# Heat content for Gaussian processes:
small-time asymptotic analysis

Kei Kobayashi and Hyunchul Park Department of Mathematics, Fordham University.
Email<EMAIL_ADDRESS>of Mathematics, State University of
New York at New Paltz, Email<EMAIL_ADDRESS>
###### Abstract

This paper establishes the small-time asymptotic behaviors of the regular heat
content and spectral heat content for general Gaussian processes in both one-
dimensional and multi-dimensional settings, where the boundary of the
underlying domain satisfies some smoothness condition. For the amount of heat
loss associated with the spectral heat content, the exact asymptotic behavior
with rate function being the expected supremum process is obtained, whereas
for the regular heat content, the exact asymptotic behavior is described in
terms of the standard deviation function.

Key words: Gaussian process, regular heat content, spectral heat content,
asymptotic behavior

2020 Mathematics Subject Classification: 60G15, 60J45

##  1 Introduction

For a Brownian motion $W=(W_{t})_{t\geq 0}$ and a bounded domain $D$ in
$\mathbb{R}^{d}$, consider the function

$\displaystyle
Q^{W}_{D}(t):=\int_{D}\mathbb{P}_{x}(\tau^{W}_{D}>t)\mathrm{d}x,\ \ \ t>0,$

where $\tau^{W}_{D}=\inf\\{t>0:W_{t}\in D^{c}\\}$ denotes the first exit time
of $W$ from $D$, and $\mathbb{P}_{x}$ is the law under which $W$ starts at the
point $x\in D$. This function, called the spectral heat content (SHC),
measures the amount of heat that has not exited the domain $D$ as of time $t$.
The intensive study of the SHC for Brownian motions that was conducted more
than three decades ago revealed that the SHC contains geometric information
about the domain and spectral information about the infinitesimal generator of
the underlying killed Brownian motions; see e.g., [24, 25, 27, 28]. Associated
with the SHC is the regular heat content (RHC) defined by

$H^{W}_{D}(t):=\int_{D}\mathbb{P}_{x}(W_{t}\in D)\mathrm{d}x,\ \ \ t>0.$

The RHC also measures the amount of heat contained in the domain $D$, but heat
particles exiting the domain do not get killed and hence are allowed to return
to $D$; see e.g., [18, 26] for discussions of the RHC. In contrast, for Lévy
processes, the investigation into the RHC and SHC started within the last
decade, where the presence of jumps requires careful analysis, especially in a
multi-dimensional setting (i.e., $d\geq 2$); see e.g., [1, 2, 9, 13, 14, 15,
19, 20, 21]. The RHC and SHC for processes with delay introduced by inverse
subordinators have been studied in [13, 14].

The purpose of this paper is to introduce the SHC and RHC for general Gaussian
processes, including fractional Brownian motions and Ornstein–Uhlenbeck
processes, and study their asymptotic behaviors as $t\downarrow 0$. We will
conduct the study in both one-dimensional and multi-dimensional settings. In
the one-dimensional setting, Theorems 3 and 5 establish the exact first-order
asymptotic behaviors of the RHC and SHC, respectively, with error bounds of
exponential decay obtained. In the multi-dimensional setting, we assume that
the domain $D$ is a $C^{1,1}$ open set and that the process has i.i.d.
Gaussian components; the latter may look somewhat restrictive, but Theorems 2
and 8 still recover the well-known first-order small-time asymptotic behaviors
of the RHC and SHC for Brownian motions, respectively. Separately, Proposition
7 establishes a one-sided bound for the SHC, with an error bound of
exponential decay valid for all small enough $t>0$; the assumption of
independent components is not essential in this proposition.

Let us now make three key remarks on our results. First and foremost, as of
now, no article can be found in the literature on the SHC and RHC for Gaussian
processes, except for the special case of Brownian motions. Therefore, our
theorems produce a number of new results as corollaries, including those for
fractional Brownian motions and fractional/non-fractional Ornstein–Uhlenbeck
processes. These examples are provided in Section 6.

Second, our statements do not assume self-similarity of the Gaussian process,
and to the authors’ knowledge, even the proofs of Theorems 3 and 5 in the one-
dimensional setting are significantly different from the proofs for the
Brownian motion case available in the literature including [1, 2, 24]. On the
other hand, the proof of Theorem 8 employs an approximate scaling argument
that is similar to the one in [15], which requires an assumption on weak
convergence of a scaled Gaussian process in the Skorokhod space. We admit that
the latter is a strong condition, but it still covers some non-self-similar
Gaussian processes.

Our third remark is that for the exact asymptotic results in Theorems 2, 3, 5
and 8, the rate functions for the RHC and SHC are described in terms of the
standard deviation function and the expected supremum function of the
underlying Gaussian process, respectively. The distinct descriptions for the
rate functions reflect the difference in the natures of the RHC and SHC.
Namely, the SHC requires tracking of particle locations from the beginning
through a given time point $t$ since whether the particles have ever exited
the domain or not matters (as they are killed upon exit), whereas the RHC
concerns particle locations at a given time $t$ only since particles never get
killed.

The organization of this paper is as follows. Section 2 reviews known facts
and introduces the RHC and SHC for Gaussian processes. Sections 3 and 4
provide our results for the RHC and SHC, respectively, as well as their
proofs, except for the proof of Theorem 8, which is postponed to Section 5.
Section 6 illustrates applications of our results to some concrete Gaussian
processes.

##  2 Preliminaries

A stochastic process $X=(X_{t})_{t\in[0,1]}$ in $\mathbb{R}$ is called a
Gaussian process if all its finite-dimensional distributions are multivariate
Gaussian. The distribution of a zero-mean Gaussian process $X$ is
characterized by its covariance function

$R_{X}(s,t):=\textrm{Cov}(X_{s},X_{t})=\mathbb{E}[X_{s}X_{t}],\ \
s,t\in[0,1].$

Throughout, we assume that a given zero-mean Gaussian process $X$ is defined
on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and has càdlàg paths
starting at the origin. Thus, $X$ can be regarded as a random element that is
defined on the measurable space $(\Omega,\mathcal{F})$ and taking values in
the space $\mathbb{D}[0,1]$ of càdlàg functions on $[0,1]$ that is equipped
with the Skorokhod $J_{1}$ topology. The associated Borel $\sigma$-algebra
$\mathcal{B}(\mathbb{D}[0,1])$ coincides with the $\sigma$-algebra generated
by the coordinate projections (or the finite-dimensional cylinder sets). For
details, see the discussion given in [29, Section 11.5.3].

The variance function of a zero-mean Gaussian process $X$ is denoted by

(1) $\displaystyle R_{t}:=R_{X}(t,t)=\mathbb{E}[X_{t}^{2}],\ \ t\in[0,1],$

and its supremum function is denoted by

$\sigma^{2}_{t}:=\sup_{s\in[0,t]}R_{s},\ \ t\in[0,1].$

The first moment of the running supremum of $X$ is denoted by

$\mu_{t}:=\mathbb{E}\biggl{[}\sup_{s\in[0,t]}X_{s}\biggr{]},\ \ t\in[0,1].$

For example, if $X$ is a standard Brownian motion $W=(W_{t})_{t\in[0,1]}$ in
$\mathbb{R}$ with characteristic function
$\mathbb{E}[e^{iuW_{t}}]=e^{-tu^{2}}$, then $R_{t}=2t$ and by [12, Problem
2.8.2],

(2) $\displaystyle\mu_{t}=\int_{0}^{\infty}\frac{x}{\sqrt{\pi
t}}e^{-\frac{x^{2}}{4t}}\mathrm{d}x=2\sqrt{\frac{t}{\pi}}.$

We assume throughout the paper that for each fixed $t\in(0,1]$, the zero-mean
Gaussian variable $X_{t}$ is non-degenerate; i.e., $\mathbb{P}(X_{t}\neq
0)>0$. This immediately implies $R_{t}>0$. Moreover, it follows that
$\mu_{t}>0$. Indeed, the assumption that $X_{0}=0$ yields $\mu_{t}\geq 0$,
whereas if $\mu_{t}=0$, then $\sup_{s\in[0,t]}X_{s}=0$ $\mathbb{P}$-a.s., so
in particular, $X_{t}\leq 0$ $\mathbb{P}$-a.s.; however, this is impossible
since the support of the non-degenerate Gaussian variable $X_{t}$ is
$\mathbb{R}$. Note also that the latter argument yields
$\sup_{s\in[0,t]}X_{s}>0$ $\mathbb{P}$-a.s., and hence, $\mu_{t}$ has the
integral representation

$\mu_{t}=\int_{0}^{\infty}\mathbb{P}\biggl{(}\sup_{s\in[0,t]}X_{s}>x\biggr{)}\mathrm{d}x.$

For each $t\in(0,1]$, assume that

(3) $\displaystyle\sup_{s\in[0,t]}X_{s}<\infty\ \ \mathbb{P}\textrm{-a.s.},$

which is known to be equivalent to the condition that $\mu_{t}<\infty$. In
fact, the following Borell–TIS inequality holds:

(4) $\displaystyle\mathbb{P}\biggl{(}\sup_{s\in[0,t]}X_{s}>x\biggr{)}\leq
2e^{-\frac{(x-\mu_{t})^{2}}{2\sigma_{t}^{2}}}\ \ \textrm{for all}\ \
x>\mu_{t}.$

See e.g., the discussion given in [23, Section 1]. Moreover, by [17, Theorem
2.5], $R_{t}\leq\sigma^{2}_{t}<\infty$. For stationary Gaussian processes, the
almost sure boundedness condition (3) is equivalent to continuity of the
sample paths; see e.g., [16, Theorem 10.2].

Under the almost sure boundedness condition (3), by the dominated convergence
theorem and the right-continuity of the paths of $X$,

(5) $\displaystyle\mu_{t}\downarrow 0\ \ \textrm{as}\ \ t\downarrow 0.$

The almost sure right-continuity of $X$ implies right-continuity in
probability, and since convergence in probability in the space of zero-mean
Gaussian variables coincides with convergence in $L^{2}$ due to [11, Theorem
1.4], it follows that the variance function $R_{t}$ defined in (1) is right-
continuous. In particular,

(6) $\displaystyle R_{t}\downarrow 0\ \ \textrm{as}\ \ t\downarrow 0,$

which is equivalent to

$\sigma^{2}_{t}\downarrow 0\ \ \textrm{as}\ \ t\downarrow 0.$

This paper also investigates a stochastic process
$X=(X_{t})_{t\in[0,1]}=(X_{t}^{1},\ldots,X_{t}^{d})_{t\in[0,1]}$ in
$\mathbb{R}^{d}$ with $d\geq 2$ whose components are i.i.d. zero-mean Gaussian
processes in $\mathbb{R}$. Such processes are simply referred to as Gaussian
processes in $\mathbb{R}^{d}$ and are considered to be random elements taking
values in the space $\mathbb{D}[0,1]$ of $\mathbb{R}^{d}$-valued càdlàg
functions on $[0,1]$ that is equipped with the $J_{1}$ topology.

Given a Gaussian process $X=(X_{t})_{t\in[0,1]}$ starting at the origin under
$\mathbb{P}$ and a bounded open set $D$ in $\mathbb{R}^{d}$, let
$\\{\mathbb{P}_{x}:x\in D\\}$ be a family of probability measures on
$(\mathbb{D}[0,1],\mathcal{B}(\mathbb{D}[0,1]))$ defined by

(7) $\displaystyle\mathbb{P}_{x}(F):=\mathbb{P}(F-x),\ \
F\in\mathcal{B}(\mathbb{D}[0,1]),$

where $F-x:=\\{\omega\in\mathbb{D}[0,1]:\omega(\cdot)+x\in F\\}$. Under
$\mathbb{P}_{x}$, the Gaussian process $X$ is considered to start at the point
$x$. The random time

$\tau^{X}_{D}=\inf\\{t>0:X_{t}\in D^{c}\\}$

represents the first exit time of $X$ from the domain $D$. Since $D$ is an
open set and $X$ is right-continuous at 0, it follows that $\tau^{X}_{D}>0$
$\mathbb{P}_{x}$-a.s. for any $x\in D$. The spectral heat content for the
Gaussian process $X$ at time $t\in(0,1]$ is defined by

(8) $\displaystyle
Q^{X}_{D}(t):=\int_{D}\mathbb{P}_{x}(\tau^{X}_{D}>t)\mathrm{d}x.$

The regular heat content of $X$ in $D$ at time $t\in(0,1]$ is defined by

(9) $\displaystyle H^{X}_{D}(t):=\int_{D}\mathbb{P}_{x}(X_{t}\in
D)\mathrm{d}x.$

Since $\\{\tau^{X}_{D}>t\\}\subset\\{X_{t}\in D\\}$, it follows that

(10) $\displaystyle Q^{X}_{D}(t)\leq H^{X}_{D}(t).$

The next lemma confirms that the above definitions of the spectral heat
content and regular heat content are well-defined.

######  Lemma 1.

Let $X=(X_{t})_{t\in[0,1]}$ be a stochastic process in $\mathbb{R}^{d}$ with
càdlàg paths starting at the origin under the probability measure
$\mathbb{P}=\mathbb{P}_{0}$. Let $D$ be a bounded open set in $\mathbb{R}^{d}$
and let $F\in\mathcal{B}(\mathbb{D}[0,1])$. Let a family of probability
measures $\\{\mathbb{P}_{x}:x\in D\\}$ be defined as in (7). Then the mapping
$x\mapsto\mathbb{P}_{x}(F)$ is $\mathcal{B}(D)$-measurable, and consequently,
the integrals defining the spectral and regular heat contents in (8) and (9)
are well-defined.

###### Proof.

For a finite-dimensional cylinder set of the form
$F=\\{\omega\in\mathbb{D}[0,1]:\omega(t_{i})\in A_{i}\ \textrm{for}\
i=0,1,\ldots,k\\}$, where $k\in\mathbb{N}$, $0=t_{0}<t_{1}<\cdots<t_{k}=1$,
and $A_{i}\in\mathcal{B}(\mathbb{R}^{d})$ for $i=0,1,\ldots,k$,

$\displaystyle\mathbb{P}_{x}(F)$ $\displaystyle=\mathbb{P}(x\in
A_{0},X_{t_{1}}+x\in A_{1},\,\ldots,\,X_{t_{k}}+x\in A_{k})$
$\displaystyle=\mathbf{1}_{A_{0}}(x)\int_{\mathbb{R}^{kd}}\mathbf{1}_{A_{1}}(z_{1}+x)\cdots\mathbf{1}_{A_{k}}(z_{k}+x)\mathbb{P}(X_{t_{1}}\in\mathrm{d}z_{1},\ldots,X_{t_{k}}\in\mathrm{d}z_{k}).$

Since the mapping

$(x,z_{1},\ldots,z_{k})\mapsto\mathbf{1}_{A_{0}}(x)\mathbf{1}_{A_{1}}(z_{1}+x)\cdots\mathbf{1}_{A_{k}}(z_{k}+x)$

is nonnegative and
$\mathcal{B}(D)\times\mathcal{B}(\mathbb{R}^{kd})$-measurable, it follows as
part of the statement of the Fubini theorem that the mapping
$x\mapsto\mathbb{P}_{x}(F)$ is $\mathcal{B}(D)$-measurable.

Now, all the finite-dimensional cylinder sets $F$ of the above form generate
the $\sigma$-algebra $\mathcal{B}(\mathbb{D}[0,1])$, and the family of sets
$F\in\mathcal{B}(\mathbb{D}[0,1])$ for which the mapping
$x\mapsto\mathbb{P}_{x}(F)$ is $\mathcal{B}(D)$-measurable forms a Dynkin
system. Therefore, by the Dynkin system theorem ([12, Chapter 2, Theorem
1.3]), the mapping $x\mapsto\mathbb{P}_{x}(F)$ is $\mathcal{B}(D)$-measurable
for any set $F\in\mathcal{B}(\mathbb{D}[0,1])$.

Finally, for a fixed $t\in(0,1]$, since
$\\{\tau_{D}^{X}>t\\}=\cap_{s\in\mathbb{Q}\cap[0,t]}\\{X_{s}\in D\\}$ and
$\\{X_{s}\in D\\}\in\mathcal{B}(\mathbb{D}[0,1])$ for each $s\in[0,t]$, it
follows that $\\{\tau_{D}^{X}>t\\}\in\mathcal{B}(\mathbb{D}[0,1])$. Hence, the
mapping $x\mapsto\mathbb{P}_{x}(\tau^{X}_{D}>t)$ is
$\mathcal{B}(D)$-measurable; consequently, the integral in (8) is well-
defined. Similarly, the integral in (9) is well-defined. ∎

An open set $D$ in $\mathbb{R}^{d}$ is said to be $C^{1,1}$ if there exist a
localization radius $R>0$ and a constant $\Lambda>0$ that satisfy the
following condition: for every $z\in\partial D$, there exist a $C^{1,1}$
function $\phi=\phi_{z}:\mathbb{R}^{d-1}\to\mathbb{R}$ satisfying $\phi(0)=0$,
$\nabla\phi(0)=(0,\cdots,0)$, $\|\nabla\phi\|_{\infty}\leq\Lambda$, and
$|\nabla\phi(x_{1})-\nabla\phi(x_{2})|\leq\Lambda|x_{1}-x_{2}|$ for
$x_{1},x_{2}\in\mathbb{R}^{d-1}$, and an orthonormal coordinate system
$CS_{z}:y=(y^{1},\ldots,y^{d-1},y^{d})=(\widetilde{y},y^{d})$ with origin at
$z$ such that

$B(z,R)\cap D=B(z,R)\cap\\{y=(\widetilde{y},y^{d})\mbox{ in
}CS_{z}:y^{d}>\phi(\widetilde{y})\\}.$

The pair $(R,\Lambda)$ is called the $C^{1,1}$ characteristics of the
$C^{1,1}$ open set $D$. It is well-known that any $C^{1,1}$ open set $D$ with
$C^{1,1}$ characteristics $(R,\Lambda)$ satisfies the uniform interior and
exterior $R$-ball condition (see [4, Lemma 2.2]): for any $z\in\partial D$,
there exist open balls $B_{1}$ and $B_{2}$ of the same radius $R$ such that

(11) $\displaystyle B_{1}\subset D,\
B_{2}\subset\mathbb{R}^{d}\setminus\overline{D},\ \textrm{and}\ \partial
B_{1}\cap\partial B_{2}=\\{z\\}.$

Given a bounded $C^{1,1}$ domain $D$ and a point $x\in D$, let $\delta_{D}(x)$
denote the Euclidean distance from $x$ to the boundary $\partial D$ of $D$;
i.e.,

(12) $\displaystyle\delta_{D}(x)=\inf\\{|x-y|:y\in\partial D\\}.$

For each $a\in(0,R/2]$, let

(13) $\displaystyle D_{a}=\\{x\in D:\delta_{D}(x)>a\\},$

which is a region obtained by removing points near the boundary $\partial D$
from $D$. Note that the $C^{1,1}$ condition with $C^{1,1}$ characteristics
$(R,\Lambda)$ implies that the set $D_{a}$ is non-empty. It also follows from
[24, Lemma 6.7] that

(14) $\displaystyle|\partial D|\left(\frac{R-a}{R}\right)^{d-1}\leq|\partial
D_{a}|\leq|\partial D|\left(\frac{R}{R-a}\right)^{d-1},$

where $|\partial D|$ and $|\partial D_{a}|$ are the $(d-1)$-dimensional
Lebesgue measures of the respective sets. In particular,

(15) $\displaystyle|\partial D_{a}|\leq 2^{d-1}|\partial D|\ \ \textrm{for
any}\ a\in(0,R/2].$

If $X$ is given by a Brownian motion $W$ with
$\mathbb{E}[e^{-i\langle\xi,W_{t}\rangle}]=e^{-t|\xi|^{2}}$, then for a
bounded open interval $D$ in $\mathbb{R}$ or a bounded connected $C^{1,1}$
open set $D$ in $\mathbb{R}^{d}$ with $d\geq 2$ (see [24, Theorem 6.2] and
[26, Theorem 2]),

(16) $\displaystyle\lim_{t\downarrow 0}\frac{|D|-Q_{D}^{W}(t)}{\sqrt{t}}$
$\displaystyle=\frac{2|\partial D|}{\sqrt{\pi}};$ (17)
$\displaystyle\lim_{t\downarrow 0}\frac{|D|-H^{W}_{D}(t)}{\sqrt{t}}$
$\displaystyle=\frac{|\partial D|}{\sqrt{\pi}},$

where $|D|$ denotes the $d$-dimensional Lebesgue measure of $D$.

##  3 The short-time behavior of the regular heat content

We first derive the asymptotic behavior of the regular heat content for
general Gaussian processes. Simply put, the amount of heat loss from the
domain $D$ behaves like the standard deviation function $\sqrt{R_{t}}$ in
small time. Note that the result below immediately reduces to (17) if $X$ is
taken to be a Brownian motion $W$ with
$\mathbb{E}[e^{-i\langle\xi,W_{t}\rangle}]=e^{-t|\xi|^{2}}$.

######  Theorem 2.

Let $D=(a,b)$ be a non-empty, bounded interval in $\mathbb{R}$ or a non-empty,
bounded, connected $C^{1,1}$ open set in $\mathbb{R}^{d}$ with $d\geq 2$. Let
$X=(X_{t})_{t\in[0,1]}$ be a stochastic process in $\mathbb{R}^{d}$ whose
components are i.i.d. zero-mean Gaussian processes with càdlàg paths starting
at the origin and common variance function $R_{t}$ under the probability
measure $\mathbb{P}=\mathbb{P}_{0}$ such that $X_{t}$ is non-degenerate for
all $t\in(0,1]$. Then

$\lim_{t\downarrow 0}\frac{|D|-H^{X}_{D}(t)}{\sqrt{R_{t}}}=\frac{|\partial
D|}{\sqrt{2\pi}}.$

###### Proof.

Note the identity $|D|-H^{X}_{D}(t)=\int_{D}\mathbb{P}_{x}(X_{t}\in
D^{c})\mathrm{d}x$. Applying this identity to a Brownian motion $W$ with
$\mathbb{E}[e^{-i\langle\xi,W_{t}\rangle}]=e^{-t|\xi|^{2}}$, we can rewrite
the statement in (17) as

$\lim_{t\downarrow
0}\frac{1}{\sqrt{t}}\int_{D}\biggl{(}\int_{D^{c}}\dfrac{1}{(4\pi
t)^{d/2}}e^{-\frac{|x-y|^{2}}{4t}}\mathrm{d}y\biggr{)}\mathrm{d}x=\frac{|\partial
D|}{\sqrt{\pi}}.$

This, together with (6), yields

$\lim_{t\downarrow
0}\sqrt{\frac{2}{R_{t}}}\int_{D}\biggl{(}\int_{D^{c}}\dfrac{1}{(2\pi
R_{t})^{d/2}}e^{-\frac{|x-y|^{2}}{2R_{t}}}\mathrm{d}y\biggr{)}\mathrm{d}x=\frac{|\partial
D|}{\sqrt{\pi}}.$

This is the desired result since the components of $X$ are assumed to be
i.i.d. zero-mean Gaussian processes with common variance function $R_{t}$. ∎

In [2], the author derived the small-time asymptotic behavior of the regular
heat content and spectral heat content for stable Lévy processes. The key idea
of the argument was to express a given stable process as a Brownian motion
time-changed by an independent stable subordinator and to utilize the self-
similarity of the Brownian motion; i.e., $(W_{at})_{t\geq
0}=^{\mathrm{d}}(a^{1/2}W_{t})_{t\geq 0}$. However, it turns out that the use
of the self-similarity is not essential. To show this, we provide an
alternative proof of Theorem 2 in the one-dimensional case that comes with an
associated error bound of exponential decay. Namely:

######  Theorem 3.

Let $D=(a,b)$ be a non-empty, bounded interval in $\mathbb{R}$. Let
$X=(X_{t})_{t\in[0,1]}$ be a Gaussian process in $\mathbb{R}$ with càdlàg
paths starting at the origin and variance function $R_{t}$ under the
probability measure $\mathbb{P}=\mathbb{P}_{0}$ such that $X_{t}$ is non-
degenerate for all $t\in(0,1]$. Then for any $t\in[0,1]$,

$|D|-H_{D}^{X}(t)=\sqrt{\frac{2R_{t}}{\pi}}-\mathrm{Error}_{H}(t),$

where

(18) $\displaystyle
0<\mathrm{Error}_{H}(t)\leq\frac{2R_{t}^{\frac{3}{2}}}{|D|^{2}\sqrt{2\pi}}e^{-\frac{|D|^{2}}{2R_{t}}}.$

Consequently,

(19) $\displaystyle\lim_{t\downarrow
0}\frac{|D|-H^{X}_{D}(t)}{\sqrt{R_{t}}}=\sqrt{\frac{2}{\pi}}.$

###### Proof.

By changing variables via $u:=b-x-c$ with $c:=\frac{b-a}{2}$,

$\displaystyle|D|-H^{X}_{D}(t)$
$\displaystyle=\int_{a}^{b}\left(\mathbb{P}_{x}(X_{t}\leq
a)+\mathbb{P}_{x}(X_{t}\geq b)\right)\mathrm{d}x$
$\displaystyle=\int_{a}^{b}\left(\mathbb{P}(X_{t}+x\leq
a)+\mathbb{P}(X_{t}+x\geq b)\right)\mathrm{d}x$
$\displaystyle=\int_{-c}^{c}\left(\mathbb{P}(X_{t}\leq
u-c)+\mathbb{P}(X_{t}\geq u+c)\right)\mathrm{d}u.$

Since $X_{t}$ has a zero-mean Gaussian distribution, $-X_{t}$ has the same
distribution as $X_{t}$. Therefore,

$\int_{-c}^{c}\mathbb{P}(X_{t}\leq
u-c)\mathrm{d}u=\int_{-c}^{c}\mathbb{P}(-X_{t}\leq
u-c)\mathrm{d}u=\int_{-c}^{c}\mathbb{P}(X_{t}\geq v+c)\mathrm{d}v,$

and hence, it follows that

$|D|-H^{X}_{D}(t)=2\int_{-c}^{c}\mathbb{P}(X_{t}\geq
u+c)\mathrm{d}u=2\int_{0}^{2c}\mathbb{P}(X_{t}\geq u)\mathrm{d}u.$

Since the distribution function $u\mapsto\mathbb{P}(Y\leq u)$ of any random
variable $Y$ is right-continuous, $\mathbb{P}(Y\leq u)=\mathbb{P}(Y<u)$ for
almost every $u$. Also, the equality
$\\{X_{t}>u\\}=\\{X_{t}\mathbf{1}_{\\{X_{t}\geq 0\\}}>u\\}$ holds for $u\geq
0$, so

(20)
$\displaystyle|D|-H^{X}_{D}(t)=2\int_{0}^{2c}\mathbb{P}(X_{t}>u)\mathrm{d}u=2\mathbb{E}[X_{t}\mathbf{1}_{\\{X_{t}\geq
0\\}}]-2\int_{2c}^{\infty}\mathbb{P}(X_{t}>u)\mathrm{d}u.$

The first term is calculated as

(21) $\displaystyle\mathbb{E}[X_{t}\mathbf{1}_{\\{X_{t}\geq
0\\}}]=\int_{0}^{\infty}\frac{x}{\sqrt{2\pi
R_{t}}}e^{-\frac{x^{2}}{2R_{t}}}\mathrm{d}x=\sqrt{\frac{R_{t}}{2\pi}},$

As for the second term, for any $u\geq x>0$, by the estimate
$\mathbb{P}(Z>z)\leq\frac{1}{z\sqrt{2\pi}}e^{-z^{2}/2}$ valid for a standard
Gaussian variable $Z$ and any constant $z>0$ (see e.g., [22, Theorem 1.2.6]),

$\displaystyle\mathbb{P}(X_{t}>u)$
$\displaystyle=\mathbb{P}\biggl{(}\frac{X_{t}}{\sqrt{R_{t}}}>\frac{u}{\sqrt{R_{t}}}\biggr{)}\leq\frac{1}{\sqrt{2\pi}}\frac{\sqrt{R_{t}}}{u}e^{-\frac{u^{2}}{2R_{t}}}$
$\displaystyle\leq\biggl{(}\frac{u}{x}\biggr{)}^{2}\frac{1}{\sqrt{2\pi}}\frac{\sqrt{R_{t}}}{u}e^{-\frac{u^{2}}{2R_{t}}}=\frac{\sqrt{R_{t}}}{x^{2}\sqrt{2\pi}}ue^{-\frac{u^{2}}{2R_{t}}}.$

Integrating both sides from $x$ to $\infty$ yields

(22)
$\displaystyle\int_{x}^{\infty}\mathbb{P}(X_{t}>u)\mathrm{d}u\leq\frac{R_{t}^{\frac{3}{2}}}{x^{2}\sqrt{2\pi}}e^{-\frac{x^{2}}{2R_{t}}}.$

Combining (20), (21) and (22) yields the error bound in (18), from which (19)
follows immediately. ∎

######  Remark 4.

A simple modification of the above proof leads to a more general statement on
the regular heat content for a symmetric, zero-mean (but not necessarily
Gaussian) process $X$ in $\mathbb{R}$ with finite first moment. Indeed, if the
regular heat content is defined as in (9), and if

$\int_{b-a}^{\infty}\mathbb{P}(X_{t}>u)\mathrm{d}u=o(\mathbb{E}[X_{t}\mathbf{1}_{\\{X_{t}\geq
0\\}}])\ \ \textrm{as}\ t\downarrow 0,$

then

(23) $\displaystyle\lim_{t\downarrow
0}\frac{|D|-H_{D}^{X}(t)}{\mathbb{E}[X_{t}\mathbf{1}_{\\{X_{t}\geq 0\\}}]}=2.$

In particular, this result holds for any symmetric stable Lévy process with
stability index $\alpha\in(1,2)$ and any symmetric tempered stable Lévy
process with underlying stability index $\alpha\in(0,2)$. On the other hand,
for any symmetric, zero-mean, self-similar process of index $H>0$ with finite
first moment, expression (23) can be rewritten as

$\lim_{t\downarrow
0}\frac{|D|-H_{D}^{X}(t)}{t^{H}}=2\mathbb{E}[X_{1}\mathbf{1}_{\\{X_{1}\geq
0\\}}].$

##  4 The short-time behavior of the spectral heat content

We now turn our attention to the spectral heat content for general Gaussian
processes. We first establish an analogue of Theorem 3 in the one-dimensional
case that comes with an associated error bound of exponential decay.

######  Theorem 5.

Let $D=(a,b)$ be a non-empty, bounded interval in $\mathbb{R}$. Let
$X=(X_{t})_{t\in[0,1]}$ be a zero-mean Gaussian process in $\mathbb{R}$ with
càdlàg paths starting at the origin under the probability measure
$\mathbb{P}=\mathbb{P}_{0}$ such that the almost sure boundedness condition
(3) holds and $X_{t}$ is non-degenerate for all $t\in(0,1]$. Then for any
$t\in(0,1]$ satisfying $\mu_{t}<|D|/2$,

$|D|-Q_{D}^{X}(t)=2\mu_{t}-\mathrm{Error}_{Q}(t),$

where

(24) $\displaystyle
0<\mathrm{Error}_{Q}(t)\leq\frac{4\sigma_{t}^{2}}{|D|-\mu_{t}}e^{-\frac{1}{2}\bigl{(}\frac{|D|-\mu_{t}}{\sigma_{t}}\bigr{)}^{2}}+\frac{4\sigma_{t}^{2}}{\frac{1}{2}|D|-\mu_{t}}e^{-\frac{1}{2}\bigl{(}\frac{\frac{1}{2}|D|-\mu_{t}}{\sigma_{t}}\bigr{)}^{2}}.$

Consequently,

(25) $\displaystyle\lim_{t\downarrow 0}\frac{|D|-Q_{D}^{X}(t)}{\mu_{t}}=2.$

###### Proof.

Observe that

$\displaystyle\mathbb{P}_{x}(\tau^{X}_{D}>t)=\mathbb{P}_{x}(a<X_{s}<b\
\textrm{for all}\ s\leq t)=\mathbb{P}(a-x<X_{s}<b-x\ \textrm{for all}\ s\leq
t).$

Since $|D|-Q^{X}_{D}(t)=\int_{a}^{b}\mathbb{P}_{x}(\tau^{X}_{D}\leq
t)\mathrm{d}x$,

$\displaystyle|D|-Q^{X}_{D}(t)$
$\displaystyle=\int_{a}^{b}\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq
b-x)\mathrm{d}x+\int_{a}^{b}\mathbb{P}(\inf_{0\leq s\leq t}X_{s}\leq
a-x)\mathrm{d}x$ $\displaystyle\ \ \ \ -\int_{a}^{b}\mathbb{P}(\sup_{0\leq
s\leq t}X_{s}\geq b-x,\,\inf_{0\leq s\leq t}X_{s}\leq a-x)\mathrm{d}x$
$\displaystyle=:I_{1}(t)+I_{2}(t)-I_{3}(t).$

Since the distribution function $u\mapsto\mathbb{P}(Y\leq u)$ of any random
variable $Y$ is right-continuous, $\mathbb{P}(Y\leq u)=\mathbb{P}(Y<u)$ for
almost every $u$ with respect to the Lebesgue measure, and in particular,
$\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq u)=\mathbb{P}(\sup_{0\leq s\leq
t}X_{s}>u)$ for almost every $u$ as well. Thus, a simple change of variables
yields

$I_{1}(t)=\int_{0}^{b-a}\mathbb{P}(\sup_{0\leq s\leq
t}X_{s}>u)\mathrm{d}u=\mu_{t}-I_{4}(t),$

where

$I_{4}(t):=\int_{b-a}^{\infty}\mathbb{P}(\sup_{0\leq s\leq
t}X_{s}>u)\mathrm{d}u.$

On the other hand, the process $-X=(-X_{t})_{t\geq 0}$ is Gaussian and has the
same covariance function as $X$, so $-X$ and $X$ have the same distribution;
therefore,

$I_{2}(t)=\int_{a}^{b}\mathbb{P}(\inf_{0\leq s\leq t}(-X_{s})\leq
a-x)\mathrm{d}x=\int_{a}^{b}\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq
x-a)\mathrm{d}x=I_{1}(t).$

Thus, we have obtained

$\displaystyle|D|-Q^{X}_{D}(t)=2\mu_{t}-2I_{4}(t)-I_{3}(t).$

With (5) in mind, take $t>0$ small enough so that $\mu_{t}<c:=\frac{b-a}{2}$.
We now show that the error term

$\mathrm{Error}_{Q}(t):=2I_{4}(t)+I_{3}(t)$

has the bound in (24). As for $I_{4}(t)$, by the Borell–TIS inequality (4) and
the inequality $\mathbb{P}(Z>z)\leq\frac{1}{z\sqrt{2\pi}}e^{-z^{2}/2}$ valid
for a standard Gaussian variable $Z$ and any constant $z>0$ (see e.g., [22,
Theorem 1.2.6]),

(26) $\displaystyle I_{4}(t)$ $\displaystyle\leq
2\int_{2c}^{\infty}e^{-\frac{(u-\mu_{t})^{2}}{2\sigma^{2}_{t}}}\mathrm{d}u=2\sigma_{t}\int_{\frac{2c-\mu_{t}}{\sigma_{t}}}^{\infty}e^{-z^{2}/2}\mathrm{d}z\leq\frac{2\sigma_{t}^{2}}{2c-\mu_{t}}e^{-\frac{1}{2}\bigl{(}\frac{2c-\mu_{t}}{\sigma_{t}}\bigr{)}^{2}}.$

On the other hand, in terms of $I_{3}(t)$, by the change of variables
$u=b-x-c$,

$I_{3}(t)=\int_{-c}^{c}f_{c,t}(u)\mathrm{d}u,$

where

$f_{c,t}(u)=\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq u+c,\,\inf_{0\leq s\leq
t}X_{s}\leq u-c).$

Since $-X$ is another Gaussian process having the same distribution as $X$,

$\displaystyle f_{c,t}(u)$ $\displaystyle=\mathbb{P}(\sup_{0\leq s\leq
t}(-X_{s})\geq u+c,\,\inf_{0\leq s\leq t}(-X_{s})\leq u-c)$
$\displaystyle=\mathbb{P}(-\inf_{0\leq s\leq t}X_{s}\geq u+c,\,-\sup_{0\leq
s\leq t}X_{s}\leq u-c)$ $\displaystyle=\mathbb{P}(\inf_{0\leq s\leq
t}X_{s}\leq-u-c,\,\sup_{0\leq s\leq t}X_{s}\geq-u+c)$
$\displaystyle=f_{c,t}(-u).$

Hence, $f_{c,t}$ is an even function, and therefore,

$I_{3}(t)=2\int_{0}^{c}f_{c,t}(u)\mathrm{d}u\leq
2\int_{0}^{c}\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq u+c)\mathrm{d}u\leq
2\int_{c}^{\infty}\mathbb{P}(\sup_{0\leq s\leq t}X_{s}\geq v)\mathrm{d}v.$

This, together with part of the estimate in (26) (with $2c$ replaced by $c$),
yields

(27) $\displaystyle
I_{3}(t)\leq\frac{4\sigma_{t}^{2}}{c-\mu_{t}}e^{-\frac{1}{2}\bigl{(}\frac{c-\mu_{t}}{\sigma_{t}}\bigr{)}^{2}}.$

Putting together (26) and (27) gives the bound in (24) valid for any small
$t>0$ satisfying $\mu_{t}<c$.

Finally, to obtain (25) from (24), we must verify that
$\mathrm{Error}_{Q}(t)=o(\mu_{t})$ as $t\downarrow 0$. To this end, observe
first that the error bound in (24) gives

(28) $\displaystyle\mathrm{Error}_{Q}(t)\leq C\sigma_{t}^{2}$

for all small enough $t>0$, where $C>0$ is some constant independent of $t$.
Next, for each small $t>0$, take $s_{\ast}(t)\in(0,t]$ such that

$\sigma^{2}_{t}=\sup_{s\in[0,t]}R_{s}<2R_{s_{\ast}(t)}.$

(For each $t>0$, such a number $s_{\ast}(t)$ exists since, otherwise,
$\sigma^{2}_{t}\geq 2R_{s}$ for all $s\in(0,t]$, which would give
$\sigma^{2}_{t}\geq 2\sigma^{2}_{t}$, and hence, $\sigma^{2}_{t}=0$, which is
impossible since $X_{t}$ is assumed non-degenerate for all $t>0$.) Recall
Sudakov’s lower bound (see [16, Theorem 10.5] and its proof, or [8,
Proposition 1.3]), which states that for any zero-mean Gaussian random vector
$(X_{1},\ldots,X_{n})$,

$\mathbb{E}\Bigl{[}\max_{1\leq i\leq n}X_{i}\Bigr{]}\geq c_{n}\min_{i\neq
j}d(i,j),\ \ \textrm{where}\ \
d(i,j):=\bigl{(}\mathbb{E}[|X_{i}-X_{j}|^{2}]\bigr{)}^{1/2}$

for some constant $c_{n}>0$. Applying this statement to the vector
$(X_{0},X_{s_{\ast}(t)})$ with $X_{0}=0$ yields

$\mu_{t}\geq\mathbb{E}\bigl{[}\max\\{X_{0},X_{s_{\ast}(t)}\\}\bigr{]}\geq
c_{2}(R_{s_{\ast}(t)})^{1/2}.$

Hence,

(29)
$\displaystyle\frac{\sigma^{2}_{t}}{\mu_{t}}\leq\frac{2R_{s_{\ast}(t)}}{c_{2}(R_{s_{\ast}(t)})^{1/2}}=\frac{2}{c_{2}}(R_{s_{\ast}(t)})^{1/2},$

which approaches 0 since $s_{\ast}(t)\to 0$ as $t\downarrow 0$ due to the
condition $s_{\ast}(t)\in(0,t]$. The latter, together with (28), yields
$\mathrm{Error}_{Q}(t)=o(\mu_{t})$, as desired. ∎

######  Remark 6.

1) If $X$ is given by a Brownian motion $W$ on $\mathbb{R}$ with
$\mathbb{E}_{0}[e^{-i\xi W_{t}}]=e^{-t\xi^{2}}$, then
$R_{t}=\sigma^{2}_{t}=2t$ and $\mu_{t}$ is given by (2), which implies that
$\mathrm{Error}_{Q}(t)\to 0$ as $t\downarrow 0$. Consequently, Theorem 5
recovers the statement in (16) when $d=1$.

2) Theorem 5 does not require that the Gaussian process $X$ be self-similar.
In fact, the argument given in the above proof is significantly different from
that given for a Brownian motion in [1], which relies heavily on the self-
similarity of the Brownian motion.

3) As discussed in Remark 4 for regular heat content, a simple modification of
the above proof leads to a statement on the spectral heat content for more
general symmetric, zero-mean (but not necessarily Gaussian) processes $X$ in
$\mathbb{R}$ with $\mu_{t}<\infty$ for all $t\in(0,1]$. Indeed, if the
spectral heat content is defined as in (8), then (25) follows as long as

$\int_{|D|/2}^{\infty}\mathbb{P}(\sup_{0\leq s\leq
t}X_{s}>u)\mathrm{d}u=o(\mu_{t})\ \ \textrm{as}\ t\downarrow 0.$

In particular, under the additional assumption that $X$ is self-similar with
index $H>0$, (25) can be rewritten as

(30) $\displaystyle\lim_{t\downarrow
0}\frac{|D|-Q_{D}^{X}(t)}{t^{H}}=2\mu_{1}.$

We now investigate the spectral heat content in a multi-dimensional setting,
which requires a careful discussion. Our main result in this case is Theorem
8, which provides the exact small-time asymptotic behavior of the amount of
heat loss $|D|-Q_{D}^{X}(t)$; however, the statement is given in a somewhat
restricted setting. Instead, in Proposition 7 below, we establish an upper
bound for the heat loss that is valid for a wide class of Gaussian processes
and for any fixed $t>0$ small enough.

In the remainder of the paper, we use the notation
$X^{j}=(X^{j}_{t})_{t\in[0,1]}$ to denote the $j$th component of a process
$X=(X_{t})_{t\in[0,1]}$ in $\mathbb{R}^{d}$, and $x^{j}$ to denote the $j$th
component of any given vector $x\in\mathbb{R}^{d}$. Recall the notations
$\delta_{D}(x)$ and $D_{a}$ defined in (12)–(13).

######  Proposition 7.

Let $D$ be a non-empty, bounded, connected $C^{1,1}$ open set in
$\mathbb{R}^{d}$ with $d\geq 2$ with $C^{1,1}$ characteristics $(R,\Lambda)$.
Let $X=(X_{t})_{t\in[0,1]}=(X_{t}^{1},\ldots,X_{t}^{d})_{t\in[0,1]}$ be a
stochastic process in $\mathbb{R}^{d}$ whose components are identically
distributed zero-mean Gaussian processes with càdlàg paths starting at 0 and
common expected running supremum $\mu_{t}$ under the probability measure
$\mathbb{P}=\mathbb{P}_{0}$ such that the almost surely boundedness condition
(3) holds and $X_{t}$ is non-degenerate for all $t\in(0,1]$. Then for any
$t\in(0,1]$ and $a\in(0,R/2]$ with $\mu_{t}<a/\sqrt{d}$,

(31) $\displaystyle|D|-Q_{D}^{X}(t)\leq d^{3/2}2^{d}|\partial
D|\mu_{t}+4d|D_{a}|e^{-\frac{(\frac{a}{\sqrt{d}}-\mu_{t})^{2}}{2\sigma_{t}^{2}}}.$

Consequently,

(32) $\displaystyle\limsup_{t\downarrow 0}\frac{|D|-Q^{X}_{D}(t)}{\mu_{t}}\leq
d^{3/2}2^{d}|\partial D|.$

###### Proof.

Let $B(b,r)$ denote the open ball of radius $r$ centered at $b$. Then for any
fixed $x\in D$, since $D\supset B(x,\delta_{D}(x))$, it follows that

$\displaystyle\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)\leq\mathbb{P}_{x}(\tau_{B(x,\delta_{D}(x))}^{X}\leq
t)=\mathbb{P}(\tau_{B(0,\delta_{D}(x))}^{X}\leq t).$

On the other hand, since $\max_{1\leq j\leq
d}|x^{j}|\leq|x|=\sqrt{\sum_{j=1}^{d}(x^{j})^{2}}\leq\sqrt{d}\max_{1\leq j\leq
d}|x^{j}|$ for any $x=(x^{1},\ldots,x^{d})\in\mathbb{R}^{d}$, for any fixed
$r>0$,

$\displaystyle\mathbb{P}(\tau^{X}_{B(0,r)}\leq t)$
$\displaystyle=\mathbb{P}(\sup_{0\leq s\leq t}|X_{s}|\geq
r)\leq\mathbb{P}\biggl{(}\sqrt{d}\sup_{0\leq s\leq t}\max_{1\leq j\leq
d}|X^{j}_{s}|\geq r\biggr{)}$
$\displaystyle\leq\mathbb{P}\biggl{(}\bigcup_{j=1}^{d}\biggl{\\{}\sup_{0\leq
s\leq t}|X^{j}_{s}|\geq\frac{r}{\sqrt{d}}\biggr{\\}}\biggr{)}$
$\displaystyle\leq d\cdot\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}|X^{1}_{s}|\geq\frac{r}{\sqrt{d}}\biggr{)}=2d\cdot\mathbb{P}\biggl{(}\sup_{0\leq
s\leq t}X^{1}_{s}\geq\frac{r}{\sqrt{d}}\biggr{)},$

where the last equality follows from the symmetry of the process $X^{1}$.
Combining the above two observations yields

$|D|-Q_{D}^{X}(t)=\int_{D}\mathbb{P}_{x}(\tau_{D}^{X}\leq t)\,\mathrm{d}x\leq
2d\int_{D}\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}X^{1}_{s}\geq\frac{\delta_{D}(x)}{\sqrt{d}}\biggr{)}\,\mathrm{d}x.$

For any $a\in(0,R/2]$, with (15) in mind, split the latter integral into
$\int_{D\setminus D_{a}}$ and $\int_{D_{a}}$. In terms of the first integral,

$\displaystyle I_{1}(t)$ $\displaystyle:=\int_{D\setminus
D_{a}}\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}X^{1}_{s}\geq\frac{\delta_{D}(x)}{\sqrt{d}}\biggr{)}\,\mathrm{d}x$
$\displaystyle=\int_{0}^{a}\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}X^{1}_{s}\geq\frac{r}{\sqrt{d}}\biggr{)}|\partial D_{r}|\,\mathrm{d}r$
$\displaystyle\leq 2^{d-1}|\partial
D|\int_{0}^{\infty}\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}X^{1}_{s}\geq\frac{r}{\sqrt{d}}\biggr{)}\,\mathrm{d}r$
$\displaystyle=\sqrt{d}\hskip 1.0pt2^{d-1}|\partial D|\mu_{t}.$

In terms of the second integral, note that $\delta_{D}(x)>a$ for all $x\in
D_{a}$, and hence, for any small $t$ satisfying $\mu_{t}<\frac{a}{\sqrt{d}}$
(so that $\frac{\delta_{D}(x)}{\sqrt{d}}>\frac{a}{\sqrt{d}}>\mu_{t}$), the
Borell–TIS inequality (4) yields

$\displaystyle I_{2}(t)$
$\displaystyle:=\int_{D_{a}}\mathbb{P}\biggl{(}\sup_{0\leq s\leq
t}X^{1}_{s}\geq\frac{\delta_{D}(x)}{\sqrt{d}}\biggr{)}\,\mathrm{d}x$
$\displaystyle\leq
2\int_{D_{a}}e^{-\frac{(\frac{\delta_{D}(x)}{\sqrt{d}}-\mu_{t})^{2}}{2\sigma_{t}^{2}}}\,\mathrm{d}x$
$\displaystyle\leq
2|D_{a}|e^{-\frac{(\frac{a}{\sqrt{d}}-\mu_{t})^{2}}{2\sigma_{t}^{2}}}.$

Combining the above estimates yields (31). Moreover, putting together (31),
(29), and the trivial inequality $e^{-x}<x^{-1}$ valid for $x>0$ yields (32)
immediately. ∎

Proposition 7 establishes an upper bound only. As for the lower bound, a
simple argument would be to combine Theorem 2 with the inequality in (10),
which immediately gives

$\frac{|\partial D|}{\sqrt{2\pi}}\leq\liminf_{t\downarrow
0}\frac{|D|-Q^{X}_{D}(t)}{\sqrt{R_{t}}}.$

However, this does not involve $\mu_{t}$ as observed in the one-dimensional
case in Theorem 5. In contrast, Theorem 8 below establishes the exact
asymptotic behavior with rate function $\mu_{t}$ under some additional
assumptions.

In the remainder of the paper, let $\mathbb{D}([0,1];\mathbb{R})$ denote the
space of all càdlàg functions on $[0,1]$ taking values in $\mathbb{R}$. The
space is equipped with the standard Skorokhod $J_{1}$ topology, under which
$x_{n}=(x_{n}(t))_{t\in[0,1]}\to x=(x(t))_{t\in[0,1]}$ as $n\to\infty$ if and
only if there exists a sequence of continuous, strictly increasing functions
$\lambda_{n}:[0,1]\to[0,1]$ whose inverses are continuous such that

(33) $\displaystyle\sup_{t\in[0,1]}|\lambda_{n}(t)-t|\to 0\ \ \ \textrm{and}\
\ \ \sup_{t\in[0,1]}|x_{n}(\lambda_{n}(t))-x(t)|\to 0$

as $n\to\infty$. Convergence in the $J_{1}$ topology coincides with
convergence in the uniform topology if $x_{n}$ and $x$ belong to the space
$C([0,1];\mathbb{R})$ of continuous functions on $[0,1]$. For details about
the $J_{1}$ topology, see e.g., [29].

######  Theorem 8.

Let $D$ be a non-empty, bounded, connected $C^{1,1}$ open set in
$\mathbb{R}^{d}$ with $d\geq 2$. Let
$X=(X_{t})_{t\in[0,1]}=(X_{t}^{1},\ldots,X_{t}^{d})_{t\in[0,1]}$ be a
stochastic process in $\mathbb{R}^{d}$ whose components are i.i.d. zero-mean
Gaussian processes with càdlàg paths starting at 0 and common expected running
supremum $\mu_{t}$ under the probability measure $\mathbb{P}=\mathbb{P}_{0}$
such that the almost surely boundedness condition (3) holds and $X_{t}$ is
non-degenerate for all $t\in(0,1]$. For each $t\in(0,1]$, define a scaled
process ${Y}^{(t)}=({Y}^{(t)}_{u})_{u\in[0,1]}$ of $X^{1}$ by

(34) $\displaystyle{Y}_{u}^{(t)}=\frac{X_{tu}^{1}}{\mu_{t}}.$

If $Y^{(t)}$ converges weakly as $t\downarrow 0$ to a continuous Gaussian
process $Y=(Y_{s})_{s\in[0,1]}$ in the space $\mathbb{D}([0,1];\mathbb{R})$
equipped with the Skorokhod $J_{1}$ topology, then

$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]\leq 1,$

and

(35) $\displaystyle\lim_{t\downarrow
0}\frac{|D|-Q^{X}_{D}(t)}{\mu_{t}}=|\partial
D|\cdot\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

The following lemma gives sufficient conditions for the weak convergence of
$Y^{(t)}$ to a Gaussian process $Y$ as stated in Theorem 8.

######  Lemma 9.

Let $X$ be as in Theorem 8. Assume that there exists a continuous function $g$
on $[0,1]^{2}$ such that for each $s,r\in[0,1]$,

(36) $\displaystyle\frac{R_{X^{1}}(ts,tr)}{\mu_{t}^{2}}\to g(s,r)\ \
\textrm{as}\ \ t\downarrow 0,$

where $R_{X^{1}}(s,r)=\mathbb{E}[X^{1}_{s}X^{1}_{r}]$. Assume further that
there exist a constant $\kappa>1$ and a continuous, nondecreasing function $h$
on $[0,1]$ such that for any $t\in(0,1]$ and $0\leq r<s<u\leq 1$, the scaled
moment condition

(37)
$\displaystyle\frac{1}{\mu_{t}^{4}}\mathbb{E}\Bigl{[}(X^{1}_{tu}-X^{1}_{ts})^{2}(X^{1}_{ts}-X^{1}_{tr})^{2}\Bigr{]}\leq(h(u)-h(r))^{\kappa}$

holds. Then as $t\downarrow 0$, $Y^{(t)}$ defined in (34) converges weakly to
a Gaussian process $Y$ with covariance function $R_{Y}(s,r)=g(s,r)$ in the
space $\mathbb{D}([0,1];\mathbb{R})$ equipped with the Skorokhod $J_{1}$
topology.

###### Proof.

Observe that $Y^{(t)}$ is a zero-mean Gaussian process in $\mathbb{R}$ with
covariance function $R_{Y^{(t)}}(s,r)=\mu_{t}^{-2}R_{X^{1}}(ts,tr),$ which
converges to $g(s,r)$ as $t\downarrow 0$ due to assumption (36). In
particular, since the function $(s,r)\mapsto R_{Y^{(t)}}(s,r)$ is nonnegative
definite for each fixed $t$, so is the limiting function $(s,r)\mapsto
g(s,r)$; thus, the latter is the covariance function of some Gaussian process
$Y=(Y_{s})_{s\in[0,1]}$ in $\mathbb{R}$. Now, the convergence of the
covariance function as $t\downarrow 0$ implies the weak convergence of any
linear combination of the form $\sum_{k=1}^{n}a_{k}Y^{(t)}_{s_{k}}$ to
$\sum_{k=1}^{n}a_{k}Y_{s_{k}}$, where $0\leq s_{1}<s_{2}<\cdots<s_{n}\leq 1$
and $a_{1},\ldots,a_{n}\in\mathbb{R}$. Therefore, by the Cramér–Wold device
(see e.g., [22, Theorem 3.10.6]), it follows that

$Y^{(t)}\to^{\textrm{f.d.d.}}Y\ \ \textrm{as}\ \ t\downarrow 0,$

where the symbol $\to^{\textrm{f.d.d.}}$ means the convergence of finite-
dimensional distributions. Thus, it now suffices to verify the tightness of
the family $\\{Y^{(t)}:t\in(0,1]\\}$ to obtain the desired weak convergence in
the Skorokhod space.

Note that since the covariance function $g(s,r)$ is continuous, due to [11,
Theorem 8.12], the Gaussian process $Y$ is continuous in $L^{2}$, and in
particular,

$Y_{1}-Y_{1-\delta}\to^{\mathrm{d}}0\ \ \textrm{as}\ \ \delta\downarrow 0.$

On the other hand, the scaled moment condition (37) implies that

$\mathbb{E}\Bigl{[}(Y^{(t)}_{u}-Y^{(t)}_{s})^{2}(Y^{(t)}_{s}-Y^{(t)}_{r})^{2}\Bigr{]}\leq(h(u)-h(r))^{\kappa}$

for any $t\in(0,1]$ and $0\leq u<s<u\leq 1$. Therefore, by [6, Theorem 13.5],
$Y^{(t)}\to^{\mathrm{d}}Y$ in the space $\mathbb{D}([0,1];\mathbb{R})$
equipped with the Skorokhod $J_{1}$ topology. ∎

######  Remark 10.

1) In Lemma 9, assumption (37) can be replaced by any condition guaranteeing
the tightness of the family $\\{Y^{(t)}:t\in(0,1]\\}$. For example, if $X$ has
continuous paths (and hence, weak convergence is discussed on the $C$ space
instead), then (37) can be replaced by the simpler moment condition

$\frac{1}{\mu_{t}^{2}}\mathbb{E}\Bigl{[}(X^{1}_{ts}-X^{1}_{tr})^{2}\Bigr{]}\leq(h(s)-h(r))^{\kappa};$

see [29, Theorem 11.6.5].

2) If the Gaussian process $X$ in Theorem 8 is continuous and self-similar
with index $H>0$, then $Y^{(t)}=^{\mathrm{d}}Y^{(1)}$ for all $t\in(0,1]$, and
in particular, the weak convergence of $Y^{(t)}$ to $Y:=Y^{(1)}$ trivially
follows. Moreover, the constant appearing on the right-hand side of (35)
simplifies to 1:

$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]=\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(1)}]=1.$

Consequently, (35) takes the form

$\displaystyle\lim_{t\downarrow 0}\frac{|D|-Q^{X}_{D}(t)}{t^{H}}=|\partial
D|\cdot\mu_{1},$

which coincides with (30) when $d=1$.

The proof of Theorem 8, which requires several lemmas, is postponed to Section
5. Before closing the current section, we provide an example of a class of
non-self-similar Gaussian processes that satisfy assumptions (36) and (37) (so
the weak convergence of $Y^{(t)}$ occurs).

######  Example 11.

Let $X$ be a stochastic process in $\mathbb{R}^{d}$ with i.i.d. components
that are equal in distribution to a one-dimensional Brownian motion $B$ (with
$\mathbb{E}[B_{t}^{2}]=t$) that is time-changed by a deterministic,
continuous, non-decreasing function $\alpha:[0,1]\to[0,\infty)$ with
$\alpha(0)=0$ and $\alpha(t)>0$ for $t\in(0,1]$; i.e.,
$(X^{1}_{t})_{t\in[0,1]}=^{\textrm{d}}(B_{\alpha(t)})_{t\in[0,1]}$. For
example, $X^{1}$ can be taken to be an Itô integral of a deterministic
integrand due to Lévy’s characterization theorem:

$X^{1}_{t}=\int_{0}^{t}f(s)\mathrm{d}\tilde{B}_{s}=^{\textrm{d}}B_{\alpha(t)},\
\ \ \textrm{where}\ \ \alpha(t)=[X^{1}]_{t}=\int_{0}^{t}f^{2}(s)\mathrm{d}s.$

The process $X^{1}$ is a continuous Gaussian process that is not necessarily
self-similar. Fix $t\in(0,1]$. Observe that for $s,r\in[0,1]$,

$R_{X^{1}}(ts,tr)=\mathbb{E}[B_{\alpha(ts)}B_{\alpha(tr)}]=\alpha(ts)\wedge\alpha(tr),$

and for $0\leq r<s<u\leq 1$, since $\alpha$ is non-decreasing,

$\displaystyle\mathbb{E}\Bigl{[}(X^{1}_{tu}-X^{1}_{ts})^{2}(X^{1}_{ts}-X^{1}_{tr})^{2}\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}(B_{\alpha(tu)}-B_{\alpha(ts)})^{2}(B_{\alpha(ts)}-B_{\alpha(tr)})^{2}\Bigr{]}$
$\displaystyle=(\alpha(tu)-\alpha(ts))(\alpha(ts)-\alpha(tr))$
$\displaystyle\leq(\alpha(tu)-\alpha(tr))^{2}.$

(Or, since this is a continuous process, we could instead discuss the quantity
$\mathbb{E}\bigl{[}(X^{1}_{tu}-X^{1}_{tr})^{4}\bigr{]}=\mathbb{E}\bigl{[}(B_{\alpha(tu)}-B_{\alpha(tr)})^{4}\bigr{]}=3(\alpha(tu)-\alpha(tr))^{2}.$)
On the other hand, since $\alpha$ is continuous, non-decreasing, and starting
at 0,

$\mu_{t}=\mathbb{E}[\sup_{0\leq s\leq t}B_{\alpha(s)}]=\mathbb{E}[\sup_{0\leq
u\leq 1}B_{\alpha(t)u}]=(\alpha(t))^{1/2}\mu_{B},\ \ \ \textrm{where}\ \
\mu_{B}=\mathbb{E}[\sup_{0\leq u\leq 1}B_{u}]=\sqrt{\frac{2}{\pi}}.$

Let us further assume that $\alpha(t)$ is regularly varying at 0 with index
$\rho>0$, which satisfies $\frac{\alpha(ts)}{\alpha(t)}\to s^{\rho}$ as
$t\downarrow 0$ for any fixed $s\in[0,1]$ (see [7, Chapter 1]). Then

$\frac{R_{X^{1}}(ts,tr)}{\mu_{t}^{2}}=\frac{\alpha(ts)\wedge\alpha(tr)}{\alpha(t)\mu_{B}^{2}}\to\frac{(s\wedge
r)^{\rho}}{\mu_{B}^{2}},$

so (36) holds with $g(s,r)=\frac{(s\wedge r)^{\rho}}{\mu_{B}^{2}}$. On the
other hand, for $0\leq r<s<u\leq 1$,

$\frac{1}{\mu_{t}^{4}}\mathbb{E}\Bigl{[}(X^{1}_{tu}-X^{1}_{ts})^{2}(X^{1}_{ts}-X^{1}_{tr})^{2}\Bigr{]}\leq\frac{1}{\mu_{B}^{4}}\biggl{(}\frac{\alpha(tu)-\alpha(tr)}{\alpha(t)}\biggr{)}^{2}.$

Although the latter approaches $\frac{1}{\mu_{B}^{4}}(u^{\rho}-r^{\rho})^{2}$,
we rather need a uniform bound valid for all $0\leq r<u\leq 1$ and for all
small $t$. To achieve this, suppose additionally that $\alpha(t)$ has
representation $\alpha(t)=t^{\rho}\ell(t)$ in terms of a slowly varying
function $\ell(t)$ at 0 satisfying the following condition: there exist
constants $0<m<M<\infty$ and a continuous, non-decreasing function $H$ such
that for all $t\in(0,1]$,

$m\leq\ell(t)\leq M$

and

$\ell(tu)-\ell(tr)\leq H(u)-H(r)\ \ \textrm{for all}\ \ 0\leq r<u\leq 1.$

(The second condition is satisfied if $\ell$ has a derivative with positive
upper bound on $(0,1]$ due to the mean value theorem.) Then for $0\leq r<u\leq
1$,

$\displaystyle 0\leq\frac{\alpha(tu)-\alpha(tr)}{\alpha(t)}$
$\displaystyle=\frac{u^{\rho}(\ell(tu)-\ell(tr))+\ell(tr)(u^{\rho}-r^{\rho})}{\ell(t)}$
$\displaystyle\leq\frac{(H(u)-H(r))+M(u^{\rho}-r^{\rho})}{m}$
$\displaystyle=\frac{(H(u)+Mu^{\rho})-(H(r)+Mr^{\rho})}{m}.$

As a result,

$\frac{1}{\mu_{t}^{4}}\mathbb{E}\Bigl{[}(X^{1}_{tu}-X^{1}_{ts})^{2}(X^{1}_{ts}-X^{1}_{tr})^{2}\Bigr{]}\leq\frac{1}{m^{2}\mu_{B}^{4}}\bigl{(}(H(u)+Mu^{\rho})-(H(r)+Mr^{\rho})\bigr{)}^{2},$

so (37) is satisfied with $\kappa=2$. Therefore, by Lemma 9, $Y^{(t)}$ defined
in (34) converges weakly to a Gaussian process $Y$ with covariance function
$R_{Y}(s,r)=\frac{(s\wedge r)^{\rho}}{\mu_{B}^{2}}$ in the space
$\mathbb{D}([0,1];\mathbb{R})$. Moreover, for any $0\leq r<s\leq 1$,

$\mathbb{E}[|Y_{s}-Y_{r}|^{2}]=\frac{1}{\mu_{B}^{2}}(s^{\rho}-r^{\rho})\leq\begin{cases}\frac{\rho}{\mu_{B}^{2}}(s-r)&\textrm{if}\
\rho>1;\\\ \frac{1}{\mu_{B}^{2}}(s-r)^{\rho}&\textrm{if}\ 0<\rho\leq
1,\end{cases}$

where we used the mean value theorem when $\rho>1$ and the sub-additivity of
$x\mapsto x^{\rho}$ when $0<\rho\leq 1$. In either case, for some $\delta>0$,
the latter upper bound is dominated above by $(\log(s-r))^{-2}$ as long as
$0<s-r<\delta$. Hence, criterion (1.4.4) in [3, Theorem 1.4.1] is satisfied,
and consequently, $Y$ is a continuous Gaussian process, as stated in Theorem
8.

Examples of time changes $\alpha(t)$ that satisfy the conditions assumed above
include $\alpha(t)=t^{\rho}\log(t+c)$ with $c>1$, and
$\alpha(t)=\sum_{k=0}^{n}c_{k}t^{\eta_{k}}$ with $c_{0}>0$, $c_{k}\geq 0$ for
all $k\geq 1$, and $0<\eta_{0}<\eta_{1}<\eta_{2}<\cdots<\eta_{n}$. In fact,
for the latter time change, one can take $\rho=\eta_{0}$, $m=c_{0}$,
$M=\sum_{k=0}^{n}c_{k}$, and $H(u)=\sum_{k=0}^{n}c_{k}u^{\eta_{k}-\eta_{0}}$.

##  5 Proof of Theorem 8

Throughout this section, we assume that $X$, $Y$, $Y^{(t)}$, and $D$ are those
appearing in the statement of Theorem 8. The first lemma guarantees that the
right-hand side of (35) is finite.

######  Lemma 12.

$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]\leq 1$.

###### Proof.

For each fixed $t>0$, observe that

$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(t)}]=\frac{\mathbb{E}[\sup_{s\in[0,1]}X_{ts}^{1}]}{\mu_{t}}=1.$

For fixed $\varepsilon>0$ and $n\geq 1$, the weak convergence
$(Y^{(t)}_{s})_{s\in[0,1]}\to^{\mathrm{d}}(Y_{s})_{s\in[0,1]}$ in the space
$\mathbb{D}([0,1];\mathbb{R})$ implies that
$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(t)}\wedge
n]\to\mathbb{E}[\sup_{s\in[0,1]}Y_{s}\wedge n]$ as $t\downarrow 0$ (as the
supremum and minimum are both continuous functions on the Skorokhod space), so
there exists $t_{0}(n)=t_{0}(n,\varepsilon)>0$ such that

$\left|\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(t_{0}(n))}\wedge
n]-\mathbb{E}[\sup_{s\in[0,1]}Y_{s}\wedge n]\right|<\varepsilon.$

By Fatou’s lemma,

$\displaystyle\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{E}[\sup_{s\in[0,1]}Y_{s}\wedge
n]$
$\displaystyle\leq\liminf_{n\to\infty}\left(\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(t_{0}(n))}\wedge
n]+\varepsilon\right)$
$\displaystyle\leq\liminf_{n\to\infty}\left(\mathbb{E}[\sup_{s\in[0,1]}Y_{s}^{(t_{0}(n))}]+\varepsilon\right)$
$\displaystyle=1+\varepsilon.$

Letting $\varepsilon\downarrow 0$ completes the proof. ∎

The next lemma shows that the weak convergence of $Y^{(t)}$ in Theorem 8
implies the weak convergence of a scaled version of the process $X$ in
$\mathbb{R}^{d}$. Here, the reader should be alerted that two kinds of the
$J_{1}$ topology can be considered in the multi-dimensional setting. One is
the product topology on the Cartesian product
$\prod_{j=1}^{d}\mathbb{D}([0,1];\mathbb{R}):=\mathbb{D}([0,1];\mathbb{R})\times\cdots\times\mathbb{D}([0,1];\mathbb{R})$,
with which the convergence $x_{n}=(x^{1}_{n},\ldots,x^{d}_{n})\to
x=(x^{1},\ldots,x^{d})$ is equivalent to the convergence $x^{j}_{n}\to x^{j}$
in $\mathbb{D}([0,1];\mathbb{R})$ for each $1\leq j\leq d$. The other is the
stronger (finer) topology that is obtained by using the maximum norm of
$\mathbb{R}^{d}$, $\|a\|:=\max_{1\leq j\leq d}|a^{j}|$, in (33). Below, the
$J_{1}$ topology for the space $\mathbb{D}([0,1];\mathbb{R}^{d})$ means this
stronger topology. In addition, $C([0,1];\mathbb{R}^{d})$ denotes the space of
$\mathbb{R}^{d}$-valued continuous functions on $[0,1]$.

######  Lemma 13.

For each $t\in(0,1]$, define a scaled process
$\widetilde{X}^{(t)}=(\widetilde{X}^{(t)}_{u})_{u\in[0,1]}$ of $X$ by

(38) $\displaystyle\widetilde{X}_{u}^{(t)}=\frac{X_{tu}}{\mu_{t}}.$

Then as $t\downarrow 0$, $\widetilde{X}^{(t)}$ converges weakly to $Z$ in
$\mathbb{D}([0,1];\mathbb{R}^{d})$ equipped with the Skorokhod $J_{1}$
topology, where $Z$ is a process whose components are independent copies of
the Gaussian process $Y$.

###### Proof.

Express $\widetilde{X}^{(t)}$ as

$\widetilde{X}^{(t)}=(\widetilde{X}^{(t)1},\ldots,\widetilde{X}^{(t)d}),$

where $\widetilde{X}^{(t)j}$’s are i.i.d. and have the same distribution as
$Y^{(t)}$. By assumption, for each fixed $j$, $\widetilde{X}^{(t)j}$ converges
weakly to a continuous process $Z^{j}$ in $\mathbb{D}([0,1];\mathbb{R})$ with
the $J_{1}$ topology, where $Z^{j}$’s have the same distribution as $Y$. Since
$\widetilde{X}^{(t)j}$’s are independent, so are $Z^{j}$’s, and one can
consider the process

$Z=(Z^{1},\ldots,Z^{d})$

on a single probability space. Since $\widetilde{X}^{(t)j}$ converges weakly
to $Z^{j}$ in $\mathbb{D}([0,1];\mathbb{R})$ with the $J_{1}$ topology for
each $j$, $\widetilde{X}^{(t)}$ converges weakly to $Z$ in
$\prod_{j=1}^{d}\mathbb{D}([0,1];\mathbb{R})$ equipped with the product
topology. However, $\widetilde{X}^{(t)}$ in fact converges weakly to $Z$ in
$\mathbb{D}([0,1];\mathbb{R}^{d})$ with the stronger topology. To verify this,
consider the identity map
$\textrm{id}:\prod_{j=1}^{d}\mathbb{D}([0,1];\mathbb{R})\to\mathbb{D}([0,1];\mathbb{R}^{d})$.
Then the $J_{1}$ version of [29, Theorem 12.6.1] implies that id is continuous
at any $x=(x^{1},\ldots,x^{d})\in C([0,1];\mathbb{R}^{d})$; in other words,
$C([0,1];\mathbb{R}^{d})\subset(\textrm{Disc}(\textrm{id}))^{c}$, where
$\textrm{Disc}(\textrm{id})$ denotes the set of all discontinuities of id.
Since $Z$ has continuous paths,
$\mathbb{P}(Z\in\textrm{Disc}(\textrm{id}))\leq\mathbb{P}(Z\in(C([0,1];\mathbb{R}^{d}))^{c})=0$.
Therefore, by the continuous mapping theorem (see e.g., [29, Theorem 3.4.3]),
the weak convergence $\widetilde{X}^{(t)}\to^{\mathrm{d}}Z$ in
$\prod_{j=1}^{d}\mathbb{D}([0,1];\mathbb{R})$ yields the weak convergence
$\widetilde{X}^{(t)}\to^{\mathrm{d}}Z$ in $\mathbb{D}([0,1];\mathbb{R}^{d})$,
as desired. ∎

With Lemmas 12 and 13 at hand, we will prove Theorem 8 following the approach
taken in [15] for general Lévy processes with regularly varying characteristic
exponents. The next lemma shows that the amount of heat loss from deep inside
the domain $D$ is negligible when compared to $\mu_{t}$. Recall the notation
$D_{a}$ defined in (13).

######  Lemma 14.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$, as $t\downarrow 0$,

$\int_{D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq t)\mathrm{d}x=o(\mu_{t}).$

###### Proof.

Let $t\in(0,1]$ be small so that $a>\mu_{t}$. For any $x\in D_{a}$,

$\displaystyle\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)\leq\mathbb{P}_{x}\left(\sup_{s\in[0,t]}|X_{s}-x|\geq
a\right)=\mathbb{P}\left(\sup_{s\in[0,t]}|X_{s}|\geq
a\right)=2\mathbb{P}\left(\sup_{s\in[0,t]}X_{s}\geq a\right)\leq
4e^{-\frac{(a-\mu_{t})^{2}}{2\sigma_{t}^{2}}},$

where we used the Borell–TIS inequality (4). The inequality $e^{-x}<x^{-1}$
valid for $x>0$ together with the asymptotic relation of
$\sigma^{2}_{t}/\mu_{t}$ obtained from (29) yields the desired statement. ∎

Now we start investigating the quantity $\int_{D\setminus
D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq t)\mathrm{d}x$, which is the amount of
heat loss from points near the boundary of the domain $D$. Throughout, the
notation

$H=\\{x=(\tilde{x},x^{d})=(x^{1},\ldots,x^{d-1},x^{d})\in\mathbb{R}^{d}:x^{d}>0\\}$

denotes the upper half-space, and $\tilde{0}$ represents the zero vector in
$\mathbb{R}^{d-1}$; in particular, a given point in $H$ that is on the last
coordinate axis can be represented as $(\tilde{0},u)$ with $u>0$.

######  Lemma 15.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$,

$\lim_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{H}\leq
t)\mathrm{d}u=\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

###### Proof.

Recall the family of scaled processes $\\{\widetilde{X}^{(t)}:t\in(0,1]\\}$
defined in (38). Note that

$\displaystyle\tau_{H}^{X}$ $\displaystyle=\inf\\{u:X_{u}\notin
H\\}=\inf\\{tu:\mu_{t}^{-1}X_{tu}\notin\mu_{t}^{-1}H\\}$
$\displaystyle=\inf\\{tu:\mu_{t}^{-1}X_{tu}\notin
H\\}=t\inf\\{u:\widetilde{X}_{u}^{(t)}\notin
H\\}=t\tau_{H}^{\widetilde{X}^{(t)}},$

where we used the fact that the upper half-space $H$ is invariant under the
multiplication by any positive constant. The latter shows that the law of
$\tau_{H}^{X}$ under $\mathbb{P}_{x}$ is equal to the law of
$t\tau_{H}^{\widetilde{X}^{(t)}}$ under $\mathbb{P}_{\mu_{t}^{-1}x}$. Hence,
for a fixed $a\in(0,R/2]$, by the change of variables $v=\mu_{t}^{-1}u$,

(39) $\displaystyle\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u=\int_{0}^{a}\mathbb{P}_{(\tilde{0},\mu_{t}^{-1}u)}(\tau_{H}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}u=\mu_{t}\int_{0}^{a\mu_{t}^{-1}}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v.$

By Lemma 12, for any $\varepsilon>0$, there exists $N=N(\varepsilon)>0$ such
that

$\int_{0}^{N}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}>v)\mathrm{d}v>\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]-\varepsilon,$

Since $\mu_{t}^{-1}\to\infty$ as $t\downarrow 0$, there exists $t_{0}>0$ such
that $a\mu_{t}^{-1}\geq N$ for all $0<t\leq t_{0}$. Hence, it follows from
(39), Fatou’s lemma, and the assumption on weak convergence of $Y^{(t)}$ to
$Y$ that

$\displaystyle\liminf_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u$ $\displaystyle\geq\liminf_{t\downarrow
0}\int_{0}^{N}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v=\liminf_{t\downarrow
0}\int_{0}^{N}\mathbb{P}_{v}(\inf_{s\in[0,1]}Y_{s}^{(t)}\leq 0)\mathrm{d}v$
$\displaystyle=\liminf_{t\downarrow
0}\int_{0}^{N}\mathbb{P}(\inf_{s\in[0,1]}(Y_{s}^{(t)}+v)\leq 0)\mathrm{d}v$
$\displaystyle=\liminf_{t\downarrow
0}\int_{0}^{N}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}>v)\mathrm{d}v$
$\displaystyle\geq\int_{0}^{N}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}>v)\mathrm{d}v>\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]-\varepsilon.$

Since $\varepsilon>0$ is arbitrary, we obtain the lower bound

$\displaystyle\liminf_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u\geq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

Derivation of the upper bound requires a delicate discussion. In fact, a
simple modification of the above argument would be to notice by (39) that

$\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u=\limsup_{t\downarrow
0}\int_{0}^{\infty}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}\geq
v)\mathbf{1}_{(0,a\mu_{t}^{-1}]}(v)\mathrm{d}v$

and use the reverse version of Fatou’s lemma for the latter limit. However,
that would require the integrand $\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}\geq
v)\mathbf{1}_{(0,a\mu_{t}^{-1}]}(v)$ to be bounded above by a function of $v$
that is both independent of $t$ and integrable on the unbounded interval
$(0,\infty)$, and finding such an upper bound is a non-trivial task. To
overcome this hurdle, fix $M>2$ and $t\in(0,1]$ so that $a\mu_{t}^{-1}>M$, and
note by the Borell–TIS inequality (4) that

$\displaystyle\int_{M}^{a\mu_{t}^{-1}}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}>v)\mathrm{d}v$
$\displaystyle=\int_{M}^{a\mu_{t}^{-1}}\mathbb{P}(\sup_{u\in[0,t]}X_{u}^{1}>v\mu_{t})\mathrm{d}v$
$\displaystyle\leq\int_{M}^{a\mu_{t}^{-1}}2e^{-\frac{(v-1)^{2}\mu_{t}^{2}}{2\sigma^{2}_{t}}}\mathrm{d}v$
$\displaystyle\leq\int_{M}^{\infty}2(v-1)e^{-\frac{(v-1)^{2}\mu_{t}^{2}}{2\sigma^{2}_{t}}}\mathrm{d}v$
$\displaystyle=\frac{2\sigma_{t}^{2}}{\mu_{t}^{2}}e^{-\frac{(M-1)^{2}\mu_{t}^{2}}{2\sigma_{t}^{2}}}\leq\frac{4\sigma_{t}^{4}}{(M-1)^{2}\mu_{t}^{4}},$

where we used the inequality $e^{-x}<x^{-1}$ valid for $x>0$. A simple
modification of the argument used to obtain (29) yields

$\frac{\sigma_{t}^{2}}{\mu_{t}^{2}}\leq\frac{2}{c_{2}^{2}}\ \ \textrm{for
all}\ t>0.$

Therefore,

(40)
$\displaystyle\int_{M}^{a\mu_{t}^{-1}}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}>v)\mathrm{d}v\leq\frac{16}{c_{2}^{4}(M-1)^{2}}.$

Now, using (39) and (40), applying the reverse version of Fatou’s lemma on the
bounded interval $(0,M]$, and using the weak convergence of $Y^{(t)}$ to $Y$,
we obtain

$\displaystyle\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u$ $\displaystyle=\limsup_{t\downarrow
0}\int_{0}^{a\mu_{t}^{-1}}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$ $\displaystyle\leq\limsup_{t\downarrow
0}\int_{0}^{M}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}\geq v)\mathrm{d}v$
$\displaystyle\quad+\limsup_{t\downarrow
0}\int_{M}^{a\mu_{t}^{-1}}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}^{(t)}\geq
v)\mathrm{d}v$
$\displaystyle\leq\int_{0}^{M}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}\geq
v)\mathrm{d}v+\frac{16}{c_{2}^{4}(M-1)^{2}}.$

Letting $M\uparrow\infty$ yields the upper bound

$\displaystyle\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u\leq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}],$

as desired. ∎

In the lemma below, $B((\tilde{0},R),R)$ represents the ball of radius $R$
centered at $(\tilde{0},R)\in H$.

######  Lemma 16.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$,

$\lim_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u=\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

###### Proof.

Since $B((\tilde{0},R),R)\subset H$, it follows that
$\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\leq\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq t)$. This,
together with Lemma 15, yields the lower bound

$\liminf_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u\geq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

To derive the upper bound, recall the definition of $\widetilde{X}^{(t)}$ in
(38) and observe that

$\displaystyle\tau_{B((\tilde{0},R),R)}^{X}$
$\displaystyle=\inf\\{u:X_{u}\notin
B((\tilde{0},R),R)\\}=\inf\\{tu:X_{tu}\notin B((\tilde{0},R),R)\\}$
$\displaystyle=\inf\\{tu:\mu_{t}^{-1}X_{tu}\notin\mu_{t}^{-1}B((\tilde{0},R),R)\\}$
$\displaystyle=t\inf\\{u:\widetilde{X}_{u}^{(t)}\notin
B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)\\}$
$\displaystyle=t\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}.$

Hence, by the change of variables $v=\mu_{t}^{-1}u$,

(41)
$\displaystyle\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u$
$\displaystyle=\int_{0}^{a}\mathbb{P}_{(\tilde{0},\mu_{t}^{-1}u)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}u$
$\displaystyle=\mu_{t}\int_{0}^{a\mu_{t}^{-1}}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v.$

Note that if $v\in(0,a\mu_{t}^{-1}]$, then $B((\tilde{0},v),v)\subset
B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)$ since $a\leq R/2$. Moreover,
$\max_{1\leq j\leq
d}|x^{j}|\leq|x|=\sqrt{\sum_{j=1}^{d}(x^{j})^{2}}\leq\sqrt{d}\max_{1\leq j\leq
d}|x^{j}|$ for any $x=(x^{1},\ldots,x^{d})\in\mathbb{R}^{d}$. Therefore, for
any $v\in(0,a\mu_{t}^{-1}]$,

$\displaystyle\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)$
$\displaystyle\leq\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},v),v)}^{\widetilde{X}^{(t)}}\leq
1)=\mathbb{P}\left(\sup_{u\in[0,1]}|\widetilde{X}_{u}^{(t)}|\geq v\right)$
$\displaystyle\leq
d\cdot\mathbb{P}\left(\sup_{u\in[0,1]}|Y_{u}^{(t)}|\geq\frac{v}{\sqrt{d}}\right)=2d\cdot\mathbb{P}\left(\sup_{u\in[0,1]}Y_{u}^{(t)}\geq\frac{v}{\sqrt{d}}\right)$

where $Y^{(t)}$ has the same distribution as the i.i.d. components of
$\widetilde{X}^{(t)}$. As in the discussion given in Lemma 15, by the
Borell–TIS inequality (4), for any fixed $M>2$,

(42) $\displaystyle\limsup_{t\downarrow
0}\int_{M\sqrt{d}}^{a\mu_{t}^{-1}}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$ $\displaystyle\leq 2d\limsup_{t\downarrow
0}\int_{M\sqrt{d}}^{a\mu_{t}^{-1}}e^{-\frac{(\frac{v}{\sqrt{d}}-1)^{2}\mu_{t}^{2}}{2\sigma_{t}^{2}}}\mathrm{d}v$
$\displaystyle\leq\frac{32d\sqrt{d}}{c_{2}^{4}(M-1)^{2}}.$

Now, to deal with the integral
$\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$, we construct an increasing sequence of domains $D(n)$ by

$D(n)=\left\\{x=(\tilde{x},x^{d})\in\mathbb{R}^{d-1}\times\mathbb{R}:|\tilde{x}|<n,\,0<x^{d}<n,\,\cos^{-1}(\vec{\bf{n}}\cdot\frac{x}{|x|})<\frac{\pi}{2}-\frac{1}{n}\right\\},\quad\vec{\bf{n}}=(\tilde{0},1).$
$n$$-n$0$D(n)$

For the limiting process $Z$ in Lemma 13, since $D(n)$ increases to $H$ as
$n\to\infty$, it follows from the bounded convergence theorem that

$\lim_{n\to\infty}\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{D(n)}^{Z}\leq
1)\mathrm{d}v=\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{Z}\leq
1)\mathrm{d}v,$

where the sequence converges decreasingly. Therefore, for a given
$\varepsilon>0$, we may take an integer $N=N(\varepsilon)$ such that

(43)
$\displaystyle\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{D(N)}^{Z}\leq
1)\mathrm{d}v<\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{Z}\leq
1)\mathrm{d}v+\varepsilon.$

Since $B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)$ increases to $H$ as
$t\downarrow 0$, we can take $t_{0}=t_{0}(N)\in(0,1]$ such that for any
$t\in(0,t_{0}]$,

$D(N)\subset B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R),$

and hence,

(44)
$\displaystyle\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\leq\mathbb{P}_{(\tilde{0},v)}(\tau_{D(N)}^{\widetilde{X}^{(t)}}\leq 1).$

Combining (43) and (44), applying the reverse Fatou’s lemma on the bounded
interval $(0,M\sqrt{d}]$, and using the weak convergence of
$\widetilde{X}^{(t)}$ to $Z$ in Lemma 13, we obtain

(45) $\displaystyle\limsup_{t\downarrow
0}\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},\mu_{t}^{-1}R),\mu_{t}^{-1}R)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$ $\displaystyle\leq\limsup_{t\downarrow
0}\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{D(N)}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$
$\displaystyle\leq\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{D(N)}^{Z}\leq
1)\mathrm{d}v$
$\displaystyle<\int_{0}^{M\sqrt{d}}\mathbb{P}_{(\tilde{0},v)}(\tau_{H}^{Z}\leq
1)\mathrm{d}v+\varepsilon$
$\displaystyle=\int_{0}^{M\sqrt{d}}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}\geq
1)\mathrm{d}v+\varepsilon.$

Finally, putting together (41), (42) and (45) yields

$\displaystyle\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u<\int_{0}^{M\sqrt{d}}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}\geq
1)\mathrm{d}v+\varepsilon+\frac{32d\sqrt{d}}{c_{2}^{4}(M-1)^{2}}.$

Letting $\varepsilon\downarrow 0$ and $M\uparrow\infty$ yields

$\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u\leq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}],$

which is the desired upper bound. ∎

Recall the notations $\delta_{D}(x)$ and $D_{a}$ defined in (12)–(13). For a
starting point $x\in D\setminus D_{R/2}$ (so that $\delta_{D}(x)\leq R/2$),
let $z_{x}$ denote the unique point in $\partial D$ such that
$|x-z_{x}|=\delta_{D}(x)$, and let
$\mathbf{n}_{z_{x}}=\frac{z_{x}-x}{|z_{x}-x|}$ denote the outward unit normal
vector to $\partial D$ at the point $z_{x}$. With the interior $R$-ball
condition in (11) in mind, let $H_{x}$ denote the unique half-space containing
the interior $R$-ball at the point $z_{x}$ whose normal vector is
$\textbf{n}_{z_{x}}$.

######  Lemma 17.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$, as $t\downarrow 0$,

$\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau^{X}_{D}\leq
t<\tau^{X}_{H_{x}})\mathrm{d}x=o(\mu_{t}).$

###### Proof.

By (15),

$\displaystyle\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau^{X}_{D}\leq
t<\tau^{X}_{H_{x}})\mathrm{d}x$ $\displaystyle\leq 2^{d-1}|\partial
D|\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t<\tau^{X}_{H})\mathrm{d}u$ $\displaystyle=2^{d-1}|\partial
D|\left(\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},R),R)}\leq
t)\mathrm{d}u-\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{H}\leq
t)\mathrm{d}u\right).$

Applying Lemmas 15 and 16 gives the desired conclusion. ∎

Now we are ready to derive the upper bound for Theorem 8.

###### Derivation of the upper bound for Theorem 8..

Suppose the $C^{1,1}$ characteristics of the domain $D$ are $(R,\Lambda)$. It
follows from (14) that for any $\varepsilon>0$, there exists
$a=a(\varepsilon)\in(0,R/2]$ such that

(46) $\displaystyle|\partial D|-\varepsilon<|\partial D_{u}|<|\partial
D|+\varepsilon\ \ \textrm{for all}\ u\leq a.$

For this particular $a$, note that

$|D|-Q_{D}^{X}(t)=\int_{D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)\mathrm{d}x+\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)\mathrm{d}x.$

The first term on the right-hand side is negligible when compared to $\mu_{t}$
due to Lemma 14. On the other hand, for any $x\in D$ with $\delta_{D}(x)\leq
a$ (in other words, $x\in D\setminus D_{a}$),

$\\{\tau_{D}^{X}\leq t\\}\subset\\{\tau^{X}_{H_{x}}\leq
t\\}\cup\\{\tau_{D}^{X}\leq t<\tau_{H_{x}}^{X}\\},$

and therefore,

(47) $\displaystyle\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)\mathrm{d}x\leq\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq
t)\mathrm{d}x+\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{D}^{X}\leq
t<\tau_{H_{x}}^{X})\mathrm{d}x.$

In terms of the first integral on the right-hand side,

$\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq
t)\mathrm{d}x=\int_{0}^{a}|\partial
D_{u}|\cdot\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq t)\mathrm{d}u,$

so (46) and Lemma 15 together gives

(48) $\displaystyle\lim_{t\downarrow 0}\mu_{t}^{-1}\int_{D\setminus
D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq t)\mathrm{d}x=|\partial
D|\cdot\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

Combining (47), (48), and Lemma 17 yields the upper bound

$\limsup_{t\downarrow 0}\mu_{t}^{-1}(|D|-Q_{D}^{X}(t))\leq|\partial
D|\cdot\mathbb{E}[\sup_{s\in[0,1]}Y_{s}],$

as desired. ∎

Establishing the lower bound for Theorem 8 requires two additional lemmas,
Lemmas 18 and 19, which are analogous to Lemmas 16 and 17, respectively. For
the lower bound, we take advantage of the exterior $R$-ball condition in (11)
for the $C^{1,1}$ open set $D$. Lemma 18 considers the first exit time
$\tau^{X}_{B((\tilde{0},-R),R)^{c}}$ from the complement of the ball
$B((\tilde{0},-R),R)$ of radius $R$ centered at the point $(\tilde{0},-R)$,
which is located in the lower half-space

$H^{\prime}=\\{x=(x^{1}\ldots,x^{d})\in\mathbb{R}^{d}:x^{d}<0\\}.$

######  Lemma 18.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$,

$\lim_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\mathrm{d}u=\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

###### Proof.

The proof is similar to the proof of Lemma 16, but we provide the details for
the reader’s convenience. Fix $a\in(0,R/2]$. Since $H\subset
B((\tilde{0},-R),R)^{c}$, it follows that
$\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\leq\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq t)$. This, together with
Lemma 15, yields the upper bound

$\limsup_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\mathrm{d}u\leq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}].$

To derive the lower bound, let $\varepsilon>0$ and recall that
$\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]\leq 1<\infty$. There exists an integer
$M=M(\varepsilon)>0$ such that

(49)
$\displaystyle\int_{0}^{M}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}>x)\mathrm{d}x>\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]-\varepsilon.$

As in the proof of Lemma 15, one can verify that the law of
$\tau_{B((\tilde{0},-R),R)^{c}}^{X}$ under $\mathbb{P}_{x}$ is equal to the
law of
$t\tau_{B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)^{c}}^{\widetilde{X}^{(t)}}$
under $\mathbb{P}_{\mu_{t}^{-1}x}$, where $\widetilde{X}^{(t)}$ is the scaled
process defined in (38). Hence, by the change of variables $v=\mu_{t}^{-1}u$,

(50)
$\displaystyle\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\mathrm{d}u$
$\displaystyle=\int_{0}^{a}\mathbb{P}_{(\tilde{0},\mu_{t}^{-1}u)}(\tau_{B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)^{c}}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}u$
$\displaystyle=\mu_{t}\int_{0}^{a\mu_{t}^{-1}/2}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)^{c}}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$
$\displaystyle\geq\mu_{t}\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)^{c}}^{\widetilde{X}^{(t)}}\leq
1)\mathrm{d}v$

for all sufficiently small $t>0$ (since $\mu_{t}^{-1}\to\infty$ as
$t\downarrow 0$).

Now we construct an increasing sequence of domains $E(n)$ by

$E(n)=\left\\{x=(\tilde{x},x^{d})\in\mathbb{R}^{d-1}\times\mathbb{R}:|\tilde{x}|<n,\,-n<x^{d}<0,\,\cos^{-1}(-\vec{\bf{n}}\cdot\frac{x}{|x|})<\frac{\pi}{2}-\frac{1}{n}\right\\},\quad-\vec{\bf{n}}=(\tilde{0},-1).$
$n$$-n$0$E(n)$

For the limiting process $Z$ in Lemma 13, since $E(n)$ increases to the lower
half-space $H^{\prime}=\\{(x^{1},\ldots,x^{d}):x^{d}<0\\}$ as $n\to\infty$, it
follows from the bounded convergence theorem that

$\lim_{n\to\infty}\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{E(n)^{c}}^{Z}<1)\mathrm{d}v=\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{(H^{\prime})^{c}}^{Z}<1)\mathrm{d}v=\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{\overline{H}}^{Z}<1)\mathrm{d}v.$

Since the latter sequence converges increasingly, we may take an integer
$N=N(\varepsilon)$ such that

(51)
$\displaystyle\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{E(N)^{c}}^{Z}<1)\mathrm{d}v>\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{\overline{H}}^{Z}<1)\mathrm{d}v-\varepsilon.$

Since $B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)$ increases to $H^{\prime}$
as $t\downarrow 0$, we can take $t_{0}=t_{0}(N)\in(0,1]$ such that for any
$t\in(0,t_{0}]$,

$E(N)\subset B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R),$

which implies that

(52)
$\displaystyle\mathbb{P}_{(\tilde{0},v)}(\tau_{B((\tilde{0},-\mu_{t}^{-1}R),\mu_{t}^{-1}R)^{c}}^{\widetilde{X}^{(t)}}\leq
1)\geq\mathbb{P}_{(\tilde{0},v)}(\tau_{E(N)^{c}}^{\widetilde{X}^{(t)}}\leq
1).$

Combining (50)–(52) and using Fatou’s lemma and the weak convergence of
$\widetilde{X}^{(t)}$ to $Z$ in Lemma 13, we obtain

$\displaystyle\liminf_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\mathrm{d}u$ $\displaystyle\geq\liminf_{t\downarrow
0}\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{E(N)^{c}}^{\widetilde{X}^{(t)}}<1)\mathrm{d}v$
$\displaystyle\geq\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{E(N)^{c}}^{Z}<1)\mathrm{d}v$
$\displaystyle>\int_{0}^{M}\mathbb{P}_{(\tilde{0},v)}(\tau_{\overline{H}}^{Z}<1)\mathrm{d}v-\varepsilon$
$\displaystyle=\int_{0}^{M}\mathbb{P}(\sup_{s\in[0,1)}Y_{s}>v)\mathrm{d}v-\varepsilon$
$\displaystyle=\int_{0}^{M}\mathbb{P}(\sup_{s\in[0,1]}Y_{s}>v)\mathrm{d}v-\varepsilon>\mathbb{E}[\sup_{s\in[0,1]}Y_{s}]-2\varepsilon,$

where the last line follows from the continuity of sample paths of $Y$ and
(49). Since $\varepsilon>0$ is arbitrary, we conclude that

$\liminf_{t\downarrow
0}\mu_{t}^{-1}\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau^{X}_{B((\tilde{0},-R),R)^{c}}\leq
t)\mathrm{d}u\geq\mathbb{E}[\sup_{s\in[0,1]}Y_{s}],$

as desired. ∎

Recall that for a starting point $x\in D\setminus D_{R/2}$, $H_{x}$ denotes
the half-space containing the interior $R$-ball at the point $z_{x}$ and
tangent to $\partial D$. Let $\tilde{B_{x}}$ denote the unique exterior
$R$-ball in $D^{c}$ touching the point $z_{x}$. (Clearly,
$H_{x}\subset\tilde{B_{x}}^{c}$.)

######  Lemma 19.

Let $(R,\Lambda)$ be the $C^{1,1}$ characteristics of the domain $D$. Then for
any fixed $a\in(0,R/2]$, as $t\downarrow 0$,

$\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq
t<\tau_{\tilde{B_{x}}^{c}}^{X})\mathrm{d}x=o(\mu_{t}).$

###### Proof.

By (15),

$\displaystyle\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq
t<\tau_{\tilde{B_{x}}^{c}}^{X})\mathrm{d}x$
$\displaystyle=\int_{0}^{a}|\partial
D_{u}|\cdot\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t<\tau_{B((\tilde{0},-R),R)^{c}}^{X})\mathrm{d}u$ $\displaystyle\leq
2^{d-1}|\partial D|\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t<\tau_{B((\tilde{0},-R),R)^{c}}^{X})\mathrm{d}u$
$\displaystyle=2^{d-1}|\partial
D|\left(\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{H}^{X}\leq
t)\mathrm{d}u-\int_{0}^{a}\mathbb{P}_{(\tilde{0},u)}(\tau_{B((\tilde{0},-R),R)^{c}}^{X}\leq
t)\mathrm{d}u\right).$

Applying Lemmas 15 and 18 gives the desired conclusion. ∎

###### Derivation of the lower bound for Theorem 8..

Suppose the $C^{1,1}$ characteristics of the domain $D$ are $(R,\Lambda)$. As
we did in the derivation of the upper bound for Theorem 8, for a fixed
$\varepsilon>0$, take $a=a(\varepsilon)\in(0,R/2]$ such that (46) holds, which
results in (48).

Under $\mathbb{P}_{x}$ with $x\in D\setminus D_{a}$,

$\displaystyle\\{\tau_{H_{x}}^{X}\leq t\\}$
$\displaystyle\subset\\{\tau_{H_{x}}^{X}\leq t,\tau_{D}^{X}\leq
t\\}\cup\\{\tau_{H_{x}}^{X}\leq t<\tau_{D}^{X}\\}$
$\displaystyle\subset\\{\tau_{D}^{X}\leq t\\}\cup\\{\tau_{H_{x}}^{X}\leq
t<\tau_{D}^{X}\\}$ $\displaystyle\subset\\{\tau_{D}^{X}\leq
t\\}\cup\\{\tau_{H_{x}}^{X}\leq t<\tau_{\tilde{B_{x}}^{c}}^{X}\\}.$

The latter implies that

$\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq t)\leq\mathbb{P}_{x}(\tau_{D}^{X}\leq
t)+\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq t<\tau_{\tilde{B_{x}}^{c}}^{X}).$

Hence,

$\displaystyle|D|-Q^{X}_{D}(t)$
$\displaystyle=\int_{D}\mathbb{P}_{x}(\tau^{X}_{D}\leq
t)\mathrm{d}x\geq\int_{D\setminus D_{a}}\mathbb{P}_{x}(\tau^{X}_{D}\leq
t)\mathrm{d}x$ $\displaystyle\geq\int_{D\setminus
D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq t)\mathrm{d}x-\int_{D\setminus
D_{a}}\mathbb{P}_{x}(\tau_{H_{x}}^{X}\leq
t<\tau_{\tilde{B_{x}}^{c}}^{X})\mathrm{d}x.$

Combining (48) and Lemma 19 gives the lower bound

$\liminf_{t\downarrow 0}\mu_{t}^{-1}(|D|-Q_{D}^{X}(t))\geq|\partial
D|\cdot\mathbb{E}[\sup_{s\in[0,1]}Y_{s}],$

as desired. ∎

##  6 Applications

The theorems established in the previous sections are applicable to a wide
class of Gaussian processes and immediately yield a number of new statements,
some of which are stated below. We focus on the one-dimensional case with the
domain $D$ being a non-empty open interval.

######  Example 20.

Let $X$ be a bi-fractional Brownian motion $B^{H,K}=(B^{H,K}_{t})_{t\in[0,1]}$
with parameters $H\in(0,1)$ and $K\in(0,1]$, which has covariance function

$R^{H,K}(s,t)=\frac{1}{2^{K}}((s^{2H}+t^{2H})^{K}-|s-t|^{2HK}).$

Note that the special case when $K=1$ yields a fractional Brownian motion with
Hurst index $H\in(0,1)$. The bi-fractional Brownian motion has the variance
function $R_{t}=t^{2HK}$ and is self-similar with index $HK$ (see e.g., [10]),
so

$\mu_{t}=t^{HK}\mu_{1}(H,K),\ \ \textrm{where}\ \
\mu_{1}(H,K):=\mathbb{E}\biggl{[}\sup_{s\in[0,1]}B^{H,K}_{s}\biggr{]}.$

(Note that even when $K=1$, the value of $\mu_{1}(H,K)$ is unknown except when
$H=1/2$.) Thus, by Theorems 2 and 5,

$\lim_{t\downarrow
0}\frac{|D|-H_{D}^{B^{H,K}}(t)}{t^{HK}}=\sqrt{\frac{2}{\pi}}$

and

$\lim_{t\downarrow 0}\frac{|D|-Q_{D}^{B^{H,K}}(t)}{t^{HK}}=2\mu_{1}(H,K).$

######  Example 21.

Recall the time-changed Brownian motion
$X^{1}_{t}=(B\circ\alpha)_{t}=B_{\alpha(t)}$ discussed in Example 11, for
which

$R_{t}=\alpha(t)\ \ \textrm{and}\ \ \mu_{t}=\sqrt{\frac{2}{\pi}\alpha(t)}.$

By Theorems 2 and 5,

$\lim_{t\downarrow
0}\frac{|D|-H_{D}^{B\circ\alpha}(t)}{\sqrt{\alpha(t)}}=\sqrt{\frac{2}{\pi}}$

and

$\lim_{t\downarrow
0}\frac{|D|-Q_{D}^{B\circ\alpha}(t)}{\sqrt{\alpha(t)}}=2\sqrt{\frac{2}{\pi}}.$

######  Example 22.

Let $X$ be a fractional Ornstein–Uhlenbeck process defined by

$U^{H}_{t}:=e^{-at}\int_{0}^{t}e^{as}\mathrm{d}B^{H}_{s}$

with $a>0$, where the integral is understood as the Skorokhod integral with
respect to a factional Brownian motion $B^{H}$ with Hurst index
$H>\frac{1}{2}$. The variance function of $U^{H}$ is given by

$R_{t}=\mathbb{E}[(U^{H}_{t})^{2}]=H(2H-1)e^{-2at}\iint_{[0,t]^{2}}e^{a(u+v)}|u-v|^{2H-2}\mathrm{d}u\mathrm{d}v$

(see [5, Equation (3.1)]). By the change of variables $u=tx$ and $v=ty$,

$R_{t}=2H(2H-1)t^{2H}\iint_{0\leq x<y\leq
1}e^{a(x+y-2)t}(y-x)^{2H-2}\mathrm{d}x\mathrm{d}y.$

Since $x+y-2<0$ whenever $0\leq x<y\leq 1$, the integrand increases as
$t\downarrow 0$, so by the monotone convergence theorem,

$\lim_{t\downarrow 0}\frac{R_{t}}{t^{2H}}=2H(2H-1)\iint_{0\leq x<y\leq
1}(y-x)^{2H-2}\mathrm{d}x\mathrm{d}y=1.$

Hence, $R_{t}\sim t^{2H}$ as $t\downarrow 0$. By Theorems 2 and 5,

$\lim_{t\downarrow 0}\frac{|D|-H_{D}^{U^{H}}(t)}{t^{H}}=\sqrt{\frac{2}{\pi}}$

and

$\lim_{t\downarrow 0}\frac{|D|-Q_{D}^{U^{H}}(t)}{\mu_{t}}=2,$

where the asymptotic behavior of $\mu_{t}$ is unknown as far as the authors
know. Note that a similar argument can be made for a standard (non-fractional)
Ornstein–Uhlenbeck process, in which case, the variance function satisfies

$R_{t}=\frac{1}{2a}(1-e^{-2at})\sim t\ \ \textrm{as}\ \ t\downarrow 0.$

######  Example 23.

Following [8], consider a continuous zero-mean Gaussian process $X$ for which
there exist some constants $C_{1},C_{2}>0$ and $H_{1},H_{2}\in(0,1)$ such that

(53) $\displaystyle
C_{1}|s-r|^{H_{1}}\leq\sqrt{\mathbb{E}[|X_{s}-X_{r}|^{2}]}\leq
C_{2}|s-r|^{H_{2}}$

for all $r,s\in[0,1]$. The process is called quasi-helix if $H_{1}=H_{2}$.
Examples of quasi-helix processes with $C_{1}<C_{2}$ include bi-fractional
Brownian motions and sub-fractional Brownian motions. Condition (53) implies
that

(54) $\displaystyle C_{1}t^{H_{1}}\leq\sqrt{R_{t}}\leq C_{2}t^{H_{2}}.$

For each fixed $t\in(0,1]$, define a continuous zero-mean Gaussian process
$V^{(t)}=(V^{(t)}_{s})_{s\in[0,1]}$ by letting $V_{s}^{(t)}:=X_{st}$. Then
$\mu_{t}=\mathbb{E}[\sup_{r\in[0,t]}X_{r}]=\mathbb{E}[\sup_{s\in[0,1]}V^{(t)}_{s}]$
and

$C^{(t)}_{1}|s-r|^{H_{1}}\leq\sqrt{\mathbb{E}[|V^{(t)}_{s}-V^{(t)}_{r}|^{2}]}\leq
C^{(t)}_{2}|s-r|^{H_{2}}$

for all $s,r\in[0,1]$, where $C^{(t)}_{i}=C_{i}t^{H_{i}}$, $i=1,2$. Therefore,
application of [8, Theorem 1] for each fixed $t\in(0,1]$ yields

$\frac{C^{(t)}_{1}}{5\sqrt{H_{1}}}\leq\mathbb{E}[\sup_{s\in[0,1]}V^{(t)}_{s}]\leq\frac{16.3C^{(t)}_{2}}{\sqrt{H_{2}}}.$

In other words,

(55)
$\displaystyle\frac{C_{1}}{5\sqrt{H_{1}}}t^{H_{1}}\leq\mu_{t}\leq\frac{16.3C_{2}}{\sqrt{H_{2}}}t^{H_{2}}.$

Theorems 2 and 5, together with (54) and (55), now give information about the
asymptotic behaviors of $H_{D}^{X}(t)$ and $Q_{D}^{X}(t)$, respectively.

## References

  * [1] L. Acuña Valverde.  On the one dimensional spectral heat content for stable processes.  J. Math. Anal. Appl., 441:11–24, 2016.
  * [2] L. Acuña Valverde.  Heat content for stable processes in domains of $\mathbb{R}^{d}$.  J. Geom. Anal., 27(1):492–524, 2017.
  * [3] R. Adler and J. Taylor.  Random Fields and Geometry.  Springer New York, 2007.
  * [4] H. Aikawa, T. Kilpeläinen, N. Shanmugalingam, and X. Zhong.  Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains.  Potential Anal., 26(3):281–301, 2007.
  * [5] G. Ascione, Y. Mishura, and E. Pirozzi.  Time-changed fractional Ornstein-Uhlenbeck process.  Fract. Calc. Appl. Anal., 23(2):450–483, 2020.
  * [6] P. Billingsley.  Convergence of Probability Measures.  Wiley, second edition, 1999.
  * [7] N. H. Bingham, C. M. Goldie, and J. L. Teugels.  Regular Variation.  Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1987.
  * [8] K. Borovkov, Y. Mishura, A. Novikov, and M. Zhitlukhin.  Bounds for expected maxima of Gaussian processes and their discrete approximations.  Stochastics, 89:21–37, 2017.
  * [9] T. Grzywny, H. Park, and R. Song.  Spectral heat content for Lévy processes.  Math. Nachr., 292(4):805–825, 2019.
  * [10] C. Houdré and J. Villa.  An example of infinite dimensional quasi-helix.  Contemporary Mathematics, 336:195–201, 2003.
  * [11] S. Jansen.  Gaussian Hilbert Spaces.  Cambridge University Press, 1997.
  * [12] I. Karatzas and S. Shreve.  Brownian Motion and Stochastic Calculus.  Springer-Verlag New York, second edition, 1998.
  * [13] K. Kobayashi and H. Park.  Large-time and small-time behaviors of the spectral heat content for time-changed stable processes.  Elect. Commun. Probab., 27:1–11, 2022.
  * [14] K. Kobayashi and H. Park.  Spectral heat content for time-changed killed brownian motions.  J. Theor. Probab., 36:1148–1180, 2023.
  * [15] K. Kobayashi and H. Park.  A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes. Available at arXiv:2306.11690.  2023\.
  * [16] M. Lifshits.  Lectures on Gaussian Processes.  Springer, 2012.
  * [17] M. B. Marcus and L. A. Shepp.  Sample behavior of Gaussian processes.  Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory. University of California, Berkeley, 1972.
  * [18] M. Miranda Jr, D. Pallara, F. Paronetto, and M. Preunkert.  Short-time heat flow and fluctuations of bounded variation in $\mathbb{R}^{N}$.  Ann. Fac. Sci. Toulouse, 16(1):125–145, 2007.
  * [19] H. Park.  Higher order terms of the spectral heat content for killed subordinate and subordinate killed Brownian motions related to symmetric $\alpha$-stable processes in $\mathbb{R}$.  Potential Anal., 57:283–303, 2022.
  * [20] H. Park and R. Song.  Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic $\alpha$-stable processes.  Bull. London Math. Soc., 51:371–384, 2019.
  * [21] H. Park and R. Song.  Spectral heat content for $\alpha$-stable processes in ${C}^{1,1}$ open sets.  Electron. J. Probab., 27(22):1–19, 2022.
  * [22] Durrett. R.  Probability: Theory and Examples.  Cambridge University Press, fifth edition, 2019.
  * [23] G. Samorodnitsky.  Probability tails of Gaussian extrema.  Stoch. Proc. Appl., 38:55–84, 1991.
  * [24] M. van den Berg and E. B. Davies.  Heat flow out of regions in ${\bf R}^{m}$.  Math. Z., 202(4):463–482, 1989.
  * [25] M. van den Berg and P. B. Gilkey.  Heat content asymptotics for a Riemannian manifold with boundary.  J. Funct. Anal., 120:48–71, 1994.
  * [26] M. van den Berg and K. Gittins.  Uniform bounds for the heat content of open sets in Euclidean space.  Differential Geom. Appl., 40:67–85, 2015.
  * [27] M. van den Berg and J. F. Le Gall.  Mean curvature and heat equation.  Math. Z., 215:437–464, 1994.
  * [28] M. van den Berg and S. Srisatkunarajah.  Heat flow and Brownian motion for a region in $\mathbb{R}^{2}$ with a polygonal boundary.  Probab. Th. Rel. Fields, 86:41–52, 1990.
  * [29] W. Whitt.  Stochastic-Process Limits.  Springer, 2002.