.s.}$

######  Remark 5.5.

Let $\delta((t,x),(s,y))=|t-s|^{1/2}+|x-y|$ be the parabolic metric in
${\mathbb{R}}^{1+N}$. Since
$\rho((t,x),(s,y))\asymp[\delta((t,x),(s,y))]^{(2-\beta)/2}$, it follows from
Theorem 5.3 that

$CL(z,T)\leq\mathcal{H}^{\psi}_{\delta}(u^{-1}(z)\cap
T)<\infty\quad\text{a.s.}$

with $\psi(r)=r^{2+N-d(2-\beta)/2}(\log\log(1/r))^{d/Q}$. In particular, the
parabolic Hausdorff dimension of the level set is $2+N-d(2-\beta)/2$.

###### Proof of Theorem 5.3.

The lower bound in (5.11) follows immediately from Theorem 4.1. To prove that
the Hausdorff measure is finite, we use the method in Xiao [35], which is
similar to Talagrand’s covering argument in [31]. First, we may assume
$T=B((t_{0},x_{0}),\eta_{0})$, where $\eta_{0}>0$ is small and
$(t_{0},x_{0})\in T$ are fixed. Let

$\displaystyle u^{1}(t,x)=u(t,x)-u^{2}(t,x)\quad\text{and}\quad
u^{2}(t,x)={\mathbb{E}}(u(t,x)|u(t_{0},x_{0})).$

Then $u^{1}$ and $u^{2}$ are independent processes.

For our current proof, it would be easier for us to work with “cubes” that are
comparable to balls in metric $\rho$ and, at the same time, have the nested
property of the ordinary dyadic cubes. For this reason, we are going to use a
family of generalized dyadic cubes $\mathscr{Q}$, which can be obtained by
Theorem 2.1 and Remark 2.2 of [16] applied to the metric space $(T,\rho)$.
More specifically, $\mathscr{Q}=\bigcup_{q=1}^{\infty}\mathscr{Q}_{q}$, where
$\mathscr{Q}_{q}=\\{I_{q,\ell}:\ell=1,\dots,n_{q}\\}$ are families of Borel
subsets of $T$, and there exist constants $c_{1},c_{2}$ such that the
following properties hold:

  1. (i)

$T=\bigcup_{\ell=1}^{n_{q}}I_{q,\ell}$ for each $q\geq 1$;

  2. (ii)

Either $I_{q,\ell}\cap I_{q^{\prime},\ell^{\prime}}=\varnothing$ or
$I_{q,\ell}\subset I_{q^{\prime},\ell^{\prime}}$ whenever $q\geq q^{\prime}$,
$1\leq\ell\leq n_{q}$, $1\leq\ell^{\prime}\leq n_{q^{\prime}}$;

  3. (iii)

For each $q,\ell$, there exists $x_{q,\ell}\in T$ such that
$B_{\rho}(x_{q,\ell},c_{1}2^{-q})\subset I_{q,\ell}\subset
B_{\rho}(x_{q,\ell},c_{2}2^{-q})$ and
$\\{x_{q,\ell}:1,\dots,n_{q}\\}\subset\\{x_{q+1,\ell}:\ell=1,\dots,n_{q+1}\\}$
for all $q\geq 1$.

For simplicity, any member of $\mathscr{Q}_{q}$ will be called a dyadic cube
of order $q$.

The main ingredient for the covering argument is the following estimate: there
exist a finite constant $K_{1}$ and $\eta_{1}>0$ small such that for all
$0<r_{0}<\eta_{1}$, and all $(t,x)\in T$, we have

$\displaystyle\begin{aligned}
{\mathbb{P}}\left\\{\exists\,r\in[r_{0}^{2},r_{0}],\sup_{(s,y)\in
B_{\rho}((t,x),2c_{2}r)}|u(t,x)-u(s,y)|\leq
K_{1}r\Big{(}\log\log\frac{1}{r}\Big{)}^{-1/Q}\right\\}\\\ \geq
1-\exp\bigg{(}-\Big{(}\log\frac{1}{r_{0}}\Big{)}^{1/2}\bigg{)}.\end{aligned}$
(5.12)

This is proved for a more general class of Gaussian random fields in Dalang et
al. [10].

Moreover, by Lemma 5.3 and Lemma 7.5 of [10], there exists a finite constant
$K_{2}$ such that for all $(t,x),(s,y)\in T$,

$|u^{2}(t,x)-u^{2}(s,y)|\leq
K_{2}\big{(}|t-s|+\sum_{j=1}^{N}|x_{j}-y_{j}|\big{)}|u(t_{0},x_{0})|.$ (5.13)

Let

$\displaystyle R_{p}=\Bigg{\\{}(t,x)\in T:\,$
$\displaystyle\exists\,r\in[2^{-2p},2^{-p}]\text{ such that }$
$\displaystyle\sup_{(s,y)\in B_{\rho}((t,x),2c_{2}r)}|u(t,x)-u(s,y)|\leq
K_{1}r\left(\log\log\frac{1}{r}\right)^{-1/Q}\Bigg{\\}}.$

Consider the events

$\displaystyle\Omega_{p,1}$
$\displaystyle=\Big{\\{}\omega:\lambda_{N}(R_{p})\geq\lambda_{N}(T)(1-\exp(-\sqrt{p}/4))\Big{\\}},$
$\displaystyle\Omega_{p,2}$
$\displaystyle=\Big{\\{}\omega:|u(t_{0},x_{0})|\leq 2^{pb}\Big{\\}},$

where $b>0$ is chosen and fixed such that $\frac{2}{2-\beta}-b>1$. By (5.12),
${\mathbb{P}}\\{(t,x)\in R_{p}\\}\geq 1-\exp(-\sqrt{p/2})$. Then by Fubini’s
theorem, $\sum_{p=1}^{\infty}{\mathbb{P}}(\Omega_{p,1}^{c})<\infty$. Moreover,
it is easy to see that
$\sum_{p=1}^{\infty}{\mathbb{P}}(\Omega_{p,2}^{c})<\infty$. Consider the event

$\displaystyle\Omega_{p,3}=\bigg{\\{}\omega:\forall\,I\in\mathscr{Q}_{2p},\sup_{(t,x),(s,y)\in
I}|u(t,x)-u(s,y)|\leq K_{2}2^{-2p}(\log 2^{2p})^{1/2}\bigg{\\}}.$

By Lemma 2.1 of Talagrand [30] (see also Lemma 3.1 in [10]), we see that for
$K_{2}$ and $p$ large,

${\mathbb{P}}\left\\{\sup_{(t,x),(s,y)\in I}|u(t,x)-u(s,y)|\leq
K_{2}2^{-2p}(\log
2^{2p})^{1/2}\right\\}\leq\exp\left(-\left(\frac{K_{2}}{c_{2}}\right)^{2}p\right).$

Since the cardinality of the family $\mathscr{Q}_{2p}$ is at most $C2^{2pQ}$,
we have $\sum_{p=1}^{\infty}{\mathbb{P}}(\Omega_{p,3}^{c})<\infty$ provided
$K_{2}$ is chosen to be a sufficiently large constant. Let
$\Omega_{p}=\Omega_{p,1}\cap\Omega_{p,2}\cap\Omega_{p,3}$. Then

$\displaystyle{\mathbb{P}}(\Omega^{*})=1,\quad\text{where
}\Omega^{*}:=\bigcup_{\ell\geq 1}\bigcap_{p\geq\ell}\Omega_{p}.$ (5.14)

Moreover, we define

$\displaystyle R_{p}^{\prime}=\Bigg{\\{}(t,x)\in T:\,$
$\displaystyle\exists\,r\in[2^{-2p},2^{-p}]\text{ such that }$
$\displaystyle\sup_{(s,y)\in
B_{\rho}((t,x),2c_{2}r)}|u^{1}(t,x)-u^{1}(s,y)|\leq
2K_{1}r\left(\log\log\frac{1}{r}\right)^{-1/Q}\Bigg{\\}}$

and the event

$\displaystyle\Omega_{p,4}=\left\\{\omega:\lambda_{N}(R_{p}^{\prime})\geq\lambda_{N}(T)(1-\exp(-\sqrt{p}/4))\right\\}.$

Note that (5.13) implies that $R_{p}\subset R_{p}^{\prime}$ on $\Omega_{p,3}$
for $p$ large enough and hence

$\displaystyle\Omega_{p,1}\cap\Omega_{p,3}\subset\Omega_{p,4}.$ (5.15)

We are going to construct a random covering for the level set $u^{-1}(z)\cap
T$. For any $p\geq 1$ and $(t,x)\in T$, let $I_{p}(t,x)\in\mathscr{Q}_{p}$ be
the unique dyadic cube of order $p$ containing $(t,x)$. We say that
$I_{q}(t,x)$ is a good dyadic cube of order $q$ if it satisfies the following
property:

$\sup_{(s,y),(s^{\prime},y^{\prime})\in
I_{q}(t,x)}|u^{1}(s,y)-u^{1}(s^{\prime},y^{\prime})|\leq 8K_{1}2^{-q}(\log\log
2^{q})^{-1/Q}.$ (5.16)

For each $(t,x)\in R_{p}^{\prime}$, there is some $r\in[2^{-q},2^{-q+1}]$ with
$p+1\leq q\leq 2p$ such that

$\displaystyle\sup_{(s,y)\in
B_{\rho}((t,x),2c_{2}r)}|u^{1}(t,x)-u^{1}(s,y)|\leq
2K_{1}r\left(\log\log\frac{1}{r}\right)^{-1/Q}.$

By property (iii) above, $I_{q}(t,x)$ is contained in some ball
$B_{\rho}(c_{2}2^{-q})$, and thus by triangle inequality, we have

$\displaystyle\sup_{(s,y),(s^{\prime},y^{\prime})\in
I_{q}(t,x)}|u^{1}(s,y)-u^{1}(s^{\prime},y^{\prime})|$
$\displaystyle\leq\sup_{(s,y)\in
B_{\rho}((t,x),2c_{2}r)}|u^{1}(s,y)-u^{1}(t,x)|+\sup_{(s^{\prime},y^{\prime})\in
B_{\rho}((t,x),2c_{2}r)}|u^{1}(t,x)-u^{1}(s^{\prime},y^{\prime})|$
$\displaystyle\leq 4K_{1}r\left(\log\log\frac{1}{r}\right)^{-1/Q}\leq
8K_{1}2^{-q}(\log\log 2^{q})^{-1/Q}.$

Hence, $I_{q}(t,x)$ is a good dyadic cube of order $q$. By property (ii), we
obtain in this way a family $\mathscr{G}_{p}^{1}$ of disjoint dyadic cubes
that cover $R_{p}^{\prime}$. On the other hand, we let $\mathscr{G}_{p}^{2}$
be the family of dyadic cubes in $T$ of order $2p$ that are not contained in
any cube of $\mathscr{G}_{p}^{1}$. In particular, the cubes in
$\mathscr{G}_{p}^{2}$ are contained in $T\setminus R_{p}^{\prime}$. Let
$\mathscr{G}_{p}=\mathscr{G}_{p}^{1}\cup\mathscr{G}_{p}^{2}$. Note that
$\mathscr{G}_{p}$ depends only on the random field $\\{u^{1}(t,x),(t,x)\in
T\\}$.

For each dyadic cube $I\in\mathscr{Q}$, choose a fixed point in $I\cap T$ and
label it by $(t_{I},x_{I})$. For any $I\in\mathscr{Q}_{q}$ of order $q$, where
$p\leq q\leq 2p$, consider the event

$\Omega_{p,I}=\\{\omega:|u(t_{I},x_{I})-z|\leq 2r_{p,I}\\}$

where

$r_{p,I}=\begin{cases}8K_{1}2^{-q}(\log\log 2^{q})^{-1/Q}&\text{if
}I\in\mathscr{G}_{p}^{1}\text{ and }I\text{ is of order }q,\\\
K_{2}2^{-2p}(\log 2^{2p})^{1/2}&\text{if }I\in\mathscr{G}_{p}^{2}.\end{cases}$

Let $\mathscr{F}_{p}$ be the subcover of $\mathscr{G}_{p}$ (depending on
$\omega$) defined by

$\mathscr{F}_{p}(\omega)=\\{I\in\mathscr{G}_{p}(\omega):\omega\in\Omega_{p,I}\\}.$

We claim that for $p$ large, on the event $\Omega_{p}$, $\mathscr{F}_{p}$
covers the set $u^{-1}(z)\cap T$. Suppose $\Omega_{p}$ occurs and $(t,x)\in
u^{-1}(z)\cap T$. Since $\mathscr{G}_{p}$ covers $T$, the point $(t,x)$ is
contained in some dyadic cube $I$ and either $I\in\mathscr{G}_{p}^{1}$ or
$I\in\mathscr{G}_{p}^{2}$.

Case 1: if $I\in\mathscr{G}_{p}^{1}$, then $I=I_{q}(t,x)$ is a good dyadic
cube of order $q$, where $p\leq q\leq 2p$, and (5.16) holds. Recall that $I$
is contained in some ball $B_{\rho}(c_{2}2^{-q})$. Since $\Omega_{p,2}$
occurs, it follows that from (5.13) and (5.16) that

$\displaystyle|u(t_{I},x_{I})-z|$
$\displaystyle\leq|u^{1}(t_{I},x_{I})-u^{1}(t,x)|+|u^{2}(t_{I},x_{I})-u^{2}(t,x)|$
$\displaystyle\leq 8K_{1}2^{-q}(\log\log
2^{q})^{-1/Q}+K_{2}\Big{(}(2c_{2}^{\frac{4}{2-\beta}}+2Nc_{2}^{\frac{2}{2-\beta}})2^{-q\frac{2}{2-\beta}}\Big{)}2^{pb}.$

This is $\leq 2r_{p,I}$ for $p$ large because $b$ is chosen such that
$\frac{2}{2-\beta}-b>1$. Hence $I\in\mathscr{F}_{p}$.

Case 2: if $I\in\mathscr{G}_{p}^{2}$, since $\Omega_{p,3}$ occurs, we have

$|u(t_{I},x_{I})-z|=|u(t_{I},x_{I})-u(t,x)|\leq K_{2}2^{-2p}p^{1/2}.$

In this case, $I\in\mathscr{F}_{p}$. Hence the claim is proved.

Let $\Sigma_{1}$ be the $\sigma$-algebra generated by $\\{u^{1}(t,x):(t,x)\in
T\\}$. To estimate the conditional probability
${\mathbb{P}}(\Omega_{p,I}|\Sigma_{1})$, we use the conditional variance
formula and (5.2) to get that for all $(t,x)\in
T=B_{\rho}((t_{0},x_{0}),\eta_{0})$,

$\displaystyle\mathrm{Var}(u^{2}(t,x))$
$\displaystyle=\mathrm{Var}({\mathbb{E}}(u(t,x)|u(t_{0},x_{0})))=\mathrm{Var}(u(t,x))-{\mathbb{E}}[\mathrm{Var}(u(t,x)|u(t_{0},x_{0}))]$
$\displaystyle\geq\inf_{(t,x)\in T}\mathrm{Var}(u(t,x))-C_{2}\sup_{(t,x)\in
T}\rho^{2}((t,x),(t_{0},x_{0}))$

which is bounded from below by a positive constant provided $\eta_{0}>0$ is
small enough. It follows that there is a constant $C<\infty$ such that for all
$(t,x)\in T$, $v\in{\mathbb{R}}^{d}$ and $r>0$,
${\mathbb{P}}\\{|u^{2}(t,x)-v|\leq r\\}\leq Cr^{d}$. Since $u^{1}$ and $u^{2}$
are independent, we have

${\mathbb{P}}(\Omega_{p,I}|\Sigma_{1})\leq Cr_{p,I}^{d}.$ (5.17)

Now, we estimate the expected value of
$\mathcal{H}_{\rho}^{\varphi}(u^{-1}(z)\cap T)$. Let $q\\{I\\}$ denote the
order of $I\in\mathscr{F}_{p}$. By conditioning, (5.15) and (5.17),

$\displaystyle{\mathbb{E}}\left[{\bf
1}_{\Omega_{p}}\sum_{I\in\mathscr{F}_{p}}\varphi(2c_{2}2^{-q\\{I\\}})\right]$
$\displaystyle\leq{\mathbb{E}}\left[{\bf
1}_{{\Omega}_{p,4}}\sum_{q=p}^{2p}\sum_{I\in\mathscr{Q}_{q}}\varphi(2c_{2}2^{-q}){\bf
1}_{\\{I\in\mathscr{G}_{p}\\}}{\bf 1}_{\Omega_{p,I}}\right]$
$\displaystyle={\mathbb{E}}\left[{\bf
1}_{{\Omega}_{p,4}}\sum_{q=p}^{2p}\sum_{I\in\mathscr{Q}_{q}}\varphi(2c_{2}2^{-q}){\bf
1}_{\\{I\in\mathscr{G}_{p}\\}}{\mathbb{E}}\left({\bf
1}_{\Omega_{p,I}}|\Sigma_{1}\right)\right]$ $\displaystyle\leq
C\,{\mathbb{E}}\left[{\bf
1}_{{\Omega}_{p,4}}\sum_{q=p}^{2p}\sum_{I\in\mathscr{Q}_{q}}\varphi(2c_{2}2^{-q})r_{p,I}^{d}{\bf
1}_{\\{I\in\mathscr{G}_{p}\\}}\right].$

If $I\in\mathscr{G}_{p}^{1}$ is of order $q$, then

$\varphi(2c_{2}2^{-q})r_{p,I}^{d}\leq C2^{-q(Q-d)}(\log\log
2^{q})^{d/Q}2^{-qd}(\log\log 2^{q})^{-d/Q}\leq C\lambda_{N}(I),$

and these $I$’s are disjoint sets contained in $T$. If
$I\in\mathscr{G}_{p}^{2}$, then

$\varphi(2c_{2}2^{-2p})r_{p,I}^{d}\leq C2^{-2pQ}(\log\log
2^{2p})^{d/Q}p^{d/2}.$

Note that there are at most $C2^{2pQ}\exp(-\sqrt{p}/4)$ many such $I$’s on the
event $\Omega_{p,4}$ since $T\setminus R_{p}^{\prime}$ has Lebesgue measure
$\leq\exp(-\sqrt{p}/4)$ and each $I\in\mathscr{G}_{p}^{2}$ has Lebesgue
measure $\sim C2^{-2pQ}$. It follows that

$\displaystyle{\mathbb{E}}\left[{\bf
1}_{\Omega_{p}}\sum_{I\in\mathscr{F}_{p}}\varphi(2^{-q\\{I\\}})\right]$
$\displaystyle\leq
C\,{\mathbb{E}}\left[\sum_{I\in\mathscr{G}_{p}^{1}}\lambda_{N}(I)+{\bf
1}_{\Omega_{p,4}}\sum_{I\in\mathscr{G}_{p}^{2}}2^{-pQ}(\log\log
2^{2p})^{d/Q}p^{d/2}\right]$ $\displaystyle\leq C\left(\lambda_{N}(T)+(\log
2p)^{d/Q}p^{d/2}\exp(-\sqrt{p}/4)\right)$

provided $p$ is large. Recall that $\mathscr{F}_{p}$ is a cover for
$u^{-1}(z)\cap T$ on $\Omega_{p}$ for large $p$ and each $I$ is contained in a
ball of radius $c_{2}2^{-q\\{I\\}}$ in metric $\rho$. Therefore, by (5.14) and
Fatou’s lemma,

$\displaystyle{\mathbb{E}}\left[\mathcal{H}^{\varphi}_{\rho}(u^{-1}(z)\cap
T)\right]$ $\displaystyle={\mathbb{E}}\left[{\bf
1}_{\Omega^{*}}\,\mathcal{H}^{\varphi}_{\rho}(u^{-1}(z)\cap T)\right]$
$\displaystyle\leq\liminf_{p\to\infty}{\mathbb{E}}\left[{\bf
1}_{\Omega_{p}}\sum_{I\in\mathscr{F}_{p}}\varphi(2c_{2}2^{-q\\{I\\}})\right]\leq
C\lambda_{N}(T)<\infty.$

This completes the proof of Theorem 5.3. ∎

Finally, we consider the solution
$\\{v(t,x)=(v_{1}(t,x),\dots,v_{d}(t,x))^{T},t\geq 0,x\in{\mathbb{R}}^{N}\\}$
of the system

$\displaystyle\begin{cases}\displaystyle\frac{\partial}{\partial
t}v_{i}(t,x)=\Delta v_{i}(t,x)+\sum_{j=1}^{d}A_{ij}\dot{W}_{j}(t,x),&t\geq
0,x\in{\mathbb{R}}^{N},\\\ v_{i}(0,x)=0,&i=1,\dots,d,\end{cases}$ (5.18)

where $A$ is a non-random $d\times d$ invertible matrix and
$(\dot{W}_{1},\dots,\dot{W}_{d})$ is the $d$-dimensional Gaussian noise as
defined in (5.1). The solution is given by the Gaussian random field

$\displaystyle
v_{i}(t,x)=\int_{0}^{t}\int_{{\mathbb{R}}^{N}}G(t-s,x,y)\sum_{j=1}^{d}A_{ij}W_{j}(ds,dy),\quad
i=1,\dots,d.$

Hence, $v=Au$, where $u(t,x)=(u_{1}(t,x),\dots,u_{d}(t,x))^{T}$ is the
solution of (5.1). In general, $v(t,x)$ has non-i.i.d. components. Although
our main results, which are based on the i.i.d. setting, cannot be applied
directly, we can still make use of the relation $v=Au$ to obtain properties
for the local times of $v$ from those of the local times of $u$.

Let $T$ be a compact interval in $(0,\infty)\times{\mathbb{R}}^{N}$. Suppose
that $d<Q:=\frac{2(2+N)}{2-\beta}$. Since $u(t,x)$ has a local time $L_{u}$ on
$T$ (by Corollary 5.2) and $A$ is an invertible matrix, it follows from the
relation $v=Au$ that the occupation measure
$\mu^{v}_{T}(\cdot)=\lambda_{1+N}\\{(t,x)\in T:v(t,x)\in\cdot\,\\}$ is
absolutely continuous with respect to the Lebesgue measure $\lambda_{d}$ on
${\mathbb{R}}^{d}$. Therefore, $v(t,x)$ also has a local time $L_{v}$ on $T$.
By the occupation density formula (2.1) and the relation $v=Au$, we obtain the
following expression for the local time of $v$:

$\displaystyle L_{v}(z,I)=|\det A|^{-1}L_{u}(A^{-1}z,I),$ (5.19)

where $I$ is any interval in $T$. As a result, all the properties of
$L_{v}(z,I)$ including joint continuity and Hölder conditions can be deduced
from those of $L_{u}(z,I)$. Moreover, since $v^{-1}(z)=u^{-1}(A^{-1}z)$,
Theorem 5.3 and (5.19) imply that

$\displaystyle C|\det A|\cdot
L_{v}(z,T)\leq\mathcal{H}^{\varphi}_{\rho}(v^{-1}(z)\cap
T)<\infty\quad\text{a.s.}$

Hence, the exact Hausdorff measure function for the level sets of $v$ is the
same as that of $u$.

Acknowledgements. The authors wish to thank Professor Davar Khoshnevisan for
stimulating discussions and encouraging the authors to publish this paper. Y.
Xiao was supported in part by the NSF grant DMS-2153846.

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Cheuk Yin Lee: School of Science and Engineering, The Chinese University of
Hong Kong, Shenzhen, China.
E-mail address<EMAIL_ADDRESS>

Yimin Xiao: Department of Statistics and Probability, C413 Wells Hall,
Michigan State University, East Lansing, MI 48824, United States.
E-mail address<EMAIL_ADDRESS>