# Caloric functions and boundary regularity for the fractional Laplacian in
Lipschitz open sets

Gavin Armstrong<EMAIL_ADDRESS>,  Krzysztof Bogdan  Faculty of
Pure and Applied Mathematics, Wrocław University of Science and Technology,
Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland<EMAIL_ADDRESS>and
Artur Rutkowski  Faculty of Pure and Applied Mathematics, Wrocław University
of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
<EMAIL_ADDRESS>
###### Abstract.

We give Martin representation of nonnegative functions caloric with respect to
the fractional Laplacian in Lipschitz open sets. The caloric functions are
defined in terms of the mean value property for the space-time isotropic
$\displaystyle\alpha$-stable Lévy process. To derive the representation, we
first establish the existence of the parabolic Martin kernel. This involves
proving new boundary regularity results for both the fractional heat equation
and the fractional Poisson equation with Dirichlet exterior conditions.
Specifically, we demonstrate that the ratio of the solution and the Green
function is Hölder continuous up to the boundary.

###### Key words and phrases:

fractional Laplacian, heat equation, Martin representation, boundary
regularity

###### 2020 Mathematics Subject Classification:

Primary 35S16, 60J50, 35C15

††footnotetext: GA was partially supported by the National Science Center of
Poland (NCN) under grant 2016/23/B/ST1/01665. KB was partially supported
through the DFG-NCN Beethoven Classic 3 programme, contract no.
2018/31/G/ST1/02252 (National Science Center, Poland) and SCHI-419/11–1 (DFG,
Germany). AR was partially supported by the National Science Center of Poland
(NCN) under grant 2019/35/N/ST1/04450.

##  1\. Introduction

Let $\displaystyle 0<\alpha<2$ and $\displaystyle d\geq 2$. For $\displaystyle
u\in C^{2}_{b}(\mathbb{R}^{d})$, define

(1.1)
$\displaystyle\displaystyle(-\Delta)^{\alpha/2}u(x):=\lim\limits_{\varepsilon\to
0^{+}}\int_{|x-y|>\varepsilon}(u(x)-u(y))\nu(x,y)\,dy,\quad
x\in\mathbb{R}^{d},$

where $\displaystyle\nu(x,y)=c_{d,\alpha}|x-y|^{-d-\alpha}$, and denote
$\displaystyle\Delta^{\alpha/2}:=-(-\Delta)^{\alpha/2}$. Let $\displaystyle
D\subset\mathbb{R}^{d}$ be a nonempty bounded open Lipschitz set with
localization radius $\displaystyle r_{0}\in(0,\infty)$ and Lipschitz constant
$\displaystyle\lambda\in(0,\infty)$. One of our goals is to investigate the
structure of nonnegative solutions to the initial-boundary value problem for
the fractional heat equation:

(1.2)
$\displaystyle\displaystyle\begin{cases}\partial_{t}u(t,x)=\Delta^{\alpha/2}u(t,x),\quad&t\in(0,T),\
x\in D,\\\ u(t,x)=g(t,x),\quad&t\in(0,T),\ x\in D^{c},\\\
u(0,x)=u_{0}(x),\quad&x\in D.\end{cases}$

Solutions to (1.2) are called caloric functions. They are defined in terms of
the mean value property for the space-time $\displaystyle\alpha$-stable Lévy
process; we refer to Section 5 for details and connections with the classical
notion of solution to (1.2). As shown by Bogdan [12] (see also Abatangelo [1]
and Bogdan, Kulczycki, and Kwaśnicki [22]), nonnegative harmonic functions for
the fractional Laplacian on $\displaystyle D$ can be decomposed into a regular
part, which can be recovered from the exterior values, and a singular part,
vanishing outside of $\displaystyle D$ and represented as an integral with
respect to a finite measure on $\displaystyle\partial D$ of the (elliptic)
Martin kernel for $\displaystyle D$ and the fractional Laplacian. Our ultimate
goal, which we complete in Section 6, is to give a counterpart of this
decomposition for caloric functions. In particular, in Theorem 6.3, we show
that nonnegative caloric functions with $\displaystyle u_{0}=g=0$ can be
expressed as integrals with respect to the parabolic Martin kernel. To obtain
the representation, we prove several new boundary regularity results for the
fractional Laplacian in Lipschitz sets, which are a significant focus of this
paper. Needless to say, the results point out directions of development for
other nonlocal operators and various classes of open sets.

Singular caloric functions were recently represented by Chan, Gómez-Castro,
and Vázquez [27] for domains more regular than Lipschitz, such as
$\displaystyle C^{1,1}$ domains. While the authors of [27] address more
general operators than our Dirichlet, or restricted, fractional Laplacian,
they do so by assuming that the (elliptic) Green function exhibits uniform
power-type decay at the boundary. Since for Lipschitz open sets, the behavior
of the Dirichlet Green function of the fractional Laplacian is more nuanced
(see Jakubowski [47]), the results of [27] are not applicable in our setting.
Another difference between [27] and our work is that we do not require any
specific regularity or integrability conditions for caloric functions, except
for assuming nonnegativity and finiteness of integrals in the mean value
property. Furthermore, in our representation, the boundary data may be a
measure; for example a Dirac delta represents represents a fixed parabolic
Martin kernel. Furthermore, in Theorem 6.5, we demonstrate that even without a
prescribed initial condition, $\displaystyle u(\varepsilon,\cdot)$ converges
to a measure on $\displaystyle D$ as $\displaystyle\varepsilon\to 0^{+}$. This
measure finitely integrates the function $\displaystyle
x\mapsto\mathbb{P}^{x}(\tau_{D}>1)$ on $\displaystyle D$ (see below), similar
to the condition used in [27].

In our development, we utilize some basic probabilistic potential theory; see,
e.g., Sato [57]. Let $\displaystyle X=(X_{t})_{t\geq 0}$ be the isotropic
$\displaystyle\alpha$-stable Lévy process in $\displaystyle\mathbb{R}^{d}$.
For $\displaystyle x\in\mathbb{R}^{d}$, we denote by
$\displaystyle\mathbb{P}^{x}$ and $\displaystyle\mathbb{E}^{x}$ the
probability and the expectation of the process starting from $\displaystyle
x$, and $\displaystyle\mathbb{P}:=\mathbb{P}^{0}$,
$\displaystyle\mathbb{E}:=\mathbb{E}^{0}$. We then consider

(1.3) $\tau_{D}:=\inf\\{s>0\ :\ X_{s}\notin D\\},$

the first exit time of the process $\displaystyle X$ from $\displaystyle D$,
and the survival probability:

$\displaystyle\mathbb{P}^{x}(\tau_{D}>t)=\int_{D}p_{t}^{D}(x,y)\,dy,$

where $\displaystyle p_{t}^{D}$ is the Dirichlet heat kernel of
$\displaystyle\Delta^{\alpha/2}$ in $\displaystyle D$ (for details see Section
2). Furthermore, let $\displaystyle G_{D}$ be the (elliptic) Green function of
$\displaystyle\Delta^{\alpha/2}$ in $\displaystyle D$. We fix arbitrary
$\displaystyle t_{0}\in(0,\infty)$ and $\displaystyle x_{0}\in D$, reference
time and point.

There are several reasonable ways to define the parabolic Martin kernel in
Lipschitz open sets. The general idea is to normalize $\displaystyle
p_{t}^{D}$ by constructing a ratio that converges to a nontrivial limit at the
boundary of $\displaystyle D$. Each of the following expressions will be
called a parabolic Martin kernel:

(1.4) $\displaystyle\displaystyle\eta_{t,Q}(x):=\lim_{D\ni y\to
Q}\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)},$ (1.5)
$\displaystyle\displaystyle\eta^{x_{0}}_{t,Q}(x):=\lim\limits_{D\ni y\to
Q}\frac{p_{t}^{D}(x,y)}{G_{D}(x_{0},y)},$ (1.6)
$\displaystyle\displaystyle\widetilde{\eta}_{t,Q}(x):=\lim\limits_{D\ni y\to
Q}\frac{p_{t}^{D}(x,y)}{p_{t_{0}}^{D}(x_{0},y)}.$

Here, $\displaystyle t>0$, $\displaystyle x\in D$, and $\displaystyle
Q\in\partial D$. We recall that the heat kernel plays the role of the Green
function for the heat equation, see, e.g., Doob [35], Watson [60], or Bogdan
and Hansen [20, Subsection 9.4]. This might indicate that
$\displaystyle\widetilde{\eta}$ is the canonical parabolic Martin kernel,
however $\displaystyle\eta$ and $\displaystyle\eta^{x_{0}}$ offer a more
explicit description of the boundary behavior of $\displaystyle p_{t}^{D}$ and
are more convenient to handle via the existing elliptic theory. If
$\displaystyle D$ is $\displaystyle C^{1,1}$, then one can also normalize
$\displaystyle p^{D}_{t}$ by using $\displaystyle\delta_{D}(y)^{\alpha/2}$
with

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

see Chen, Kim, and Song [29]; see also [27]. The next result may be considered
as a consequence and a follow-up of the approximate factorization (2.10) of
$\displaystyle p_{t}^{D}$ by Bogdan, Grzywny, and Ryznar [18].

######  Theorem 1.1.

Recall that $\displaystyle D\subset\mathbb{R}^{d}$ is open, bounded, and
Lipschitz with localization radius $\displaystyle r_{0}$, Lipschitz constant
$\displaystyle\lambda$, and reference point $\displaystyle x_{0}$ and time
$\displaystyle t_{0}$. Then, the limits in (1.4), (1.5), and (1.6) exist for
all $\displaystyle t>0$, $\displaystyle x\in D$, and $\displaystyle
Q\in\partial D$. Furthermore, they are finite, strictly positive, continuous
in $\displaystyle t$ and $\displaystyle x$, and

(1.7) $\displaystyle\displaystyle\eta_{1,Q}(x)$
$\displaystyle\displaystyle\approx\mathbb{P}^{x}(\tau_{D}>1),\quad x\in D,$
(1.8) $\displaystyle\displaystyle\eta_{t+s,Q}(x)$
$\displaystyle\displaystyle=\int_{D}\eta_{t,Q}(z)p_{s}^{D}(z,x)\,dz,\quad
0<s,t<\infty,\quad x\in D.$

The formula (1.7) is a sample of more general estimates for
$\displaystyle\eta$, which we give in Corollary 3.6 below. The proofs of
Theorem 1.1 and other results of this section are given later on. Here we note
that the mere existence of a Martin-type kernel is a deep boundary
regularity111Here and below, the term signals relative regularity, i.e.,
continuity or even Hölder continuity of ratios at the boundary. result. In the
elliptic setting, for $\displaystyle G_{D}$, it is usually proved using the
boundary Harnack principle. For solutions of parabolic equations like (1.2),
we may utilize the elliptic results after expressing the numerators and
denominators in (1.4), (1.5), and (1.6) as Green potentials. This is precisely
our approach—it was used before by Bogdan, Palmowski, and Wang [23] for
Lipschitz cones at the vertex. We further remark that an early version of
proof of Theorem 1.1 for (1.4) has appeared in the PhD thesis of the first-
named author [3].

To obtain the representation of nonnegative caloric functions, we refine
Theorem 1.1 to ensure a uniform rate of convergence in (1.4). To this end, we
extend the spatial domain of the functions in (1.4), (1.5), (1.6), by
additionally defining, for $\displaystyle t>0$, $\displaystyle x\in D$, and
$\displaystyle y\in D$,

(1.9)
$\eta^{x_{0}}_{t,y}(x):=\frac{p_{t}^{D}(x,y)}{G_{D}(x_{0},y)},\quad\eta_{t,y}(x):=\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)},\quad\widetilde{\eta}_{t,y}(x):=\frac{p_{t}^{D}(x,y)}{p_{t_{0}}^{D}(x_{0},y)}.$

######  Theorem 1.2.

Recall that $\displaystyle D\subset\mathbb{R}^{d}$ is open, bounded, and
Lipschitz with localization radius $\displaystyle r_{0}$, Lipschitz constant
$\displaystyle\lambda$, and reference point $\displaystyle x_{0}$ and time
$\displaystyle t_{0}$. Fix $\displaystyle r_{1}\in(0,\infty)$ and
$\displaystyle 0<T_{1}<T_{2}<\infty$. For $\displaystyle x\in D$ and
$\displaystyle t\in[T_{1},T_{2}]$, $\displaystyle\eta$,
$\displaystyle\eta^{x_{0}}$, and $\displaystyle\widetilde{\eta}$ are Hölder
continuous in $\displaystyle y$ on $\displaystyle\overline{D}$,
$\displaystyle\overline{D}$, and $\displaystyle\overline{D}\setminus
B(x_{0},r_{1})$, respectively. The Hölder exponents and constants depend only
on $\displaystyle d,\alpha,\underline{D},T_{1},T_{2}$ (for
$\displaystyle\eta^{x_{0}}$ also on $\displaystyle x_{0},r_{1}$; for
$\displaystyle\widetilde{\eta}$ also on $\displaystyle t_{0},x_{0}$).

Here and below, we say constants depend on $\displaystyle\underline{D}$ if
they depend only on $\displaystyle r_{0},\lambda$, and an upper bound for
$\displaystyle{\rm diam}(D)$. Theorem 1.2 yields the following boundary
regularity for the semigroup

$\displaystyle P_{t}^{D}f(x):=\int_{D}p_{t}^{D}(x,y)f(y)dy.$

######  Corollary 1.3.

Fix $\displaystyle r_{1}\in(0,\infty)$. Let $\displaystyle u_{0}\in L^{1}(D)$,
$\displaystyle 0<T_{1}<T_{2}<\infty$, and $\displaystyle t\in[T_{1},T_{2}]$.
Then, the functions

$\displaystyle\displaystyle\frac{P_{t}^{D}u_{0}(y)}{G_{D}(x_{0},y)},\quad\frac{P_{t}^{D}u_{0}(y)}{\mathbb{P}^{y}(\tau_{D}>1)},\quad\frac{P_{t}^{D}u_{0}(y)}{p_{t_{0}}^{D}(x_{0},y)}$

are Hölder continuous in $\displaystyle y$ on
$\displaystyle\overline{D}\setminus B(x_{0},r_{1})$,
$\displaystyle\overline{D}$, and $\displaystyle\overline{D}$ respectively. The
Hölder exponents and constants depend only on $\displaystyle
d,\alpha,\underline{D},T_{1},T_{2}$ (and $\displaystyle t_{0},x_{0}$,
$\displaystyle r_{1}$, where relevant).

Theorem 1.2 and Corollary 1.3 can be viewed as analogues of the boundary
regularity result for $\displaystyle C^{1,1}$ open sets by Fernández-Real and
Ros-Oton [38, Theorem 1.1 (b)], see also [39]. However, such regularity
results for nonlocal equations are quite scarce for Lipschitz and less regular
domains. That is, much is known about harmonic functions [11, 22, 47], but the
first result for the Poisson equation ($\displaystyle\Delta^{\alpha/2}u=-f$)
appeared only recently in the work of Borthagaray and Nochetto [26], who
proved optimal Besov regularity of solutions. For regularity results in
$\displaystyle C^{1,\gamma}$ domains with $\displaystyle\gamma\in(0,1)$, see,
e.g., Abels and Grubb [2] or Dong and Ryu [34] and the references therein.

Incidentally, our proof of Theorem 1.2 unveils the following integral estimate
for the Green function.

######  Theorem 1.4.

Recall that $\displaystyle D\subset\mathbb{R}^{d}$ is open, bounded, and
Lipschitz with localization radius $\displaystyle r_{0}$, Lipschitz constant
$\displaystyle\lambda$, and reference point $\displaystyle x_{0}$. Let
$\displaystyle r>0$. There exists $\displaystyle
p_{0}=p_{0}(d,\alpha,\underline{D},r)>1$ and constants $\displaystyle
C\in(0,\infty)$ and $\displaystyle\sigma\in(0,1]$ depending only on
$\displaystyle d,\alpha,\underline{D},p,r$, such that for all $\displaystyle
p\in[1,p_{0})$,

$\displaystyle\displaystyle\bigg{\|}\frac{G_{D}(y,\cdot)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},\cdot)}{G_{D}(x_{0},y^{\prime})}\bigg{\|}_{L^{p}(D)}\leq
C|y-y^{\prime}|^{\sigma},\quad y,y^{\prime}\in\overline{D}\setminus
B(x_{0},r).$

Recall that Green potentials $\displaystyle
v(x)=G_{D}f(x):=\int_{D}G_{D}(x,y)f(y)dy$ solve the Dirichlet problem for the
Poisson equation:

$\displaystyle\displaystyle\begin{cases}(-\Delta)^{\alpha/2}v(x)=f(x),\quad&x\in
D,\\\ \hskip 41.0ptv(x)=0,\quad&x\in D^{c},\end{cases}$

see [14]. Theorem 1.4 yields a boundary, or relative, Hölder estimate, as
follows.

######  Corollary 1.5.

Let $\displaystyle p>p_{0}/(p_{0}-1)$ and let $\displaystyle f\in L^{p}(D)$.
Then, $\displaystyle G_{D}f(y)/G_{D}(x_{0},y)$ is Hölder continuous in
$\displaystyle\in D\setminus B(x_{0},r)$ with Hölder constant and exponent
depending only on $\displaystyle d,\alpha,\underline{D},p,r$ and
$\displaystyle\|f\|_{L^{p}(D)}$.

A similar result for $\displaystyle C^{1,1}$ domains was obtained by Ros-Oton
and Serra [55] with explicit and sharp Hölder exponents. Our regularity
results are far from being sharp in terms of $\displaystyle p_{0}$ and
$\displaystyle\sigma$, but this is to be expected for Lipschitz sets—some
insight about precise boundary behavior can be gained from the results on
cones [5, 32, 53] or numerical considerations [37], but we do not pursue this
point here.

Let us add a few general comments. The mean-value property for fractional
caloric functions is important for our development. It was considered before,
e.g., by Chen and Kumagai [30]. Here we focus on the mean-value property in
cylinders, which seems adequate for the initial-exterior problem (1.2). The
advantage of the approach is that from the Ikeda–Watanabe formula we obtain a
semi-explicit formula for the Poisson kernel. We also have the following
stochastic interpretation: if $\displaystyle u$ satisfies the mean-value
property $\displaystyle(0,T)\times D$, then $\displaystyle u(t,x)$ can be
recovered from the space-time isotropic $\displaystyle\alpha$-stable process
$\displaystyle s\mapsto(t-s,X_{s}+x)$, which starts from $\displaystyle(t,x)$
at time $\displaystyle s=0$, by computing the expectation of $\displaystyle
u(t-s,X_{s}+x)$ at the place of the first exit of the process from
$\displaystyle(0,T)\times D$. The exit can occur when $\displaystyle x+X_{s}$
leaves $\displaystyle D$ before time $\displaystyle t$—in which case the
exterior conditions affect the expectation—or when the time coordinate
$\displaystyle t-s$ reaches $\displaystyle 0$—then the initial condition comes
into play. Singular caloric functions start to appear once we assume that the
mean-value property is satisfied only on $\displaystyle(0,T)\times U$ for all
open (relatively compact sets) $\displaystyle U\subset\subset D$. We refer to
the book of Freidlin [40, Theorem 2.3] for a counterpart of this theory for
local operators.

With a view toward applications in probability, we note that the existence of
the limit (1.4) indicates how the isotropic $\displaystyle\alpha$-stable
process in $\displaystyle D$, conditioned on surviving at least time 1,
behaves near the boundary of $\displaystyle D$. More precisely, it implies the
existence of a “Yaglom limit”, see Theorem 3.7 below. Thanks to (1.8),
$\displaystyle\eta_{t,Q}(y)$ may be understood as the entrance law for the
killed process from $\displaystyle Q$ into $\displaystyle D$, see Blumenthal
[8]. This was used in [44, 51] to describe the behavior of the process started
from a point on the boundary, e.g., the apex of a cone. Furthermore, the
boundary behavior of the heat kernel yields a measure which represents the
probability distribution of a rescaled process conditioned on non-extinction.

Let us now present an outline of the proofs and methods in this paper. In
order to prove Theorem 1.1 we obtain an explicit representation of the
survival probability as a Green potential and we show that it behaves like
$\displaystyle G_{D}(x_{0},\cdot)$ at the boundary. Then we approximate
$\displaystyle p_{t}^{D}$ by Green potentials and obtain the limit in (1.4)
with the help of Prokhorov theorem. To this end, we utilize the uniform
integrability of ratios of Green functions. The proof of Theorem 1.4 consists
in splitting the integral into one region where the boundary Harnack principle
can be applied, and another region where we use a technical interior
regularity argument adapted to possible singularities of the Green function.
In order to prove Theorem 1.2, we represent $\displaystyle p_{t}^{D}$ as a
Green potential and we apply Theorem 1.4. We make use of the spectral theory
to show that $\displaystyle p_{t}^{D}$ has regularity necessary for the proof;
some ideas here were inspired by [27]. The boundary measure in the
representation of singular caloric functions is obtained from an approximating
sequence constructed via the so-called lateral Poisson kernel. Our
construction is quite different than the one in [27], in particular it does
not use the inhomogeneous fractional heat equation.

The structure of the rest of the paper is as follows. Section 2 contains basic
definitions and facts. In Section 3, we prove Theorem 1.1 and its
consequences. In Section 4, we prove Theorems 1.4 and 1.2. In Section 5, we
introduce the caloric functions and the parabolic Poisson kernel and study
their properties. Then in Section 6, we discuss the representation of
nonnegative parabolic functions in Lipschitz cylinders.

##  2\. Preliminaries

We assume throughout that the considered sets, measures, and functions are
Borel. For nonnegative functions $\displaystyle f$ and $\displaystyle g$, we
write $\displaystyle f(x)\lesssim g(x)$, $\displaystyle x\in A$, if there is a
number $\displaystyle C\in(0,\infty)$, referred to as constant, such that
$\displaystyle f(x)\leq Cg(x)$, $\displaystyle x\in A$. We write
$\displaystyle C=C(d,\alpha,\ldots)$ if $\displaystyle C$ is a constant
depending only on $\displaystyle d,\alpha,\ldots$, that is, $\displaystyle C$
may be considered as a function of the parameters $\displaystyle
d,\alpha,\ldots$, but not of $\displaystyle x\in A$. We say that
$\displaystyle f$ and $\displaystyle g$ are comparable and write
$\displaystyle f\approx g$ if $\displaystyle f\lesssim g$ and $\displaystyle
g\lesssim f$ (this notation was used in Section 1). We often use
$\displaystyle:=$ and occasionally employ cursive for definitions.

###  2.1. Geometry

Let $\displaystyle B(x,r):=\\{y\in\mathbb{R}^{d}:|y-x|<r\\}$. Recall that
$\displaystyle D$ is a Lipschitz open set with constant
$\displaystyle\lambda\in(0,\infty)$ and localization radius $\displaystyle
r_{0}\in(0,\infty)$. This means that for every $\displaystyle Q\in\partial D$
there is a rigid motion $\displaystyle R_{Q}$ and a Lipschitz function
$\displaystyle f_{Q}\colon\mathbb{R}^{d-1}\to\mathbb{R}$ with Lipschitz
constant $\displaystyle\lambda$, such that $\displaystyle R_{Q}(Q)=0$ and
$\displaystyle D\cap
B(Q,r_{0})=R_{Q}^{-1}(B(0,r_{0})\cap\\{y_{d}>f_{Q}(y_{1},\ldots,y_{d-1})\\})$.
For $\displaystyle r>0$, we let

(2.1) $\displaystyle\displaystyle D_{r}:=\\{x\in D:\delta_{D}(x)>1/r\\}.$

Let $\displaystyle\kappa=1/(4\sqrt{1+\lambda^{2}})$. Of course,
$\displaystyle\kappa<1$. For $\displaystyle y\in\overline{D}$ and
$\displaystyle r>0$, we define

(2.2) $\displaystyle\displaystyle\mathcal{A}_{r}(y):=\begin{cases}\\{A\in
D:B(A,\kappa r)\subseteq D\cap B(y,r)\\},\quad&r\leq r_{0}/2,\\\
\\{x_{0}\\},\quad&r>r_{0}/2.\end{cases}$

######  Lemma 2.1.

If $\displaystyle D$ is Lipschitz, then $\displaystyle\mathcal{A}_{r}(y)$ is
nonempty for every $\displaystyle r>0$ and $\displaystyle y\in\overline{D}$.

###### Proof.

Obviously, it suffices to consider $\displaystyle r\leq r_{0}/2$. For
$\displaystyle y\in\partial D$ the statement is true even with
$\displaystyle\kappa$ replaced by $\displaystyle
2\kappa=1/(2\sqrt{1+\lambda^{2}})$. Indeed, if we consider the interior right-
circular cone with angle $\displaystyle{\rm arccot}(\lambda)$ and vertex at
$\displaystyle y$, then the point $\displaystyle A\in D$ on the axis of the
cone such that $\displaystyle|A-y|=r$ satisfies $\displaystyle
B(A,r/(2\sqrt{1+\lambda^{2}}))\subseteq D\cap B(y,r)$. If $\displaystyle y\in
D$ and $\displaystyle y\notin\mathcal{A}_{r}(y)$, then there is $\displaystyle
Q\in\partial D$ with $\displaystyle|y-Q|=\delta_{D}(y)<\kappa r$ and
$\displaystyle A\in D$ with

$\displaystyle\displaystyle B(A,r/(4\sqrt{1+\lambda^{2}}))\subseteq D\cap
B(Q,r/2)\subseteq D\cap B(y,r).$

∎

Thus, by definition (see, e.g., [18]), $\displaystyle D$ is
$\displaystyle\kappa$-fat at each scale $\displaystyle r\in(0,r_{0}/2)$. We
will denote by $\displaystyle A_{r}(y)$ an arbitrary point in
$\displaystyle\mathcal{A}_{r}(y)$. The actual choice is unimportant in the
sense that if $\displaystyle A_{1},A_{2}\in\mathcal{A}_{r}(y)$ and
$\displaystyle u\geq 0$ is harmonic in $\displaystyle B(A_{1},\kappa r)$ and
$\displaystyle B(A_{2},\kappa r)$—see Definition 2.4 below—then we have the
comparability $\displaystyle C^{-1}u(A_{1})\leq u(A_{2})\leq Cu(A_{1})$, where
$\displaystyle C=C(d,\alpha)$; see the Harnack inequality in [13, Lemma 1],
see also [14, Lemma 4.4].

For $\displaystyle x,y\in D$, let $\displaystyle
r_{x,y}:=|x-y|\vee\delta_{D}(x)\vee\delta_{D}(y)$. Let
$\displaystyle\mathcal{A}_{x,y}:=\\{x_{0}\\}$ if $\displaystyle
r_{x,y}>r_{0}/32$, and otherwise let

$\displaystyle\displaystyle\mathcal{A}_{x,y}:=\\{A\in D:B(A,\kappa
r_{x,y})\subset D\cap B(x,3r_{x,y})\cap B(y,3r_{x,y})\\}.$

Then, $\displaystyle\mathcal{A}_{x,y}$ is nonempty, [47]. We denote by
$\displaystyle A_{x,y}$ any point in $\displaystyle\mathcal{A}_{x,y}$. The
actual choice is unimportant in the sense that under suitable assumptions on
functions $\displaystyle u\geq 0$, there exists $\displaystyle
C=C(d,\alpha,\underline{D})$ such that for all $\displaystyle
A_{1},A_{2}\in\mathcal{A}_{x,y}$, $\displaystyle C^{-1}u(A_{1})\leq
u(A_{2})\leq Cu(A_{1})$. See Remark 2.2, following (2.14).

###  2.2. Potential theory

As stated in the introduction, we denote by
$\displaystyle(X_{t},\mathbb{P}^{x})$ the standard rotation invariant
$\displaystyle\alpha$-stable Lévy process in $\displaystyle\mathbb{R}^{d}$.
The process is determined by the jump measure with density function

(2.3)
$\nu(y)=\frac{2^{\alpha}\Gamma((d+\alpha)/2)}{\pi^{d/2}|\Gamma(-\alpha/2)|}|y|^{-d-\alpha}=:c_{d,\alpha}|y|^{-d-\alpha},\quad
y\in\mathbb{R}^{d}.$

It is a process with independent and stationary increments and characteristic
function
$\displaystyle\mathbb{E}^{x}e^{i\langle\xi,X_{t}-x\rangle}=e^{-t|\xi|^{\alpha}}$,
$\displaystyle t>0$, $\displaystyle x,\xi\in\mathbb{R}^{d}$. It is strong
Markov with the following time-homogeneous transition probability

$P_{t}(x,A):=\int_{A}p_{t}(x,y)\,dy,\quad t>0,\ x\in\mathbb{R}^{d},\
A\subseteq\mathbb{R}^{d}.$

Here $\displaystyle p_{t}(x,y):=p_{t}(x-y)$ and $\displaystyle p_{t}$ is the
smooth real-valued function on $\displaystyle\mathbb{R}^{d}$ with the Fourier
transform:

(2.4) $\int_{\mathbb{R}^{d}}p_{t}(x)e^{i\langle
x,\xi\rangle}\,dx=e^{-t|\xi|^{\alpha}},\quad\xi\in\mathbb{R}^{d}.$

The associated semigroup of operators acts on, e.g., $\displaystyle u\in
C_{0}(\mathbb{R}^{d})$ as follows:

$P_{t}u(x):=\int_{\mathbb{R}^{d}}u(y)p_{t}(x,y)\,dy,\quad x\in\mathbb{R}^{d},\
t\geq 0.$

We have the following scaling property as a consequence of (2.4):

(2.5) $p_{t}(x)=t^{-d/\alpha}p_{1}(t^{-1/\alpha}x),\quad x\in\mathbb{R}^{d},\
t>0.$

Furthermore, there exists a constant $\displaystyle c$ such that

(2.6) $c^{-1}\bigg{(}t^{-d/\alpha}\wedge\frac{t}{|x|^{d+\alpha}}\bigg{)}\leq
p_{t}(x)\leq
c\bigg{(}t^{-d/\alpha}\wedge\frac{t}{|x|^{d+\alpha}}\bigg{)},\quad
x\in\mathbb{R}^{d},\ t>0,$

see, e.g., [10, 25]. Thus, in short,

(2.7) $p_{t}(x)\approx t^{-d/\alpha}\wedge\frac{t}{|x|^{d+\alpha}},\quad
x\in\mathbb{R}^{d},\ t>0.$

Recall that $\displaystyle\tau_{D}$ is the first exit time from $\displaystyle
D$ defined in (1.3). If $\displaystyle D$ is bounded, then
$\displaystyle\tau_{D}<\infty$ almost surely, see, e.g., Pruitt [54]. The
Dirichlet heat kernel $\displaystyle p_{t}^{D}(x,y)$ of $\displaystyle D$ is
defined by the Hunt’s formula:

(2.8)
$p_{t}^{D}(x,y)=p_{t}(x,y)-\mathbb{E}^{x}\big{[}p_{t-\tau_{D}}(X_{\tau_{D}},y)\,;\,\tau_{D}<t\big{]},$

where $\displaystyle x,y\in\mathbb{R}^{d}$ and $\displaystyle t>0$. Here, as
usual,

$\mathbb{E}^{x}\big{[}p_{t-\tau_{D}}(X_{\tau_{D}},y)\,;\,\tau_{D}<t\big{]}:=\int_{\\{\tau_{D}<t\\}}p_{t-\tau_{D}}(X_{\tau_{D}},y)\,d\mathbb{P}^{x}.$

Since $\displaystyle D$ is Lipschitz, it satisfies the exterior cone
condition. Therefore, $\displaystyle\mathbb{P}^{x}(\tau_{D}=0)=1$ for all
$\displaystyle x\in D^{c}$ by Blumenthal’s zero-one law. In particular
$\displaystyle p_{t}^{D}(x,y)=0$ when $\displaystyle x$ or $\displaystyle y$
are outside of $\displaystyle D$. For bounded or nonnegative functions
$\displaystyle f$ we define

$P_{t}^{D}f(x):=\mathbb{E}^{x}\big{[}f(X_{t})\,;\,\tau_{D}>t\big{]}=\int_{\mathbb{R}^{d}}f(y)p_{t}^{D}(x,y)\,dy,$

see [31, Section 2]. We also note that

$0\leq p_{t}^{D}(x,y)=p_{t}^{D}(y,x)\leq p_{t}(y-x)$

and $\displaystyle p_{t}$ satisfies the Chapman–Kolmogorov equations:

$\int p_{s}^{D}(x,y)p_{t}^{D}(y,z)\,dy=p_{t+s}^{D}(x,z),\quad s,t>0,\
x,z\in\mathbb{R}^{d},$

see [16, 29]. The following scaling property follows from (2.5),

(2.9)
$p_{t}^{D}(x,y)=t^{-d/\alpha}p_{1}^{t^{-1/\alpha}D}(t^{-1/\alpha}x,t^{-1/\alpha}y),\quad
x,y\in\mathbb{R}^{d},\ t>0.$

By [18, Theorem 1], for every $\displaystyle T>0$ we have the approximate
factorization:

(2.10)
$p_{t}^{D}(x,y)\approx\mathbb{P}^{x}(\tau_{D}>t)p_{t}(x,y)\mathbb{P}^{y}(\tau_{D}>t),\quad
x,y\in D,\ t\in(0,T).$

If $\displaystyle D$ is (open, bounded, and) $\displaystyle C^{1,1}$, then the
estimate takes on a more explicit form [29]:

(2.11)
$p_{t}^{D}(x,y)\approx\bigg{(}1\wedge\frac{\delta_{D}(x)^{\alpha/2}}{\sqrt{t}}\bigg{)}p_{t}(x,y)\bigg{(}1\wedge\frac{\delta_{D}(y)^{\alpha/2}}{\sqrt{t}}\bigg{)},\quad
x,y\in D,\ t\in(0,T).$

We also recall the large time estimates. Let
$\displaystyle\lambda_{1}=\lambda_{1}(D)>0$ be the first eigenvalue and
$\displaystyle\varphi_{1}$ the first eigenfunction of the Dirichlet fractional
Laplacian on $\displaystyle D$, see Section 2.3 below for more details. By the
intrinsic ultracontractivity due to Kulczycki [48], for every $\displaystyle
T>0$ we have

(2.12) $p_{t}^{D}(x,y)\approx
e^{-\lambda_{1}t}\varphi_{1}(x)\varphi_{1}(y),\quad x,y\in D,\
t\in(T,\infty).$

If $\displaystyle D$ is (open, bounded, and) $\displaystyle C^{1,1}$, then we
even have

(2.13) $p_{t}^{D}(x,y)\approx
e^{-\lambda_{1}t}\delta_{D}(x)^{\alpha/2}\delta_{D}(y)^{\alpha/2},\quad x,y\in
D,\ t\in(T,\infty),$

see [29, Theorem 1.1 (ii)]. We define the killing intensity of $\displaystyle
X$ on $\displaystyle D$ as

$\kappa_{D}(z):=\int_{D^{c}}\nu(z-y)\,dy,\quad z\in D.$

By [57, Theorem 31.5], $\displaystyle\Delta^{\alpha/2}$ coincides with the
generator of $\displaystyle X_{t}$ for the class $\displaystyle
C^{2}_{c}(\mathbb{R}^{d})$ of real-valued twice continuously differentiable
functions with compact support in $\displaystyle\mathbb{R}^{d}$.

The Green function of $\displaystyle D$ is given by the formula:

$G_{D}(x,y):=\int_{0}^{\infty}p_{t}^{D}(x,y)\,dt,\quad x,y\in\mathbb{R}^{d}.$

In particular, $\displaystyle G_{D}(x,y)=0$ if either $\displaystyle x\in
D^{c}$ or $\displaystyle y\in D^{c}$. We note that $\displaystyle G_{D}$ is
finite for all $\displaystyle x\neq y$ and by (2.8), $\displaystyle
G_{D}(x,y)\leq G_{\mathbb{R}^{d}}(x,y)=c|x-y|^{\alpha-d}$. For further
reference, we recall the Green function estimates of Jakubowski [47, Theorem
1]: If we let

$\displaystyle\Phi(x):=G_{D}(x_{0},x)\wedge 1,$

then there exists $\displaystyle C(d,\alpha,\underline{D})>0$ such that

(2.14) $\displaystyle\displaystyle
C^{-1}|x-y|^{\alpha-d}\frac{\Phi(x)\Phi(y)}{\Phi(A_{x,y})^{2}}\leq
G_{D}(x,y)\leq
C|x-y|^{\alpha-d}\frac{\Phi(x)\Phi(y)}{\Phi(A_{x,y})^{2}}\,,\qquad x,y\in D,$

see Subsection 2.1 for notation and the following remark.

######  Remark 2.2.

We note that if $\displaystyle A_{1},A_{2}\in\mathcal{A}_{x,y}$, then
$\displaystyle\Phi(A_{1})\approx\Phi(A_{2})$; see [47, Lemma 13]. We also note
that [47] uses an extra reference point $\displaystyle x_{1}$ to define
$\displaystyle A_{x,y}$ for $\displaystyle r_{x,y}\geq r_{0}/32$, but the
resulting values of $\displaystyle\Phi(A_{x,y})$ are trivially comparable in
both settings. In particular, (2.14) remains true in the present (simplified)
setting.

######  Remark 2.3.

It is implicit in (2.10) and (2.12) that
$\displaystyle\varphi_{1}(y)\approx\mathbb{P}^{y}(\tau_{D}>1)$, $\displaystyle
y\in D$. Furthermore, by [18, Theorem 2],
$\displaystyle\mathbb{P}^{y}(\tau_{D}>1)\approx\mathbb{E}^{y}\tau_{D}$,
$\displaystyle y\in D$, and, by [47, Lemma 17],
$\displaystyle\mathbb{E}^{y}\tau_{D}\approx\Phi(y)$, $\displaystyle y\in D$.
Therefore,

(2.15)
$\varphi_{1}(y)\approx\mathbb{P}^{y}(\tau_{D}>1)\approx\mathbb{E}^{y}\tau_{D}\approx\Phi(y),\quad
y\in D.$

In our proofs, we mostly use the survival probability and $\displaystyle\Phi$,
but we also refer to results stated in terms of $\displaystyle\varphi_{1}$ and
the expected exit time.

We define the Green operator (or Green potential)

$(G_{D}f)(x):=\int_{D}G_{D}(x,y)f(y)\,dy,\quad x\in\mathbb{R}^{d},$

for integrable or nonnegative functions $\displaystyle f$. For $\displaystyle
f\in L^{1}(D)$, the function $\displaystyle u:=G_{D}f$ is a distributional
solution of $\displaystyle(-\Delta)^{\alpha/2}u=f$ in $\displaystyle D$, see
[14, Proposition 3.13].

######  Definition 2.4.

Let $\displaystyle u\geq 0$ be a Borel measurable function on
$\displaystyle\mathbb{R}^{d}$.

  * •

We say that $\displaystyle u$ is $\displaystyle\alpha$-harmonic in an open set
$\displaystyle V\subseteq\mathbb{R}^{d}$ if for every open (relatively
compact) $\displaystyle B\subset\subset D$,

$\displaystyle u(x)=\mathbb{E}^{x}u(X_{\tau_{B}})<\infty,\quad x\in B.$

  * •

We say that $\displaystyle u$ is regular $\displaystyle\alpha$-harmonic in
$\displaystyle D\subset\mathbb{R}^{d}$ if

$\displaystyle u(x)=\mathbb{E}^{x}u(X_{\tau_{D}})<\infty,\quad x\in D.$

  * •

We say that $\displaystyle u$ is singular $\displaystyle\alpha$-harmonic in
$\displaystyle D\subset\mathbb{R}^{d}$, if $\displaystyle u$ is
$\displaystyle\alpha$-harmonic in $\displaystyle D$ and $\displaystyle u=0$ on
$\displaystyle D^{c}$.

We will often write ‘harmonic’ instead of ‘$\displaystyle\alpha$-harmonic’.
Since $\displaystyle\tau_{B}\leq\tau_{V}$ for $\displaystyle B\subset V$, by
the strong Markov property it follows that regular harmonic functions are
harmonic. Also by the strong Markov property, $\displaystyle G_{D}(\cdot,y)$
is harmonic in $\displaystyle D\setminus\\{y\\}$, see [31, Theorem 2.5] or
[49, (2.1)].

For $\displaystyle x\in\mathbb{R}^{d}$, the
$\displaystyle\mathbb{P}^{x}$-distribution of $\displaystyle X_{\tau_{D}}$ is
called the $\displaystyle\alpha$-harmonic measure, denoted by
$\displaystyle\omega^{x}_{D}$. This measure is concentrated on $\displaystyle
D^{c}$ and for $\displaystyle u$ regular harmonic in $\displaystyle D$, we
have

$u(x)=\int_{D^{c}}u(z)\,\omega_{D}^{x}(dz),\quad x\in D.$

The $\displaystyle\alpha$-harmonic measure of a Lipschitz open set is
absolutely continuous with respect to the Lebesgue measure. Its density
function is given by the Poisson kernel:

(2.16) $P_{D}(x,z):=\int_{D}G_{D}(x,y)\nu(y,z)\,dy,\quad x\in D,\,z\in D^{c},$

see [11, Lemma 6]. Therefore, for every regular harmonic $\displaystyle u$ we
have the representation

$u(x)=\int_{D^{c}}P_{D}(x,z)u(z)\,dz,\quad x\in D.$

We also recall the Ikeda–Watanabe formula from [46]:

(2.17) $\mathbb{P}^{x}\big{[}\tau_{D}\in I,X_{\tau_{D}-}\in A,X_{\tau_{D}}\in
B\big{]}=\int_{I}\int_{B}\int_{A}\nu(y,z)p_{u}^{D}(x,dy)\,dz\,du,$

where $\displaystyle I\subset(0,\infty)$, $\displaystyle A\subset D$, and
$\displaystyle B\subset(\overline{D})^{c}$. See also [6, Lemma 1], [11], [24,
(4.13)], or [59, Theorem 2.4].

Recall that $\displaystyle x_{0}\in D$ is an arbitrary but fixed (reference)
point. We define the Martin kernel, $\displaystyle M_{D}^{x_{0}}(y,Q)$ as
follows: for every $\displaystyle Q\in\partial D$ and $\displaystyle y\in D$
we let

(2.18) $M_{D}^{x_{0}}(y,Q)=\lim_{D\ni x\to
Q}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}.$

In [12, Lemma 6] it is shown that the Martin kernel exists, the mapping
$\displaystyle(y,Q)\mapsto M_{D}^{x_{0}}(y,Q)$ is continuous on $\displaystyle
D\times\partial D$, and for every $\displaystyle Q\in\partial D$ the function
$\displaystyle M_{D}^{x_{0}}(\cdot,Q)$ is singular
$\displaystyle\alpha$-harmonic in $\displaystyle D$.

###  2.3. Auxiliary results on $\displaystyle P_{t}^{D}$ and its spectral
decomposition

We recall that the operators $\displaystyle P_{t}^{D}$ are compact on
$\displaystyle L^{2}(D)$, see, e.g., [15, Chapter 4]. Therefore there exist a
nondecreasing sequence of nonnegative numbers $\displaystyle\lambda_{n}$
diverging to infinity and an orthonormal sequence of functions
$\displaystyle\varphi_{n}\in C_{0}(D)$ such that for every
$\displaystyle\phi\in L^{2}(D)$, we have

(2.19) $\displaystyle\displaystyle
P_{t}^{D}\phi(x)=\sum\limits_{n=1}^{\infty}e^{-\lambda_{n}t}\langle\phi,\varphi_{n}\rangle\varphi_{n}(x)$

and

(2.20) $\displaystyle\displaystyle
p_{t}^{D}(x,y)=\sum\limits_{n=1}^{\infty}e^{-\lambda_{n}t}\varphi_{n}(x)\varphi_{n}(y),\quad
x,y\in D,\;t>0.$

The fractional Weyl bounds [9, 41] read

(2.21) $\displaystyle\displaystyle\lambda_{n}\approx n^{\alpha/d}.$

Note that $\displaystyle P_{t}^{D}\varphi_{n}=e^{-\lambda_{n}t}\varphi_{n}$
pointwise. Therefore,

(2.22) $\displaystyle\displaystyle
G_{D}\varphi_{n}=\int_{0}^{\infty}P_{t}^{D}\varphi_{n}\,dt=\lambda_{n}^{-1}\varphi_{n}.$

By iterating (2.22) and using the regularity results for the fractional
Laplacian [56, 43], we find that $\displaystyle\varphi_{n}$ are smooth in
$\displaystyle D$. Furthermore, by [38, Proposition 3.1], there exist
$\displaystyle C>0$ and $\displaystyle w\geq 1$, such that

(2.23) $\displaystyle\displaystyle\|\varphi_{n}\|_{\infty}\leq
C\lambda_{n}^{w-1},\quad n\in\mathbb{N}.$

We say that $\displaystyle\phi$ belongs to $\displaystyle D(L^{D})$, the
domain of the $\displaystyle L^{2}$-generator of $\displaystyle P_{t}^{D}$, if
the following limit exists in $\displaystyle L^{2}$:

$\displaystyle\displaystyle L^{D}\phi:=\lim\limits_{t\to
0^{+}}\frac{P_{t}^{D}\phi-\phi}{t}.$

Furthermore, if the limit exists for a function $\displaystyle\phi$ and some
$\displaystyle x\in D$, we denote it as $\displaystyle L^{D}\phi(x)$.

######  Lemma 2.5.

  1. (1)

We have $\displaystyle\varphi_{n}\in D(L^{D})$ and $\displaystyle
L^{D}\varphi_{n}(x)=-\lambda_{n}\varphi_{n}(x)$ for all $\displaystyle x\in
D$.

  2. (2)

We have

$\displaystyle\displaystyle A:=\\{\phi\in
L^{2}(D):\sum\limits_{n=1}^{\infty}\lambda_{n}^{2}|\langle\phi,\varphi_{n}\rangle|^{2}<\infty\\}\subseteq
D(L^{D}),$

and for each $\displaystyle\phi\in A$,

$\displaystyle\displaystyle
L^{D}\phi=\sum\limits_{n=1}^{\infty}\lambda_{n}\langle\phi,\varphi_{n}\rangle\varphi_{n}.$

  3. (3)

For every $\displaystyle y\in D$ and $\displaystyle t>0$, $\displaystyle
p_{t}^{D}(\cdot,y)\in A$.

  4. (4)

For every $\displaystyle x,y\in D$, we have $\displaystyle
L^{D}_{x}p_{t}^{D}(x,y)=\Delta^{\alpha/2}_{x}p_{t}^{D}(x,y)$.

###### Proof.

Statements (1) and (2) follow quite easily from (2.19) and (2.21). In order to
prove (3), we first let $\displaystyle m\in\mathbb{N}$. Then, by (2.20) and
(2.23),

$\displaystyle\displaystyle|\langle
p_{t}^{D}(\cdot,y),\varphi_{m}\rangle|=|e^{-\lambda_{m}t}\varphi_{m}(x)|\leq
e^{-\lambda_{m}t}\|\varphi_{m}\|_{\infty}\leq
Ce^{-\lambda_{m}t}\lambda_{m}^{w-1}.$

Using (2.21), we get (3).

We now prove (4). Note that $\displaystyle x\mapsto p_{t}^{D}(x,y)\in
C^{2}(D)\cap C_{c}(\mathbb{R}^{d})$. Let $\displaystyle\phi\in
C^{2}_{c}(B(x,\delta_{D}(x)/2))$ (extended by 0 to the whole of
$\displaystyle\mathbb{R}^{d}$) and $\displaystyle g\in C_{c}(\mathbb{R}^{d})$
be such that $\displaystyle\phi(x)+g(x)=p_{t}^{D}(x,y)$ and $\displaystyle
g(x)=0$ on $\displaystyle B(x,\delta_{D}(x)/4)$. Note that by (2.7),

(2.24)
$\displaystyle\displaystyle\frac{p_{t}^{D}(x,z)}{t}\leq\frac{p_{t}(x,z)}{t}\lesssim\nu(x,z),$

which for $\displaystyle|x-z|>\delta_{D}(x)/4$ is uniformly bounded.
Furthermore, since $\displaystyle p_{t}(x,z)/t\to\nu(x,z)$ for all
$\displaystyle x,z\in\mathbb{R}^{d}$, $\displaystyle x\neq z$, by (2.8) we
find that for fixed $\displaystyle z\in D\setminus\\{x\\}$,

$\displaystyle\displaystyle\lim\limits_{t\to
0^{+}}\frac{p_{t}^{D}(x,z)}{t}=\nu(x,y)+\lim\limits_{t\to
0^{+}}\frac{1}{t}\mathbb{E}^{x}[p_{t-\tau_{D}}(X_{\tau_{D}},z)\,;\,\tau_{D}<t].$

Since $\displaystyle x$ and $\displaystyle z$ are fixed we have $\displaystyle
p_{t-\tau_{D}}(X_{\tau_{D}},z)\lesssim t$, so the limit on the right hand side
is equal to 0, hence $\displaystyle p_{t}^{D}(x,z)/t\to\nu(x,z)$ as well. By
this, (2.24), and the dominated convergence theorem, we get
$\displaystyle\Delta^{\alpha/2}g(x)=L^{D}g(x)$.

Let $\displaystyle L$ be the $\displaystyle C_{0}(\mathbb{R}^{d})$-generator
of the semigroup induced by $\displaystyle p_{t}$. By Sato [57, Theorem 31.5],
we have $\displaystyle\Delta^{\alpha/2}\phi(x)=L\phi(x)$. Therefore,

$\displaystyle\displaystyle
L^{D}\phi(x)=\Delta^{\alpha/2}\phi(x)+\lim\limits_{t\to
0^{+}}\frac{P_{t}^{D}\phi(x)-P_{t}\phi(x)}{t}.$

We will show that the last limit exists and is equal to 0. By (2.8),
Fubini–Tonelli, and the fact that $\displaystyle X_{\tau_{D}}\in D^{c}$ almost
surely,

$\displaystyle\displaystyle\frac{|P_{t}^{D}\phi(x)-P_{t}\phi(x)|}{t}\leq\|\phi\|_{\infty}\frac{1}{t}\mathbb{E}^{x}\bigg{[}\int_{B(x,\delta_{D}(x)/2)}p_{t-\tau_{D}}(X_{\tau_{D}},z)\,dz\,;\,\tau_{D}<t\bigg{]}\lesssim\mathbb{P}^{x}(\tau_{D}<t)\mathop{\longrightarrow}\limits^{t\to
0^{+}}0.$

By collecting the above results we find that

$\displaystyle\displaystyle\Delta^{\alpha/2}_{x}p_{t}^{D}(x,y)=\Delta^{\alpha/2}\phi(x)+\Delta^{\alpha/2}g(x)=L^{D}\phi(x)+L^{D}g(x)=L^{D}_{x}p_{t}^{D}(x,y),$

which ends the proof. ∎

######  Corollary 2.6.

For every $\displaystyle t>0$, $\displaystyle\Delta^{\alpha/2}_{x}p_{t}^{D}$
is bounded in $\displaystyle D\times D$.

###### Proof.

By Lemma 2.5 and (2.23), we have

$\displaystyle\displaystyle|\Delta^{\alpha/2}_{x}p_{t}^{D}(x,y)|=|L^{D}_{x}p_{t}^{D}(x,y)|=\bigg{|}\sum\limits_{n=1}^{\infty}\lambda_{n}e^{-\lambda_{n}t}\varphi_{n}(x)\varphi_{n}(y)\bigg{|}\leq\sum\limits_{n=1}^{\infty}C\lambda_{n}e^{-\lambda_{n}t}\lambda_{n}^{2w-2}\leq
C_{0}<\infty.$

∎

######  Lemma 2.7.

Let $\displaystyle\phi\in C_{c}^{\infty}(D)$. Then,

$\displaystyle\displaystyle
P_{t}^{D}L\phi(y)=\sum\limits_{n=1}^{\infty}e^{-\lambda_{n}t}\lambda_{n}\langle\phi,\varphi_{n}\rangle\varphi_{n}(y),\quad
y\in D.$

###### Proof.

Note that $\displaystyle L\phi\in L^{2}(D)$, hence

$\displaystyle\displaystyle
P_{t}^{D}L\phi(y)=\sum\limits_{n=1}^{\infty}e^{-\lambda_{n}t}\langle\varphi_{n},L\phi\rangle\varphi_{n}(y).$

By (2.22) we have $\displaystyle\varphi_{n}=G_{D}[\lambda_{n}\varphi_{n}]$.
Therefore, by [14, Proposition 3.13],

$\displaystyle\displaystyle\langle\varphi_{n},L\phi\rangle=\langle
G_{D}[\lambda_{n}\varphi_{n}],L\phi\rangle=\langle\lambda_{n}\varphi_{n},\phi\rangle,$

which ends the proof of the lemma. ∎

The following result is a weighted Hausdorff–Young type inequality.

######  Lemma 2.8.

There exist $\displaystyle c=c(d,\alpha,\underline{D})$ and $\displaystyle
w\in\mathbb{N}$ such that for any $\displaystyle p\in[2,\infty]$ and
$\displaystyle u\in L^{p}(D)$,

$\displaystyle\displaystyle\|u\|_{L^{p}(D)}\leq
c\bigg{(}\sum\limits_{n=1}^{\infty}|\langle
u,\varphi_{n}\rangle|^{p^{\prime}}\lambda_{n}^{w-1}\bigg{)}^{1/p^{\prime}},$

where $\displaystyle p^{\prime}=p/(p-1)$ is the Hölder conjugate exponent of
$\displaystyle p$.

###### Proof.

Let $\displaystyle\phi\in L^{2}(D)$. By (2.23), we have
$\displaystyle\|\varphi_{n}\|_{\infty}\leq C\lambda_{n}^{w-1}$ for some
$\displaystyle C>0$ and $\displaystyle w\geq 1$ independent of $\displaystyle
n$. Therefore for $\displaystyle x\in D$,

$\displaystyle\displaystyle\|\phi\|_{\infty}\leq\sum\limits_{n=1}^{\infty}|\langle\phi,\varphi_{n}\rangle|\|\varphi_{n}\|_{\infty}\leq
C\sum\limits_{n=1}^{\infty}|\langle\phi,\varphi_{n}\rangle|\lambda_{n}^{w-1}.$

If we let
$\displaystyle\hat{\phi}=(\langle\phi,\varphi_{1}\rangle,\langle\phi,\varphi_{2}\rangle,\ldots)$
and denote by $\displaystyle l^{p}_{\lambda}$ the space of sequences with the
$\displaystyle p$-th powers summable with the weight
$\displaystyle(\lambda_{1}^{w-1},\lambda_{2}^{w-1},\ldots)$, then the above
means that $\displaystyle\hat{\phi}\mapsto\phi$ is bounded from $\displaystyle
l^{1}_{\lambda}$ to $\displaystyle L^{\infty}(D)$. By Parseval’s identity,
this map is also bounded from $\displaystyle l^{2}$ to $\displaystyle
L^{2}(D)$, hence also from $\displaystyle l^{2}_{\lambda}$ to $\displaystyle
L^{2}(D)$. The statement of the lemma follows from the Riesz–Thorin theorem. ∎

##  3\. Yaglom limits in Lipschitz open sets

In this section we prove Theorem 1.1. We first establish the asymptotics of
Green potentials at the boundary points of $\displaystyle D$. This extends
what is already known about the asymptotics of Green potentials at the vertex
of cone [23, Lemma 3.5]; we also propose a different proof.

######  Lemma 3.1.

If $\displaystyle f$ is a measurable function bounded on $\displaystyle D$ and
$\displaystyle Q\in\partial D$, then

$\displaystyle\lim_{x\to
Q}\int_{D}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}f(y)\,dy=\int_{D}\lim_{x\to
Q}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}f(y)\,dy<\infty,\quad x\in D.$

###### Proof.

Fix two points $\displaystyle x_{1},x_{2}\in D$ and let

(3.1) $\rho=(\delta_{D}(x_{1})\wedge\delta_{D}(x_{2})\wedge|x_{1}-x_{2}|)/3,$

so that $\displaystyle B(x_{1},\rho),B(x_{2},\rho)\subset D$ and
$\displaystyle B(x_{1},\rho)\cap B(x_{2},\rho)=\emptyset$. We know that
$\displaystyle M_{D}^{x_{0}}(\cdot,Q)$ given by (2.18) is regular
$\displaystyle\alpha$-harmonic on $\displaystyle B(x_{1},\rho)$ and
$\displaystyle B(x_{2},\rho)$, and for $\displaystyle x$ sufficiently close to
$\displaystyle\partial D$ so is $\displaystyle G_{D}(x,\cdot)$. Therefore, for
$\displaystyle i=1,2$,

$\displaystyle\displaystyle\int_{B(x_{i},\rho)^{c}}\lim_{x\to
Q}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}\,\omega_{B(x_{i},\rho)}^{x_{i}}(dy)$
$\displaystyle\displaystyle=\int_{B(z_{i},\rho)^{c}}M_{D}^{x_{0}}(y,Q)\,\omega_{B(z_{i},\rho)}^{z_{i}}(dy)$
$\displaystyle\displaystyle=M_{D}^{x_{0}}(x_{i},Q)$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{G_{D}(x,x_{i})}{G_{D}(x,x_{0})}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{\int_{B(x_{i},\rho)^{c}}G_{D}(x,y)\,\omega_{B(x_{i},\rho)}^{x_{i}}(dy)}{G_{D}(x,x_{0})}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\int_{B(x_{i},\rho)^{c}}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}\,\omega_{B(x_{i},\rho)}^{x_{i}}(dy).$

The $\displaystyle\alpha$-harmonic measures
$\displaystyle\omega^{x_{i}}_{B(x_{i},\rho)}(dy)$ are absolutely continuous
and have radially decreasing densities $\displaystyle g_{i}$, see, e.g., [12].
Therefore there exists $\displaystyle C>0$ such that
$\displaystyle\omega^{x_{i}}_{B(x_{i},\rho)}(dy)=g_{i}(y)\,dy\geq C$ for
$\displaystyle y\in D\cap(B(x_{i},\rho)^{c})$. Let $\displaystyle
g=g_{1}+g_{2}$. Vitali’s theorem [58, Theorem 16.6 (i) and (iii)] yields the
following $\displaystyle L^{1}$ convergence:

$\displaystyle\lim\limits_{x\to
Q}\int_{D}\bigg{|}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}g(y)-M_{D}^{x_{0}}(y,Q)g(y)\bigg{|}\,dy=0.$

Since $\displaystyle|f|\lesssim C\lesssim g$, the result follows. ∎

We can also establish the following identity, an analogue of [23, (3.16)].

######  Lemma 3.2.

For $\displaystyle x\in\mathbb{R}^{d}$, we have

(3.2) $\mathbb{P}^{x}(\tau_{D}>1)=(G_{D}P_{1}^{D}\kappa_{D})(x).$

###### Proof.

Let $\displaystyle x\in D$. Since our set $\displaystyle D$ is Lipschitz, from
Lemma 6 and the proof of Lemma 17 in [11],

$\displaystyle\displaystyle\omega_{D}^{x}(\partial
D)=\mathbb{P}^{x}(X_{\tau_{D}}\in\partial D)$ $\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle\mathbb{P}^{x}(X_{\tau_{D}-}=X_{\tau_{D}})$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle\mathbb{P}^{x}(X_{\tau_{D}-}\in D)$
$\displaystyle\displaystyle=1.$

By the Ikeda–Watanabe formula (2.17) and the Chapman–Kolmogorov equations we
have

$\displaystyle\displaystyle\mathbb{P}^{x}(\tau_{D}>1)$
$\displaystyle\displaystyle=\mathbb{P}^{x}\big{[}\tau_{D}>1,\ X_{\tau_{D}-}\in
D,\ X_{\tau_{D}}\in D^{c}\big{]}$
$\displaystyle\displaystyle=\int_{1}^{\infty}\int_{D^{c}}\int_{D}p_{s}^{D}(x,z)\nu(z-w)\,dz\,dw\,ds$
$\displaystyle\displaystyle=\int_{\mathbb{R}^{d}}\int_{D^{c}}\int_{0}^{\infty}p_{t+1}^{D}(x,z)\nu(z-w)\,dt\,dw\,dz$
$\displaystyle\displaystyle=\int_{\mathbb{R}^{d}}\int_{D^{c}}\int_{0}^{\infty}\int_{D}p_{t}^{D}(x,y)p_{1}^{D}(y,z)\,dy\,\nu(z-w)\,dt\,dw\,dz$
$\displaystyle\displaystyle=\int_{D}\int_{0}^{\infty}p_{t}^{D}(x,y)\,dt\int_{\mathbb{R}^{d}}p_{1}^{D}(y,z)\int_{D^{c}}\nu(z-w)\,dw\,dz\,dy$
$\displaystyle\displaystyle=\int_{D}G_{D}(x,y)\int_{\mathbb{R}^{d}}p_{1}^{D}(y,z)\kappa_{D}(z)\,dz\,dy$
$\displaystyle\displaystyle=\int_{D}G_{D}(x,y)(P_{1}^{D}\kappa_{D})(y)\,dy$
$\displaystyle\displaystyle=(G_{D}P_{1}^{D}\kappa_{D})(x).$

For $\displaystyle x\in D^{c}$, both sides of (3.2) are equal to 0. This ends
the proof.

∎

We define

$\displaystyle
C_{1}:=\int_{D}\int_{D}M_{D}^{x_{0}}(y,Q)p_{1}^{D}(y,z)\kappa_{D}(z)\,dz\,dy.$

Combining the two lemmas above, we obtain the following result.

######  Lemma 3.3.

We have $\displaystyle 0<C_{1}<\infty$ and $\displaystyle\lim_{x\to
Q}\frac{\mathbb{P}^{x}(\tau_{D}>1)}{G_{D}(x,x_{0})}=C_{1}.$

###### Proof.

By Lemma 3.2,
$\displaystyle\mathbb{P}^{x}(\tau_{D}>1)=(G_{D}P_{1}^{D}\kappa_{D})(x)$. Note
that $\displaystyle(P_{1}^{D}\kappa_{D})(y)$ is bounded. Indeed, by (2.10),

$\displaystyle\displaystyle(P_{1}^{D}\kappa_{D})(y)$
$\displaystyle\displaystyle=\int_{D}p_{1}^{D}(y,z)\kappa_{D}(z)\,dz$ (3.3)
$\displaystyle\displaystyle\approx\mathbb{P}^{y}(\tau_{D}>1)\int_{D}\mathbb{P}^{z}(\tau_{D}>1)p_{1}(y,z)\kappa_{D}(z)\,dz,\quad
y\in D.$

Since $\displaystyle D$ is bounded, by (2.7)

(3.4) $p_{1}(y,z)\approx 1,\quad y,z\in D.$

Hence (3.3) becomes

(3.5)
$(P_{1}^{D}\kappa_{D})(y)\approx\mathbb{P}^{y}(\tau_{D}>1)\int_{D}\mathbb{P}^{z}(\tau_{D}>1)\kappa_{D}(z)\,dz,\quad
y\in D.$

Using (3.2), we see that for $\displaystyle x\in\mathbb{R}^{d}$,

$\int_{D}G_{D}(x,y)(P_{1}^{D}\kappa_{D})(y)\,dy=(G_{D}P_{1}^{D}\kappa_{D})(x)=\mathbb{P}^{x}(\tau_{D}>1)\leq
1.$

By (2.14), $\displaystyle G_{D}(x,y)$ is strictly positive for all
$\displaystyle x,y\in D$. Thus $\displaystyle P_{1}^{D}\kappa_{D}$ has to be
finite almost everywhere. Hence the integral in (3.5) is finite and

$\displaystyle(P_{1}^{D}\kappa_{D})(y)\approx\mathbb{P}^{y}(\tau_{D}>1),$

for $\displaystyle y\in D$. In particular,
$\displaystyle(P_{1}^{D}\kappa_{D})(y)$ is bounded on $\displaystyle D$. By
using Lemma 3.1 with $\displaystyle f(y)=(P_{1}^{D}\kappa_{D})(y)$,

$\displaystyle\displaystyle\lim_{x\to
Q}\frac{\mathbb{P}^{x}(\tau_{D}>1)}{G_{D}(x,x_{0})}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{(G_{D}P_{1}^{D}\kappa_{D})(x)}{G_{D}(x,x_{0})}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\int_{D}\frac{G_{D}(x,y)}{G_{D}(x,x_{0})}(P_{1}^{D}\kappa_{D})(y)\,dy$
$\displaystyle\displaystyle=\int_{D}M_{D}^{x_{0}}(y,Q)(P_{1}^{D}\kappa_{D})(y)\,dy=C_{1}<\infty.$

∎

We are now in a position to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let us define

(3.6)
$m_{x}(A):=\frac{\int_{A}p_{1}^{D}(x,y)dy}{\mathbb{P}^{x}(\tau_{D}>1)},\quad
x\in D,\ A\subseteq\mathbb{R}^{d}.$

First we note that the family $\displaystyle\\{m_{x}:x\in D\\}$ is tight.
Indeed, combining the factorization of $\displaystyle p_{1}^{D}(x,y)$ in
(2.10) with the equation (3.4), we get

(3.7)
$\frac{p_{1}^{D}(x,y)}{\mathbb{P}^{x}(\tau_{D}>1)}\approx\mathbb{P}^{y}(\tau_{D}>1),\qquad
x,y\in D.$

Since the densities of the measures $\displaystyle m_{x}(A)$ are bounded by an
integrable function, the tightness follows.

Next we wish to prove that the measures $\displaystyle m_{x}$ converge weakly
to a probability measure $\displaystyle m_{Q}$ on $\displaystyle D$ as
$\displaystyle x\to Q$. To this end, consider an arbitrary sequence
$\displaystyle\\{x_{n}\\}$ such that $\displaystyle x_{n}\to Q$. By tightness,
there exists a subsequence $\displaystyle\\{x_{n_{k}}\\}$ such that
$\displaystyle m_{x_{n_{k}}}\implies m_{Q}$ for some probability measure
$\displaystyle m_{Q}$, as $\displaystyle k\to\infty$. We will show that this
limit is unique.

Let $\displaystyle\phi\in C_{c}^{\infty}(D)$ and $\displaystyle
u_{\phi}=(-\Delta)^{\alpha/2}\phi$. For $\displaystyle x\in\mathbb{R}^{d}$, we
claim that

(3.8) $(P_{1}^{D}\phi)(x)=(G_{D}P_{1}^{D}u_{\phi})(x).$

To show this, we first remark that $\displaystyle u_{\phi}\in
C_{0}(\mathbb{R}^{d})$ and that $\displaystyle(G_{D}u_{\phi})(x)=\phi(x)$, see
[36, Lemma 5.7] and [22, (11)]. By (2.7) it follows that

$\displaystyle\displaystyle(P_{1}^{D}|u_{\phi}|)(x)=\int_{D}p_{1}^{D}(x,y)|u_{\phi}(y)|\,dy\leq
c<\infty.$

Therefore, since for a fixed $\displaystyle z\in D^{c}$ we have
$\displaystyle\nu(y,z)\gtrsim 1$ for $\displaystyle y\in D$, by (2.16) we get

$\displaystyle\displaystyle(G_{D}P_{1}^{D}|u_{\phi}|)(x)$
$\displaystyle\displaystyle=\int_{D}G_{D}(x,y)(P_{1}^{D}|u_{\phi}|)(y)\,dy$
$\displaystyle\displaystyle\leq c\int_{D}G_{D}(x,y)\,dy<\infty.$

As a result, we can apply Fubini–Tonelli theorem and establish (3.8) as
follows:

$\displaystyle\displaystyle(G_{D}P_{1}^{D}u_{\phi})(x)$
$\displaystyle\displaystyle=\int_{D}\int_{D}\int_{0}^{\infty}p_{t}^{D}(x,y)p_{1}^{D}(y,z)u_{\phi}(z)\,dt\,dz\,dy$
$\displaystyle\displaystyle=\int_{D}\int_{0}^{\infty}p_{t+1}^{D}(x,z)u_{\phi}(z)\,dt\,dz$
$\displaystyle\displaystyle=\int_{D}\int_{0}^{\infty}\int_{D}p_{1}^{D}(x,y)p_{t}^{D}(y,z)u_{\phi}(z)\,dy\,dt\,dz$
$\displaystyle\displaystyle=\int_{D}\int_{D}\int_{0}^{\infty}p_{1}^{D}(x,y)p_{t}^{D}(y,z)u_{\phi}(z)\,dt\,dz\,dy$
(3.9)
$\displaystyle\displaystyle=(P_{1}^{D}G_{D}u_{\phi})(x)=(P_{1}^{D}\phi)(x).$

Let us denote $\displaystyle m_{x}(\phi):=\int_{D}\phi(y)\,m_{x}(dy)$. Using
(3.8), Lemma 3.3, and Lemma 3.1, we get

$\displaystyle\displaystyle\lim_{x\to Q}m_{x}(\phi)$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{(P_{1}^{D}\phi)(x)}{\mathbb{P}^{x}(\tau_{D}>1)}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{(P_{1}^{D}G_{D}u_{\phi})(x)}{\mathbb{P}^{x}(\tau_{D}>1)}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{(G_{D}P_{1}^{D}u_{\phi})(x)}{G_{D}(x,x_{0})}\frac{G_{D}(x,x_{0})}{\mathbb{P}^{x}(\tau_{D}>1)}$
(3.10)
$\displaystyle\displaystyle=\frac{1}{C_{1}}\int_{D}M_{D}^{x_{0}}(y,Q)(P_{1}^{D}u_{\phi})(y)\,dy.$

In particular, $\displaystyle
m_{Q}(\phi):=\lim_{k\to\infty}m_{x_{n_{k}}}(\phi)$ does not depend on the
choice of the subsequence. Thus, by the Portmanteau Theorem, $\displaystyle
m_{x}\implies m_{Q}$ as $\displaystyle x\to Q$.

For $\displaystyle t>1$, we consider
$\displaystyle\phi_{t,y}(\cdot):=p_{t-1}^{D}(\cdot,y)\in
C_{0}(\mathbb{R}^{d})$, see [18] or [31, Proposition 1.19]. Using
Chapman–Kolmogorov, we get

$\displaystyle\displaystyle\eta_{t,Q}(y)$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{x}(\tau_{D}>1)}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{\int_{D}p_{t-1}^{D}(z,y)p_{1}^{D}(x,z)\,dz}{\mathbb{P}^{x}(\tau_{D}>1)}$
$\displaystyle\displaystyle=\lim_{x\to
Q}\frac{(P_{1}^{D}p_{t-1}^{D}(\cdot,y))(x)}{\mathbb{P}^{x}(\tau_{D}>1)}$
$\displaystyle\displaystyle=\lim_{x\to Q}m_{x}(p_{t-1}^{D}(\cdot,y)).$

By (3.10), the existence of $\displaystyle\eta_{t,Q}(y)$ for $\displaystyle
t>1$ follows:

$\eta_{t,Q}(y)=m_{Q}(p_{t-1}^{D}(\cdot,y)).$

Note that the threshold $\displaystyle t>1$ is arbitrary, that is,
$\displaystyle 1$ can be replaced with any $\displaystyle t_{0}>0$. Indeed,
the results of this section can be readily reformulated with $\displaystyle
t_{0}$ in place of $\displaystyle 1$, for instance, Lemma 3.3 may be
strengthened to assert that for every $\displaystyle t_{0}>0$,

$\displaystyle\lim\limits_{x\to
Q}\frac{\mathbb{P}^{x}(\tau_{D}>t_{0})}{G_{D}(x,x_{0})}=\int_{D}\int_{D}M_{D}^{x_{0}}(y,Q)p_{t_{0}}^{D}(y,z)\kappa_{D}(z)\,dz\,dy.$

Accordingly, we get the existence of the limit

(3.11) $\lim\limits_{x\to
Q}\frac{\mathbb{P}^{x}(\tau_{D}>1)}{\mathbb{P}^{x}(\tau_{D}>t_{0})}.$

We can also reuse the above arguments to get for all $\displaystyle t>t_{0}$,
the existence of

(3.12) $\lim\limits_{x\to
Q}\frac{p^{D}_{t}(x,y)}{\mathbb{P}^{x}(\tau_{D}>t_{0})}.$

Of course, (3.11) and (3.12) give the existence of
$\displaystyle\eta_{t,Q}(y)$ for $\displaystyle t>t_{0}$.

The equation (1.7) follows from equation (3.7), and the equation (1.8) follows
from the Chapman–Kolmogorov equations and the dominated convergence theorem
(see [18, (27)]):

$\eta_{t+s,Q}(y)=\lim_{x\to
Q}\int_{D}\frac{p_{t}^{D}(x,z)}{\mathbb{P}^{x}(\tau_{D}>1)}p_{s}^{D}(z,y)\,dz=\int_{D}\eta_{t,Q}(z)p_{s}^{D}(z,y)\,dz.$

The fact that $\displaystyle\widetilde{\eta}$ and $\displaystyle\eta^{x_{0}}$
exist follows from the existence of $\displaystyle\eta$ and from Lemma 3.3. ∎

######  Corollary 3.4.

The functions
$\displaystyle(t,y)\mapsto\eta_{t,Q}(y),\widetilde{\eta}_{t,Q}(y),\eta^{x_{0}}_{t,Q}(y)$
are continuous on $\displaystyle(0,\infty)\times\overline{D}$.

###### Proof.

By Theorem 1.1 and the fact that $\displaystyle p_{t}^{D}(x,y)$ and
$\displaystyle\mathbb{P}^{y}(\tau_{D}>1)$ are continuous for
$\displaystyle(t,y)\in(0,\infty)\times D$, and separated from 0 in
sufficiently small neighborhood of any point $\displaystyle(t,y)$, it suffices
to verify that for any sequence
$\displaystyle((t_{n},y_{n}))\subset(0,\infty)\times D$ such that
$\displaystyle(t_{n},y_{n})\to(t,Q)\in(0,\infty)\times\partial D$, we have

(3.13)
$\displaystyle\displaystyle\lim_{n\to\infty}\frac{p_{t_{n}}^{D}(x,y_{n})}{\mathbb{P}^{y_{n}}(\tau_{D}>1)}=\eta_{t,Q}(x).$

Furthermore, by Theorem 1.1, in order to obtain (3.13) it suffices to prove
that for any $\displaystyle t>0$ there exists a modulus of continuity
$\displaystyle\omega$ independent of $\displaystyle y$ such that

(3.14)
$\displaystyle\displaystyle\bigg{|}\frac{p_{t+\varepsilon}^{D}(x,y)-p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)}\bigg{|}\leq\omega(\varepsilon),\quad\varepsilon>0.$

By Chapman–Kolmogorov, we have

$\displaystyle\displaystyle\bigg{|}\frac{p_{t+\varepsilon}^{D}(x,y)-p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)}\bigg{|}$
$\displaystyle\displaystyle\leq\int_{D}\frac{|p_{t}^{D}(z,y)-p_{t}^{D}(x,y)|p_{\varepsilon}^{D}(x,z)}{\mathbb{P}^{y}(\tau_{D}>1)}\,dz$
$\displaystyle\displaystyle=\bigg{(}\int_{D\setminus
B(x,\delta_{D}(x)/2)}+\int_{B(x,\delta_{D}(x)/2)}\bigg{)}\frac{|p_{t}^{D}(z,y)-p_{t}^{D}(x,y)|p_{\varepsilon}^{D}(x,z)}{\mathbb{P}^{y}(\tau_{D}>1)}\,dz=:I_{1}+I_{2}.$

Then by (2.10),

$\displaystyle\displaystyle I_{1}\leq\int_{D\setminus
B(x,\delta_{D}(x)/2)}p_{\varepsilon}^{D}(x,z)\,dz\leq\int_{D\setminus
B(x,\delta_{D}(x)/2)}p_{\varepsilon}(x,z)\,dz\leq\omega(\epsilon).$

For $\displaystyle I_{2}$, we use the gradient bounds of Kulczycki and Ryznar
[50, Theorem 1.1] and (2.10):

$\displaystyle\displaystyle I_{2}$
$\displaystyle\displaystyle\leq\int_{B(x,\delta_{D}(x)/2)}\frac{|p_{t}^{D}(z,y)-p_{t}^{D}(x,y)|p_{\varepsilon}^{D}(x,z)}{\mathbb{P}^{y}(\tau_{D}>1)}\,dz$
$\displaystyle\displaystyle\leq\int_{B(x,\delta_{D}(x)/2)}|x-z|\frac{\|\nabla_{x}p_{t}^{D}(\cdot,y)\|_{L^{\infty}(B(x,\delta_{D}(x)/2)}}{\mathbb{P}^{y}(\tau_{D}>1)}p_{\varepsilon}^{D}(x,z)\,dz$
$\displaystyle\displaystyle\lesssim\int_{B(x,\delta_{D}(x)/2)}|x-z|\frac{\|p_{t}^{D}(\cdot,y)\|_{L^{\infty}(B(x,\delta_{D}(x)/2)}}{\mathbb{P}^{y}(\tau_{D}>1)}p_{\varepsilon}^{D}(x,z)\,dz$
$\displaystyle\displaystyle\lesssim\int_{B(x,\delta_{D}(x)/2)}|x-z|p_{\varepsilon}^{D}(x,z)\,dz\leq\int_{B(x,\delta_{D}(x)/2)}|x-z|p_{\varepsilon}(x,z)\,dz\leq\omega(\varepsilon).$

Thus, $\displaystyle I_{1}+I_{2}\leq\omega(\varepsilon)$, which ends the proof
for $\displaystyle\eta$. For $\displaystyle\widetilde{\eta}$ and
$\displaystyle\eta^{x_{0}}$, we use Lemma 3.2 and (1.7). ∎

Here is a rough result about the behavior of $\displaystyle\eta_{s,Q}(x)$ away
from the singularity at $\displaystyle(0,Q)$.

######  Lemma 3.5.

If $\displaystyle Q\in\partial D$ then
$\displaystyle(s,x)\mapsto\eta_{s,Q}(x)$ is locally bounded on
$\displaystyle((0,\infty)\times\mathbb{R}^{d})\setminus\\{(0,Q)\\}$.
Furthermore, if $\displaystyle t=0$ or $\displaystyle y\in\partial D$, but
$\displaystyle(t,y)\neq(0,Q)$, then $\displaystyle\eta_{s,Q}(x)\to 0$ as
$\displaystyle(s,x)\to(t,y)$.

###### Proof.

By (2.10) and (2.7), we have

$\displaystyle\displaystyle\eta_{s,Q}(x)$
$\displaystyle\displaystyle=\lim\limits_{D\ni\xi\to
Q}\frac{p_{s}^{D}(x,\xi)}{\mathbb{P}^{\xi}(\tau_{D}>1)}\lesssim\limsup\limits_{D\ni\xi\to
Q}\frac{\mathbb{P}^{\xi}(\tau_{D}>s)}{\mathbb{P}^{\xi}(\tau_{D}>1)}p_{s}(x,\xi)\mathbb{P}^{x}(\tau_{D}>s)$
$\displaystyle\displaystyle\lesssim|x-Q|^{-d-\alpha}\mathbb{P}^{x}(\tau_{D}>s)\limsup\limits_{D\ni\xi\to
Q}\frac{s\mathbb{P}^{\xi}(\tau_{D}>s)}{\mathbb{P}^{\xi}(\tau_{D}>1)}.$

If $\displaystyle|x-Q|\geq\varepsilon$ then $\displaystyle\eta_{s,Q}(x)$ is
bounded—it even converges to $\displaystyle 0$ as $\displaystyle s\to 0$—see
Lemma B.2. If $\displaystyle s>\varepsilon$ then we use the approximate
factorization of $\displaystyle p^{D}$—and the fact that
$\displaystyle\mathbb{P}^{x}(\tau_{D}>s)\to 0$ as $\displaystyle x\to
y\in\partial D$. ∎

Let us summarize estimates of $\displaystyle\eta$ that follow from the
estimates of the Dirichlet heat kernel.

######  Corollary 3.6.

If $\displaystyle D$ is $\displaystyle C^{1,1}$, then

(3.15)
$\displaystyle\displaystyle\eta_{t,Q}(x)\approx\begin{cases}\frac{1}{\sqrt{t}}\bigg{(}1\wedge\frac{\delta_{D}^{\alpha/2}(x)}{\sqrt{t}}\bigg{)}p_{t}(x,Q),\quad&t\in(0,1),\
x\in D,\ Q\in\partial D,\\\
e^{-\lambda_{1}t}\delta_{D}(x)^{\alpha/2},\quad&t\in[1,\infty),\ x\in D,\
Q\in\partial D.\end{cases}$

If $\displaystyle D$ is Lipschitz, then

(3.16) $\displaystyle\displaystyle\eta_{t,Q}(x)\approx
e^{-\lambda_{1}t}\mathbb{P}^{x}(\tau_{D}>t),\quad t\in[1,\infty),\ x\in D,\
Q\in\partial D,$

and

(3.17)
$\displaystyle\displaystyle\eta_{t,Q}(x)\approx\frac{\mathbb{P}^{x}(\tau_{D}>t)p_{t}(x,Q)}{\Phi(A_{t^{1/\alpha}}(Q))},\quad
t\in(0,1),\ x\in D,\ Q\in\partial D.$

Furthermore, there exist $\displaystyle 0<\sigma_{1}\leq\sigma_{2}<1$ such
that

(3.18) $\displaystyle\displaystyle
t^{-\sigma_{1}}\lesssim\frac{\eta_{t,Q}(x)}{\mathbb{P}^{x}(\tau_{D}>t)p_{t}(x,Q)}\lesssim
t^{-\sigma_{2}},\quad t\in(0,1),\ x\in D,\ Q\in\partial D.$

###### Proof.

The estimate (3.15) follows from (2.11) and (2.13). By [48, Theorem 1.1] and
(2.10), $\displaystyle\mathbb{P}^{y}(\tau_{D}>1)\approx\varphi_{1}(y)$, so
(3.16) is a consequence of (2.12). It remains to prove (3.17) and (3.18). By
(2.10),

(3.19)
$\displaystyle\displaystyle\eta_{t,Q}(x)\approx\mathbb{P}^{x}(\tau_{D}>t)p_{t}(x,Q)\lim\limits_{y\to
Q}\frac{\mathbb{P}^{y}(\tau_{D}>t)}{\mathbb{P}^{y}(\tau_{D}>1)}.$

By [18, Theorem 2] and (2.15),

(3.20)
$\displaystyle\displaystyle\frac{\mathbb{P}^{y}(\tau_{D}>t)}{\mathbb{P}^{y}(\tau_{D}>1)}\approx\frac{1}{\Phi(A_{t^{1/\alpha}}(y))}.$

By geometrical considerations, we can choose points $\displaystyle
A_{t^{1/\alpha}}(y)$ converging to a point in
$\displaystyle\mathcal{A}_{t^{1/\alpha}}(Q)$. This proves (3.17). By (3.19)
and Lemma B.2, we get the upper bound in (3.18). The lower bound follows from
(3.17) and [11, Lemma 3] with some $\displaystyle\sigma_{1}>0$. Of course, we
must have $\displaystyle\sigma_{1}\leq\sigma_{2}$ in (3.18). ∎

A consequence of Theorem 1.1 is the Yaglom-type limit, obtained in the thesis
of the first author [3].

######  Theorem 3.7.

Suppose that $\displaystyle D$ is a bounded Lipschitz open set such that
$\displaystyle 0\in\partial D$ and $\displaystyle D\cup\\{0\\}$ is star-shaped
at $\displaystyle 0$. If $\displaystyle x\in D$ then for every Borel
$\displaystyle A\subseteq\mathbb{R}^{d}$,

$\lim_{t\to\infty}\mathbb{P}^{x}\bigg{(}\frac{X_{t}}{t^{1/\alpha}}\in A\
\bigg{|}\ \bigg{(}\frac{X_{s}}{t^{1/\alpha}}\bigg{)}_{0\leq s\leq t}\subset
D\bigg{)}=m_{0}(A),$

where $\displaystyle\mathbb{P}^{x}(A_{1}|A_{2}):=\mathbb{P}^{x}(A_{1}\cap
A_{2})/\mathbb{P}^{x}(A_{2})$ is the conditional probability and
$\displaystyle m_{0}(A):=\int_{A}\eta_{1,{0}}(y)\,dy$.

###### Proof.

Let $\displaystyle x\in D$, $\displaystyle t\geq 1$, and let $\displaystyle
A\subset\mathbb{R}^{d}$ be Borel. Then we have

$\displaystyle\displaystyle\mathbb{P}^{x}\bigg{(}\frac{X_{t}}{t^{1/\alpha}}\in
A\ \bigg{|}\ \bigg{(}\frac{X_{s}}{t^{1/\alpha}}\bigg{)}_{0\leq s\leq t}\subset
D\bigg{)}$ $\displaystyle\displaystyle=\frac{\mathbb{P}^{x}(X_{t}\in
t^{1/\alpha}A,\ (X_{s})_{0\leq s\leq t}\subset
t^{1/\alpha}D)}{\mathbb{P}^{x}((X_{s})_{0\leq s\leq t}\subset t^{1/\alpha}D)}$
$\displaystyle\displaystyle=\frac{\int_{t^{1/\alpha}A}p_{t}^{t^{1/\alpha}D}(x,y)\,dy}{\int_{t^{1/\alpha}D}p_{t}^{t^{1/\alpha}D}(x,y)\,dy}$
$\displaystyle\displaystyle=\frac{\int_{t^{1/\alpha}A}t^{-d/\alpha}p_{1}^{D}(t^{-1/\alpha}x,t^{-1/\alpha}y)\,dy}{\int_{t^{1/\alpha}D}t^{-d/\alpha}p_{1}^{D}(t^{-1/\alpha}x,t^{-1/\alpha}y)\,dy}$
$\displaystyle\displaystyle=\frac{\int_{A}p_{1}^{D}(t^{-1/\alpha}x,y)\,dy}{\int_{D}p_{1}^{D}(t^{-1/\alpha}x,y)\,dy}=m_{t^{-1/\alpha}x}(A),$

where $\displaystyle m_{t^{-1/\alpha}x}$ is the measure defined in (3.6) above
(note that $\displaystyle t^{-1/\alpha}x\in D$). Therefore, by Theorem 1.1,
this probability approaches $\displaystyle m_{0}(A)$ as $\displaystyle
t\to\infty$. ∎

##  4\. Hölder regularity

This section is devoted to proving Theorems 1.4 and 1.2. The proof of Theorem
1.4 uses a mix of the boundary Harnack principle and interior Hölder
regularity. Then Theorem 1.2 follows by using the formulas of Section 3, which
enable us to relate the heat kernel regularity to the elliptic regularity.

Fix $\displaystyle n_{0}\geq 2$ such that the reference points $\displaystyle
x_{0}$ belongs to $\displaystyle D_{n_{0}/2}$.

######  Lemma 4.1.

There exists $\displaystyle p_{0}=p_{0}(d,\alpha,\underline{D})>1$ such that
the family $\displaystyle\\{(G_{D}(y,\cdot)/G_{D}(x_{0},y))^{p}:y\in D\\}$ is
uniformly integrable in $\displaystyle D$ for all $\displaystyle
p\in[1,p_{0})$.

###### Proof.

For $\displaystyle y\in D_{n_{0}}$ we have a crude bound:

$\displaystyle\displaystyle G_{D}(y,z)/G_{D}(x_{0},y)\leq
C(d,\alpha,\underline{D})|y-z|^{\alpha-d},\quad z\in D.$

Considering the functions on the right-hand side, we see that $\displaystyle
p_{0}=d/(d-\alpha)$ will do.

From now on assume that $\displaystyle y\in D\setminus D_{n_{0}}$. By (2.14),
there exists $\displaystyle C=C(d,\alpha,\underline{D})$ such that

$\displaystyle\displaystyle\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}\leq
C\frac{|y-z|^{\alpha-d}}{|x_{0}-y|^{\alpha-d}}\frac{\Phi(z)\Phi(A_{x_{0},y})^{2}}{\Phi(x_{0})\Phi(A_{y,z})^{2}}.$

We immediately get that

$\displaystyle\displaystyle\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}\leq
C^{\prime}|y-z|^{\alpha-d}\frac{\Phi(z)}{\Phi(A_{y,z})^{2}}.$

By the Carleson estimate [47, Lemma 13], we further find that
$\displaystyle\Phi(z)/\Phi(A_{y,z})\leq C(d,\alpha,\underline{D})$. If we let
$\displaystyle U=D_{32/r_{0}}\cup(D\setminus B(y,r_{0}/32))$, then it follows
that

$\displaystyle\displaystyle\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}\leq\begin{cases}C(d,\alpha,\underline{D})|y-z|^{\alpha-d},\quad&z\in
U,\\\ C(d,\alpha,\underline{D})|y-z|^{\alpha-d}\Phi(A_{y,z})^{-1},\quad&z\in
D\setminus U.\end{cases}$

The definition of $\displaystyle A_{y,z}$ implies that for $\displaystyle z\in
D\setminus U$ there exists $\displaystyle Q=Q(z)\in\partial D$ such that
$\displaystyle y,z\in B(Q,3r)$ and $\displaystyle B(A_{y,z},\kappa r)\subset
D\cap B(Q,6r)$. Using [11, Lemma 5], we find that there exist $\displaystyle
C=C(d,\alpha,\underline{D})$ and
$\displaystyle\gamma=\gamma(d,\alpha,\underline{D})\in(0,\alpha)$ such that

$\displaystyle\displaystyle\Phi(A_{y,z})\geq C|A_{y,z}-Q(z)|^{\gamma}\geq
C\kappa^{\gamma}r^{\gamma}\geq C\kappa^{\gamma}|y-z|^{\gamma}.$

Therefore,

(4.1) $\displaystyle\displaystyle\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}\leq
C(|y-z|^{\alpha-d}\vee|y-z|^{\alpha-\gamma-d}),\quad y\in D\setminus
D_{n_{0}},\ z\in D,$

so the statement of the lemma holds for all $\displaystyle
p\in[1,d/(d-\alpha+\gamma))$. We can take $\displaystyle
p_{0}=d/(d-\alpha+\gamma)$. ∎

The following lemma is a specific Carleson-type estimate.

######  Lemma 4.2.

Let $\displaystyle 0<r<\delta_{D}(y)$, $\displaystyle|z-y|\geq 2r$, and
$\displaystyle|v-y|\leq r$. There exists $\displaystyle
C=C(d,\alpha,\underline{D})$ such that

$\displaystyle\displaystyle G_{D}(z,v)\leq CG_{D}(z,y).$

###### Proof.

Note that $\displaystyle 2|z-v|\geq|z-y|$. By (2.14), there is $\displaystyle
c=c(d,\alpha,\underline{D})$ such that

$\displaystyle\displaystyle G_{D}(z,v)\leq
c\frac{\Phi(z)\Phi(v)}{\Phi(A_{z,v})^{2}}|z-v|^{\alpha-d}\leq
2^{d-\alpha}c\frac{\Phi(z)\Phi(v)}{\Phi(A_{z,v})^{2}}|z-y|^{\alpha-d}.$

By elementary calculations, we find that $\displaystyle 2r_{z,v}\geq r_{z,y}$.
By [47, Lemma 13] we therefore get $\displaystyle\Phi(A_{z,v})\geq
c(d,\alpha,\underline{D})\Phi(A_{z,y})$. Furthermore, by [11, Lemma 4 and 5]
we get $\displaystyle\Phi(v)\leq c(d,\alpha,\underline{D})\Phi(y)$. This ends
the proof. ∎

The next lemma can be viewed as a more concrete, quantified version of [22,
Lemma 8]. We give an interior-type Hölder regularity for ratios of Green
functions, taking into account the singularity at the diagonal. The structure
of the proof follows the boundary regularity approach of [11, Lemma 16], but
here the singularity can occur between the boundary and the arguments of the
function, see Figure 1.

$\displaystyle Q$$\displaystyle z$$\displaystyle y$$\displaystyle
y^{\prime}$$\displaystyle\partial D$$\displaystyle D\cap B(Q,3\delta_{D}(y))$
Figure 1. Illustration for Lemma 4.3. The boundary Harnack principle cannot be
used to estimate increments between $\displaystyle y$ and $\displaystyle
y^{\prime}$ because of the singularity at $\displaystyle z$. Instead we show
regularity in the smaller ball using harmonicity in the larger ball.

######  Lemma 4.3.

Let $\displaystyle y\in D$ and $\displaystyle Q\in\partial D$ satisfy
$\displaystyle|Q-y|=\delta_{D}(y)$. Assume that $\displaystyle z\in D\cap
B(Q,3\delta_{D}(y))$ and let $\displaystyle r=|z-y|/4$, so that
$\displaystyle\overline{B(y,r)}\subset D$. Then there exist constants
$\displaystyle C\geq 1$, $\displaystyle k_{0}\geq 4$,
$\displaystyle\sigma\in(0,1]$, and $\displaystyle\gamma\in(0,\alpha)$,
depending only on $\displaystyle d,\alpha,\underline{D}$, such that for every
$\displaystyle y^{\prime}\in B(y,2^{-k_{0}}r)$ we have

$\displaystyle\displaystyle\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}\leq
C\bigg{(}\frac{|y-y^{\prime}|}{r}\bigg{)}^{\sigma}r^{\alpha-d-\gamma}.$

###### Proof.

Note that $\displaystyle B(y,r)\subseteq D$. Let

$\displaystyle\displaystyle B_{k}=B(y,(2^{k_{0}})^{-k}r),\quad k=0,1,\ldots,$
$\displaystyle\displaystyle\Pi_{k}=B_{k}\setminus B_{k+1},\quad
k=0,1,\ldots,\quad\Pi_{-1}=D\setminus B_{0},$ $\displaystyle\displaystyle
u(y)=G_{D}(y,z),\quad v(y)=G_{D}(x_{0},y).$

We will show that there exist $\displaystyle c=c(d,\alpha,\underline{D})$ and
$\displaystyle\zeta=\zeta(d,\alpha,\underline{D})\in(0,1]$, such that for
$\displaystyle k=0,1,\ldots$,

(4.2)
$\displaystyle\displaystyle\sup\limits_{B_{k}}\frac{u}{v}\leq(1+c\zeta^{k})\inf\limits_{B_{k}}\frac{u}{v}.$

By virtue of (4.1), this implies the statement of the theorem.

For $\displaystyle-1\leq l<k$ we define

$\displaystyle\displaystyle
u_{k}^{l}(x)=\mathbb{E}^{x}[u(X_{\tau_{B_{k}}})\,;\,X_{\tau_{B_{k}}}\in\Pi_{l}],\quad
v_{k}^{l}(x)=\mathbb{E}^{x}[v(X_{\tau_{B_{k}}})\,;\,X_{\tau_{B_{k}}}\in\Pi_{l}],\quad
x\in\mathbb{R}^{d}.$

In order to obtain (4.2), we will prove the following two claims. Then it
suffices to repeat the final part of the proof in [11, Lemma 16]—we will skip
those details.
Claim 1. There exist $\displaystyle c=c(d,\alpha,\underline{D})$ and
$\displaystyle q=q(d,\alpha,\underline{D})\in(0,1)$ such that for
$\displaystyle-1\leq l\leq k-2$ and $\displaystyle x\in B_{k}$,

$\displaystyle\displaystyle u_{k}^{l}(x)\leq C(q^{k_{0}})^{k-l-1}u(x),$
$\displaystyle\displaystyle v_{k}^{l}(x)\leq C(q^{k_{0}})^{k-l-1}v(x).$

We define the oscillation of function $\displaystyle f$ as
$\displaystyle\operatorname{Osc}_{A}f=\sup\nolimits_{A}f-\inf\nolimits_{A}f.$
The constant $\displaystyle k_{0}$ is specified so that $\displaystyle
q^{k_{0}}$ is a sufficiently small number from the interval
$\displaystyle[1/2,1)$—we refer to [11] for details.

Claim 2. Let $\displaystyle g(x)=u_{k+1}^{k}(x)/v_{k+1}^{k}(x)$. Then there is
$\displaystyle\delta=\delta(d,\alpha,\underline{D})$ such that
$\displaystyle\operatorname{Osc}_{B_{k+2}}g\leq\delta\operatorname{Osc}_{B_{k}}g$.

We will now prove Claim 1 for $\displaystyle u$, the proof for $\displaystyle
v$ is identical. First let $\displaystyle 0\leq l\leq k-2$. By Lemma 4.2,

$\displaystyle\displaystyle
u_{k}^{l}(x)=\int_{\Pi_{l}}G_{D}(z,v)P_{B_{k}}(x,v)\,dv\leq
cG_{D}(z,y)\mathbb{P}^{x}(X_{\tau_{B_{k}}}\in\Pi_{l}).$

Furthermore, since $\displaystyle k\geq 1$, Lemma 4.2 yields $\displaystyle
G_{D}(z,y)\leq cG_{D}(z,x)$. Therefore,

(4.3) $\displaystyle\displaystyle u_{k}^{l}(x)\leq
cG_{D}(z,x)\mathbb{P}^{x}(X_{\tau_{B_{k}}}\in\Pi_{l}).$

Recall the explicit formula for the Poisson kernel of the ball—see, e.g.,
Landkof [52]:

(4.4) $\displaystyle\displaystyle
P_{B(0,r)}(x,v)=c_{d,\alpha}\frac{(r^{2}-|x|^{2})^{\alpha/2}}{(|v|^{2}-r^{2})^{\alpha/2}}|x-v|^{-d},\quad
x\in B(0,r),\ v\in B(0,r)^{c}.$

Using the formula, we find that

$\displaystyle\displaystyle\mathbb{P}^{x}(X_{\tau_{B_{k}}}\in\Pi_{l})=\int_{\Pi_{l}}P_{B_{k}}(x,v)\,dv$
$\displaystyle\displaystyle\leq
c_{d,\alpha}(r(2^{k_{0}})^{-k})^{\alpha}\int_{\Pi_{l}}(|v-y|^{2}-(r(2^{k_{0}})^{-k})^{2})^{-\alpha/2}|x-v|^{-d}\,dv$
$\displaystyle\displaystyle\leq\tilde{c}_{d,\alpha}\frac{(r(2^{k_{0}})^{-k})^{\alpha}}{(r(2^{k_{0}})^{-l-1})^{\alpha}}=\tilde{c}_{d,\alpha}(2^{-k_{0}\alpha})^{k-l-1}.$

Coming back to (4.3) we get Claim 1 for $\displaystyle 0\leq l\leq k-2$.

Now, let $\displaystyle l=-1$. Using (4.4), we get

$\displaystyle\displaystyle u_{k}^{-1}(x)$ $\displaystyle\displaystyle\leq
C(d,\alpha,\underline{D})\int_{D\setminus
B(y,r)}G_{D}(z,v)\frac{((r(2^{k_{0}})^{-k})^{2}-|x-y|^{2})^{\alpha/2}}{(|v-y|^{2}-(r(2^{k_{0}})^{-k})^{2})^{\alpha/2}}|x-v|^{-d}\,dv$
$\displaystyle\displaystyle\leq
C(d,\alpha,\underline{D})((2^{k_{0}})^{-k})^{\alpha}\int_{D\setminus
B(y,r)}G_{D}(z,v)\frac{r^{\alpha}}{(|v-y|^{2}-(r(2^{k_{0}})^{-k})^{2})^{\alpha/2}}|x-v|^{d}\,dv$
$\displaystyle\displaystyle\leq
c(d,\alpha)C(d,\alpha,\underline{D})(2^{-\alpha k_{0}})^{k}\int_{D\setminus
B(y,r)}G_{D}(z,v)\frac{(r^{2}-|x-y|^{2})^{\alpha/2}}{(|v-y|^{2}-r^{2})^{\alpha/2}}|x-v|^{d}\,dv$
$\displaystyle\displaystyle\leq\tilde{c}(d,\alpha)C(d,\alpha,\underline{D})(2^{-\alpha
k_{0}})^{k}\int_{D\setminus B(y,r)}G_{D}(z,v)P_{B(y,r)}(x,v)\,dv.$

Since $\displaystyle G_{D}(z,\cdot)$ is harmonic in $\displaystyle
D\setminus\\{z\\}$, the last integral is equal to $\displaystyle G_{D}(z,x)$,
which yields Claim 1 for $\displaystyle l=-1$. Thus, Claim 1 is proved.

It remains to prove Claim 2, which we do now. Let $\displaystyle
a_{1}=\inf\nolimits_{B_{k}}g$ and $\displaystyle
a_{2}=\sup\nolimits_{B_{k}}g$. Without any loss of generality we may assume
$\displaystyle a_{1}\neq a_{2}$. Then, we let

$\displaystyle\displaystyle g^{\prime}(x)=\frac{g(x)-a_{1}}{a_{2}-a_{1}},\quad
x\in B_{k}.$

We have $\displaystyle 0\leq g^{\prime}\leq 1$,
$\displaystyle\operatorname{Osc}_{B_{k}}g^{\prime}=1$, and
$\displaystyle\operatorname{Osc}_{B_{k+2}}g=\operatorname{Osc}_{B_{k+2}}g^{\prime}\operatorname{Osc}_{B_{k}}g$.
If $\displaystyle\sup\nolimits_{B_{k+2}}g^{\prime}\leq\tfrac{1}{2}$, then we
are done, so assume otherwise. Note that

$\displaystyle\displaystyle
g^{\prime}(x)=\frac{\frac{u_{k+1}^{k}(x)-a_{1}v_{k+1}^{k}(x)}{a_{2}-a_{1}}}{v_{k+1}^{k}(x)}=:\frac{g_{1}(x)}{g_{2}(x)},\quad
x\in B_{k+2}.$

By (4.4), we have

(4.5) $\displaystyle\displaystyle
1\leq\frac{\sup_{B_{k+2}}g_{2}}{\inf_{B_{k+2}}g_{2}}=\frac{\sup_{B_{k+2}}v_{k+1}^{k}}{\inf_{B_{k+2}}v_{k+1}^{k}}\leq
C(d,\alpha).$

Furthermore, since $\displaystyle v_{k+1}^{k}(x)\leq\sup\nolimits_{B_{0}}v\leq
C(d,\alpha,\underline{D})$ for all $\displaystyle x\in\mathbb{R}^{d}$, we get

$\displaystyle\displaystyle g_{1}(x)=v_{k+1}^{k}(x)g^{\prime}(x)\leq
C(d,\alpha,\underline{D}),\quad x\in B_{k}.$

If we extend $\displaystyle g_{1}$ to be equal to 0 on
$\displaystyle\mathbb{R}^{d}\setminus B_{k}$, then $\displaystyle g_{1}$ is
regular harmonic on $\displaystyle B_{k+1}$, nonnegative and bounded.
Therefore, by the Harnack inequality in an explicit scale invariant
formulation [13, Lemma 1]; see also Bass and Levin [7, Theorem 3.6] or Grzywny
[42],

(4.6) $\displaystyle\displaystyle
1\leq\frac{\sup_{B_{k+2}}g_{1}}{\inf_{B_{k+2}}g_{1}}\leq
C(d,\alpha,\underline{D}).$

By (4.5) and (4.6), we get

$\displaystyle\displaystyle\inf\limits_{B_{k+2}}g^{\prime}\geq
C^{-2}\sup\limits_{B_{k+2}}g^{\prime}\geq\tfrac{1}{2}C^{-2}.$

Therefore,

$\displaystyle\displaystyle\operatorname{Osc}_{B_{k+2}}g^{\prime}\leq\max(\tfrac{1}{2},1-\tfrac{1}{2}C^{-2})=1-\tfrac{1}{2}C^{-2},$

which ends the proof of Claim 2, and thus the lemma is proved. ∎

###### Proof of Theorem 1.4.

By Lemma 4.1, we can assume without loss of generality that
$\displaystyle|y-y^{\prime}|\leq 1/16$.

We first consider the case $\displaystyle
2^{k_{0}}|y^{\prime}-y|^{1/2}\geq\delta_{D}(y)$, with $\displaystyle k_{0}$
from Lemma 4.3), and let $\displaystyle Q\in\partial D$ be such that
$\displaystyle|y-Q|=\delta_{D}(y)$. Note that $\displaystyle y,y^{\prime}\in
B(Q,2^{k_{0}+1}|y-y^{\prime}|^{1/2})$—since $\displaystyle|y-y^{\prime}|<1$,
we have $\displaystyle|y-y^{\prime}|<|y-y^{\prime}|^{1/2}$. We split the
integral as follows:

(4.7)
$\displaystyle\displaystyle\int_{D}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\,dz=\int_{D\cap
B(Q,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}+\int_{D\setminus
B(Q,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}.$

By (4.1),there exist $\displaystyle c=c(d,\alpha,\underline{D})$ and
$\displaystyle\gamma=\gamma(d,\alpha,\underline{D})\in(0,\alpha)$ such that

$\displaystyle\displaystyle\int_{D\cap
B(Q,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\,dz$
$\displaystyle\displaystyle\leq 2^{p}\int_{D\cap
B(Q,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}\bigg{(}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}\bigg{|}^{p}+\bigg{|}\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\bigg{)}\,dz$
$\displaystyle\displaystyle\leq
c\int_{B(0,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}|z|^{p(\alpha-\gamma-d)}\,dz$
$\displaystyle\displaystyle=cC(d,\alpha,p)|y-y^{\prime}|^{(d+p(\alpha-\gamma-d))/2}.$

In the second integral of (4.7) we use the boundary Harnack principle given in
[12, Lemma 3]: we let $\displaystyle u(y)=G_{D}(y,z)$, $\displaystyle
v(y)=G_{D}(x_{0},y)$ and $\displaystyle r=2^{k_{0}+1}|y-y^{\prime}|^{1/2}$
there. By the Green function estimates (2.14) and arguments similar to the
proof of Lemma 4.1 we find that for $\displaystyle z\in
D\cap(B(Q,2^{k+k_{0}+3}|y-y^{\prime}|^{1/2})\setminus
B(Q,2^{k+k_{0}+2}|y-y^{\prime}|^{1/2}))$ we have $\displaystyle
u(A_{r}(Q))/v(A_{r}(Q))\leq
C(d,\alpha,\underline{D})(2^{k}|y-y^{\prime}|^{1/2})^{\alpha-\gamma-d}$, for
all $\displaystyle k\in\\{0,1,\ldots,N_{0}\\}$, where $\displaystyle
N_{0}=\lceil\log_{2}({\rm diam}(D)/2^{k_{0}+2}|y-y^{\prime}|^{1/2})\rceil$ and
we define $\displaystyle u/v$ to be 0 outside $\displaystyle D$. Therefore, by
[12, Lemma 3], there exist $\displaystyle c$ and $\displaystyle\sigma>0$
depending only on $\displaystyle d,\alpha,\underline{D}$ such that

$\displaystyle\displaystyle\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\leq
c(2^{k}|y-y^{\prime}|^{1/2})^{p(\alpha-\gamma-d)}|y-y^{\prime}|^{\sigma p/2}$

holds for all $\displaystyle z\in
D\cap(B(Q,2^{k+k_{0}+3}|y-y^{\prime}|^{1/2})\setminus
B(Q,2^{k+k_{0}+2}|y-y^{\prime}|^{1/2}))$. It follows that

$\displaystyle\displaystyle\int_{D\setminus
B(Q,2^{k_{0}+2}|y-y^{\prime}|^{1/2})}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\,dz$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{k=0}^{N_{0}}\int_{D\cap(B(Q,2^{k+k_{0}+3}|y-y^{\prime}|^{1/2})\setminus
B(Q,2^{k+k_{0}+2}|y-y^{\prime}|^{1/2}))}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\,dz$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
c|y-y^{\prime}|^{\sigma
p/2}\sum\limits_{k=0}^{N_{0}}(2^{k}|y-y^{\prime}|^{1/2})^{p(\alpha-\gamma-d)}(2^{k}|y-y^{\prime}|^{1/2})^{d}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
c|y-y^{\prime}|^{\sigma
p/2}\sum\limits_{k=0}^{N_{0}}(2^{k}|y-y^{\prime}|^{1/2})^{d+p(\alpha-\gamma-d)}.$

The last sum is comparable to $\displaystyle{\rm
diam}(D)^{d+p(\alpha-\gamma-d)}$, so the proof is complete when $\displaystyle
2^{k_{0}}|y-y^{\prime}|^{1/2}\geq\delta_{D}(y)$.

Now assume that $\displaystyle 2^{k_{0}}|y-y^{\prime}|^{1/2}<\delta_{D}(y)$.
We split the integral in the following way:

(4.8)
$\begin{split}&\int_{D}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}^{p}\,dz\\\
=&\int_{D\cap B(y,2^{k_{0}}|y-y^{\prime}|^{1/2})}+\int_{D\cap
B(y,2^{k_{0}}|y-y^{\prime}|^{1/2})^{c}\cap
B(Q,3\delta_{D}(y))^{c}}+\int_{D\cap
B(y,2^{k_{0}}|y-y^{\prime}|^{1/2})^{c}\cap B(Q,3\delta_{D}(y))}.\end{split}$

The first two integrals are handled as the ones in (4.7). In particular, in
the second one we can use the boundary Harnack principle. In the last integral
on the right-hand side of (4.8) we will apply Lemma 4.3. To this end, we split
once more:

$\displaystyle\displaystyle\int_{D\cap
B(y,2^{k_{0}}|y-y^{\prime}|^{1/2})^{c}\cap
B(Q,3\delta_{D}(y))}\leq\sum\limits_{k=0}^{M_{0}}\int_{D\cap
B(Q,3\delta_{D}(y))\cap(B(y,2^{k+k_{0}+1}|y-y^{\prime}|^{1/2})\setminus
B(y,2^{k+k_{0}}|y-y^{\prime}|^{1/2}))}=:\sum\limits_{k=0}^{M_{0}}I_{k},$

where $\displaystyle
M_{0}=\lceil\log_{2}(3\delta_{D}(y)/(2^{k_{0}}|y-y^{\prime}|^{1/2}))\rceil$.
We then use Lemma 4.3 with $\displaystyle
r=r_{k}=2^{k_{0}+k}|y-y^{\prime}|^{1/2}/4$:

$\displaystyle\displaystyle I_{k}$ $\displaystyle\displaystyle\leq
C(d,\alpha,\underline{D})|y-y^{\prime}|^{\sigma
p/2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{B(y,2^{k+k_{0}+1}|y-y^{\prime}|^{1/2})\setminus
B(y,2^{k+k_{0}}|y-y^{\prime}|^{1/2})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{k}^{p(\alpha-d-\gamma)}\,dz$
$\displaystyle\displaystyle\leq\widetilde{C}(d,\alpha,\underline{D})|y-y^{\prime}|^{\sigma
p/2}(2^{k+k_{0}}|y-y^{\prime}|^{1/2})^{d-p(\alpha-\gamma-d)},\quad
k=0,\ldots,M_{0},$

since for $\displaystyle|y-y^{\prime}|\leq 1/16$, we have
$\displaystyle|y-y^{\prime}|\leq|y-y^{\prime}|^{1/2}/4$, so $\displaystyle
y^{\prime}\in B(y,2^{-k_{0}}r)$. Therefore we get

$\displaystyle\displaystyle\sum\limits_{k=0}^{M_{0}}I_{k}\leq
C(d,\alpha,\underline{D})|y-y^{\prime}|^{\sigma
p/2}\delta_{D}(y)^{d-p(\alpha-\gamma-d)},$

which ends the proof. ∎

###### Proof of Theorem 1.2.

Fix $\displaystyle x\in D$ and $\displaystyle t\in[T_{1},T_{2}]$. First, we
investigate $\displaystyle\eta^{x_{0}}$. By the results of Section 2.3,

$\displaystyle\displaystyle
p_{t}^{D}(x,y)=G_{D}\Delta^{\alpha/2}_{y}p_{t}^{D}(x,y).$

Furthermore, by Corollary 2.6, the function $\displaystyle
f(y)=\Delta^{\alpha/2}_{y}p_{t}^{D}(x,y)$ is bounded and the bound does not
depend on $\displaystyle x\in D$. Therefore, by Theorem 1.4, for
$\displaystyle y,y^{\prime}\in D\setminus B(x_{0},r_{1})$,

$\displaystyle\displaystyle\bigg{|}\frac{p_{t}^{D}(x,y)}{G_{D}(x_{0},y)}-\frac{p_{t}^{D}(x,y^{\prime})}{G_{D}(x_{0},y^{\prime})}\bigg{|}$
$\displaystyle\displaystyle\leq\int_{D}\bigg{|}\frac{G_{D}(y,z)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},z)}{G_{D}(x_{0},y^{\prime})}\bigg{|}|f(z)|\,dz$
$\displaystyle\displaystyle\leq\bigg{\|}\frac{G_{D}(y,\cdot)}{G_{D}(x_{0},y)}-\frac{G_{D}(y^{\prime},\cdot)}{G_{D}(x_{0},y^{\prime})}\bigg{\|}_{L^{1}(D)}\|f\|_{\infty}\leq
C|y-y^{\prime}|^{\sigma},$

where the constants $\displaystyle C,\sigma$ depend only on $\displaystyle
d,\alpha,\underline{D},T_{1},T_{2},x_{0},$ and $\displaystyle r_{1}$.

We now proceed to $\displaystyle\widetilde{\eta}$. Note that there exist
$\displaystyle x_{1}\in D$ and $\displaystyle r=r(\underline{D})$ such that
$\displaystyle B(x_{1},2r)\subset D$. Without loss of generality, we can
assume that $\displaystyle|y-y^{\prime}|<r/4$. Then, for any fixed
$\displaystyle y,y^{\prime}$, there exists $\displaystyle x_{2}$ such that
$\displaystyle B(x_{2},r/4)\subset D$ and $\displaystyle y,y^{\prime}\notin
B(x_{2},r/4)$. This means that $\displaystyle
G_{D}(x_{2},y),G_{D}(x_{2},y^{\prime})\leq C$, where $\displaystyle C\geq 1$
depends only on $\displaystyle d,\alpha,$ and $\displaystyle\underline{D}$. We
then split as follows:

$\displaystyle\displaystyle\bigg{|}\frac{p_{t}^{D}(x,y)}{p_{t_{0}}^{D}(x_{0},y)}-\frac{p_{t}^{D}(x,y^{\prime})}{p_{t_{0}}^{D}(x_{0},y^{\prime})}\bigg{|}$
$\displaystyle\displaystyle=\bigg{|}\frac{p_{t}^{D}(x,y)}{G_{D}(x_{2},y)}\frac{G_{D}(x_{2},y)}{p_{t_{0}}^{D}(x_{0},y)}-\frac{p_{t}^{D}(x,y^{\prime})}{G_{D}(x_{2},y^{\prime})}\frac{G_{D}(x_{2},y^{\prime})}{p_{t_{0}}^{D}(x_{0},y^{\prime})}\bigg{|}$
(4.9)
$\displaystyle\displaystyle\leq\frac{p_{t}^{D}(x,y)}{G_{D}(x_{2},y)}\bigg{|}\frac{G_{D}(x_{2},y)}{p_{t_{0}}^{D}(x_{0},y)}-\frac{G_{D}(x_{2},y^{\prime})}{p_{t_{0}}^{D}(x_{0},y^{\prime})}\bigg{|}+\frac{G_{D}(x_{2},y^{\prime})}{p_{t_{0}}^{D}(x_{0},y^{\prime})}\bigg{|}\frac{p_{t}^{D}(x,y^{\prime})}{G_{D}(x_{2},y^{\prime})}-\frac{p_{t}^{D}(x,y)}{G_{D}(x_{2},y)}\bigg{|}.$

By using Lemma 3.2 and (2.10), we find that

(4.10)
$\displaystyle\displaystyle\frac{p_{t}^{D}(x,y)}{G_{D}(x_{2},y)}\lesssim\frac{\mathbb{P}^{y}(\tau_{D}>t)}{G_{D}(x_{2},y)}=\frac{G_{D}P_{t}^{D}\kappa_{D}(y)}{G_{D}(x_{2},y)}\leq
C(d,\alpha,\underline{D},T_{1},T_{2})<\infty.$

By similar arguments,

(4.11)
$\displaystyle\displaystyle\frac{p_{t}^{D}(x_{0},y)}{G_{D}(x_{2},y)}\geq
c(d,\alpha,\underline{D},T_{1},T_{2},x_{0})>0.$

From (4.9), (4.10), (4.11), and the Hölder regularity of
$\displaystyle\eta^{x_{0}}$ obtained above, we arrive at

$\displaystyle\displaystyle\bigg{|}\frac{p_{t}^{D}(x,y)}{p_{t_{0}}^{D}(x_{0},y)}-\frac{p_{t}^{D}(x,y^{\prime})}{p_{t_{0}}^{D}(x_{0},y^{\prime})}\bigg{|}\leq
C|y-y^{\prime}|^{\sigma},$

with $\displaystyle C$ and $\displaystyle\sigma$ depending on $\displaystyle
d,\alpha,\underline{D},T_{1},T_{2},x_{0}$.

The arguments for $\displaystyle\eta$ are similar to the ones for
$\displaystyle\widetilde{\eta}$, with no dependence on $\displaystyle x_{0}$.
The proof is complete. ∎

##  5\. Space-time stable processes and caloric functions

###  5.1. Preliminaries

Recall that $\displaystyle(X_{s})_{s\geq 0}$ is the isotropic
$\displaystyle\alpha$-stable Lévy process. Like for the space-time Brownian
motion [35], we define the space-time $\displaystyle\alpha$-stable process as
the following Lévy process on $\displaystyle\mathbb{R}^{d+1}$:

$\displaystyle\dot{X}_{s}:=(-s,X_{s}),\quad s\geq 0.$

Since $\displaystyle\dot{X}$ is a Lévy process, it has the strong Markov
property. Many properties of the space-time process are inherited from the
$\displaystyle\alpha$-stable process. Thus, for a (Borel) set $\displaystyle
A\subseteq\mathbb{R}^{d+1}$, we let

$\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{s}\in
A):=\mathbb{P}((t-s,X_{s}+x)\in A),$

and for a (Borel) function $\displaystyle
f\colon\mathbb{R}^{d+1}\to\mathbb{R}^{d}$, we have

$\displaystyle\mathbb{E}^{(t,x)}[f(\dot{X}_{s})]=\mathbb{E}[f(t-s,X_{s}+x)].$

It can be easily verified that the transition probability of
$\displaystyle\dot{X}$ takes on the following form

$\displaystyle\widetilde{p}_{s}(t,x,du,dy)=p_{s}(x,y)\,dy\otimes\delta_{\\{t-s\\}}(du),\quad
s\geq 0,\ (t,x),(u,y)\in\mathbb{R}\times\mathbb{R}^{d}.$

The corresponding semigroup will be denoted by $\displaystyle\widetilde{P}$.

######  Lemma 5.1.

The pointwise generator of the semigroup of the space-time
$\displaystyle\alpha$-stable process coincides with the fractional heat
operator $\displaystyle\Delta^{\alpha/2}-\partial_{t}$ for functions
$\displaystyle u\in C^{1,2}_{b}([0,\infty)\times\mathbb{R}^{d})$.

###### Proof.

Let $\displaystyle u\in C^{1,2}_{b}([0,\infty)\times\mathbb{R}^{d})$. For all
$\displaystyle(t,x)\in[0,\infty)\times\mathbb{R}^{d}$ and $\displaystyle
s\in(0,t)$, we have

$\displaystyle\displaystyle\frac{1}{s}(\widetilde{P}_{s}u(t,x)-u(t,x))$
$\displaystyle\displaystyle=\frac{1}{s}\int_{\mathbb{R}^{d}}\int_{[0,\infty)}(u(r,y)-u(t,x))\widetilde{p}_{s}(t,x,dy,dr)$
$\displaystyle\displaystyle=\frac{1}{s}\int_{\mathbb{R}^{d}}(u(t-s,y)-u(t,x))p_{s}(x,y)\,dy$
(5.1)
$\displaystyle\displaystyle=\frac{1}{s}\int_{\mathbb{R}^{d}}(u(t-s,y)-u(t-s,x))p_{s}(x,y)\,dy$
(5.2) $\displaystyle\displaystyle\,+\frac{1}{s}(u(t-s,x)-u(t,x)).$

Clearly, (5.2) converges to $\displaystyle-\partial_{t}u(t,x)$ as
$\displaystyle s\to 0^{+}$, so it suffices to show that (5.1) converges to
$\displaystyle\Delta^{\alpha/2}_{x}u(t,x)$. To this end, we will prove that

(5.3)
$\displaystyle\displaystyle\frac{1}{s}\int_{\mathbb{R}^{d}}((u(t,y)-u(t,x))-(u(t-s,y)-u(t-s,x)))p_{s}(x,y)\,dy$

converges to 0 as $\displaystyle s\to 0^{+}$. Let $\displaystyle\varepsilon>0$
and let $\displaystyle\delta>0$ be so small that $\displaystyle
p_{s}(x,B(x,\delta)^{c})<\varepsilon$. Then we also have $\displaystyle
p_{s^{\prime}}(x,B(x,\delta)^{c})<\varepsilon$ for $\displaystyle
s^{\prime}\in(0,s)$. By Lagrange’s mean value theorem, we get

$\displaystyle\bigg{|}\frac{1}{s}\int_{B(0,\delta)^{c}}((u(t,y)-u(t,x))-(u(t-s,y)-u(t-s,x)))p_{s}(x,y)\,dy\bigg{|}<2\varepsilon\|u\|_{C^{1,2}}.$

By Taylor’s expansion, $\displaystyle
u(t-s,x)=u(t,x)-s\partial_{t}u(t,x)+o(s)$ as $\displaystyle s\to 0$, and
similarly for $\displaystyle y$, so

$\displaystyle\displaystyle\bigg{|}\frac{1}{s}\int_{B(0,\delta)}((u(t,y)-u(t,x))-(u(t-s,y)-u(t-s,x)))p_{s}(x,y)\,dy\bigg{|}$
$\displaystyle\displaystyle=\,$
$\displaystyle\displaystyle\bigg{|}\int_{B(0,\delta)}(\partial_{t}u(t,x)-\partial_{t}u(t,y)+\frac{o(s)}{s})p_{s}(x,y)\,dy\bigg{|}$
$\displaystyle\displaystyle\leq\,$
$\displaystyle\displaystyle\delta\|u\|_{C^{1,2}}+o(1).$

This ends the proof. ∎

In the next result we exhibit a space-time Poisson kernel for cylindrical
domains. As usual, for arbitrary (open) $\displaystyle
G\subseteq\mathbb{R}\times\mathbb{R}^{d}$, we let

$\displaystyle\tau_{G}:=\inf\\{t>0:\dot{X}_{t}\notin G\\}.$

######  Lemma 5.2.

Recall that $\displaystyle D\subseteq\mathbb{R}^{d}$ is Lipschitz open set and
let $\displaystyle\dot{D}=(r,t)\times D$ for some (arbitrary)
$\displaystyle-\infty\leq r<t$. Then the distribution of
$\displaystyle\dot{X}_{\tau_{\dot{D}}}$—the first exit place of
$\displaystyle\dot{X}$ from $\displaystyle\dot{D}$—is given by the formula

$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{\tau_{\dot{D}}}\in(ds,dy))=\begin{cases}{\textrm{\bf
1}}_{[r,t)}(s)\,ds\otimes J^{D}(t,x,s,y)\,dy+\delta_{t-r}(ds)\otimes
p_{t-s}^{D}(x,y)\,dy,\quad&r>-\infty,\\\ \textrm{\bf
1}_{(-\infty,t)}(s)\,ds\otimes J^{D}(t,x,s,y)\,dy,\quad&r=-\infty,\end{cases}$

where

$\displaystyle
J^{D}(t,x,s,y):=\int_{D}p_{t-s}^{D}(x,\xi)\nu(\xi,y)\,d\xi,\quad s<t,\ x\in
D,\ y\in D^{c}.$

We call $\displaystyle J^{D}$ the lateral Poisson kernel.

######  Remark 5.3.

For the cylinder $\displaystyle\dot{D}=(r,t)\times D$, if the process
$\displaystyle\dot{X}$ starts at $\displaystyle(t,x)$ with some $\displaystyle
x\in D$, then it immediately enters $\displaystyle\dot{D}$, so
$\displaystyle\tau_{\dot{D}}>0$ almost surely, although
$\displaystyle(t,x)\notin\dot{D}$. In the language of potential theory, the
points on the top of the cylinder are irregular.

###### Proof of Lemma 5.2..

Let $\displaystyle r>-\infty$. We have

(5.4)
$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{\tau_{\dot{D}}}\in(ds,dy))=\
$
$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{\tau_{\dot{D}}}\in(ds,dy),\tau_{\dot{D}}>\tau_{D})$
(5.5) $\displaystyle\displaystyle+\,$
$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{\tau_{\dot{D}}}\in(ds,dy),\tau_{\dot{D}}=\tau_{D})$
(5.6) $\displaystyle\displaystyle+\,$
$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(\dot{X}_{\tau_{\dot{D}}}\in(ds,dy),\tau_{\dot{D}}<\tau_{D}).$

Note that (5.4) vanishes, because
$\displaystyle\mathbb{P}^{(t,x)}(\tau_{\dot{D}}>\tau_{D})=0$.

By the Ikeda–Watanabe formula (2.17), the term (5.5) is equal to

$\displaystyle\displaystyle\mathbb{P}^{(t,x)}(X_{\tau_{D}}\in A,\,\tau_{D}\leq
t-r,\,\tau_{D}\in ds)={\textrm{\bf 1}}_{[r,t)}(s)\,ds\otimes
J^{D}(t,x,s,y)\,dy.$

In (5.6) we have $\displaystyle\tau_{D}>\tau_{\dot{D}}=t-r$, so by the
definition of the Dirichlet heat kernel, this term is equal to

$\displaystyle\displaystyle\delta_{t-r}(ds)\otimes p_{t-r}^{D}(x,y),$

see [31, Chapter 2]. The case of $\displaystyle r=-\infty$ is left to the
reader. ∎

We see that $\displaystyle J^{D}(t,x,s,y)$ represents the scenario of
$\displaystyle\dot{X}$ starting at $\displaystyle(t,x)$ and leaving to
$\displaystyle(s,y)$, where, recall, $\displaystyle x\in D$, $\displaystyle
y\in D^{c}$, and $\displaystyle s<t$. Another way to express the result in
Lemma 5.2, is as follows:

(5.7)
$\displaystyle\displaystyle\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{\dot{D}}})=\int_{r}^{t}\int_{D^{c}}J^{D}(t,x,s,z)u(s,z)\,dz\,ds+\int_{D}p_{t-r}^{D}(x,y)u(r,y)\,dy,$

whenever this integral makes sense, e.g., for nonnegative $\displaystyle u$.
By analogy to the elliptic equations, we call the right-hand side of (5.7) the
Poisson integral, and the first term on the right-hand side of (5.7)—the
lateral Poisson integral.

######  Remark 5.4.

Another motivation for calling $\displaystyle J^{D}(t,x,s,z)$ the lateral
Poisson kernel comes from the fact that it is the nonlocal normal derivative
of $\displaystyle p_{t-s}^{D}$, whereas $\displaystyle p_{t-s}^{D}$ serves as
the Green function for the fractional heat equation. Indeed, using the
definition of the nonlocal normal derivative from [33]:

$[\partial_{\vec{n}}u](x):=\int_{D}(u(y)-u(x))\nu(x,y)\,dy,\quad x\in D^{c},$

we see that for every $\displaystyle z\in D^{c}$,

$\partial_{\vec{n}}p_{t-s}^{D}(x,\cdot)(z)=\int_{D}p_{t-s}^{D}(x,y)\nu(y,z)\,dy=J^{D}(t,x,s,z),\quad
x\in D.$

###  5.2. Caloric functions

We define the caloric functions in terms of the mean value property. We stress
that we only consider finite nonnegative functions.

######  Definition 5.5.

Let $\displaystyle-\infty<T_{1}<T_{2}<\infty$. We say that $\displaystyle
u\colon(T_{1},T_{2})\times\mathbb{R}^{d}\to[0,\infty)$ is caloric in
$\displaystyle(T_{1},T_{2})\times D$, if the mean value property:

(5.8)
$u(t,x)=\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{G}}),\qquad(t,x)\in(T_{1},T_{2})\times
D,$

holds for every open set $\displaystyle G\subset\subset(T_{1},T_{2})\times D$.

We say that $\displaystyle
u\colon[T_{1},T_{2})\times\mathbb{R}^{d}\to[0,\infty)$ is caloric in
$\displaystyle[T_{1},T_{2})\times D$ if (5.8) holds for every open
$\displaystyle G\subset\subset[T_{1},T_{2})\times D$. If $\displaystyle u$ is
caloric in $\displaystyle[T_{1},T_{2})\times D$ and satisfies (5.8) for
$\displaystyle G=(T_{1},T_{2})\times D$, then we say that $\displaystyle u$ is
regular caloric. If $\displaystyle u$ is caloric in
$\displaystyle[T_{1},T_{2})\times D$ and $\displaystyle u\equiv 0$ on the
parabolic boundary

$\displaystyle D^{p}:=(\\{T_{1}\\}\times D)\cup((T_{1},T_{2})\times D^{c}),$

then we say that $\displaystyle u$ is singular caloric.

######  Remark 5.6.

  1. (a)

Our caloric functions are just harmonic functions of the space-time isotropic
stable Lévy process.

  2. (b)

We may also consider $\displaystyle T_{1}=-\infty$ or $\displaystyle
T_{2}=\infty$, where appropriate, in particular when defining functions
caloric on $\displaystyle(T_{1},T_{2})\times D$.

  3. (c)

The condition $\displaystyle G\subset\subset[T_{1},T_{2})\times D$ allows
$\displaystyle G$ to touch $\displaystyle\\{T_{1}\\}\times D$. Caloricity in
$\displaystyle[T_{1},T_{2})\times D$ may be considered as a (new) relaxation
of regular caloricity, localized near the part $\displaystyle\\{T_{1}\\}\times
D$ of the boundary of $\displaystyle(T_{1},T_{2})\times D$, see also Lemma
5.7. Both notions are meant to facilitate discussion of boundary conditions
(they generalize to harmonic functions of other strong Markov processes).

  4. (d)

The caloricity in $\displaystyle[T_{1},T_{2})\times D$ helps to handle initial
conditions which are functions, but also rules out some interesting cases,
e.g., $\displaystyle(t,y)\mapsto p_{t}^{D}(x,y)$. See also [22]. Remarkably,
every (nonnegative) function caloric in $\displaystyle(T_{1},T_{2})\times D$
has a certain initial condition which is a measure, see Section 6.

  5. (e)

A caloric function need not satisfy the fractional heat equation pointwise,
due to lack of time regularity. This can be seen using the counterexample
given by Chang-Lara and Dávila [28, Section 2.4.1] for viscosity solutions.
See also Remark 5.12 below.

######  Lemma 5.7.

Regular caloricity implies caloricity in $\displaystyle[T_{1},T_{2})\times D$,
which in turn implies caloricity in $\displaystyle(T_{1},T_{2})\times D$.
Furthermore, (5.8) only needs to be verified for cylinders $\displaystyle G$.

###### Proof.

Assume that (5.8) holds for $\displaystyle G$. By the strong Markov property
of $\displaystyle\dot{X}$, (5.8) then holds for every open $\displaystyle
G^{\prime}\subset G$:

$\displaystyle
u(t,x)=\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{G}})=\mathbb{E}^{(t,x)}\mathbb{E}^{\dot{X}_{\tau_{G^{\prime}}}}u(\dot{X}_{\tau_{G}})=\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{G^{\prime}}}).$

This first two assertions follow immediately. To clarify the third one, note
that every open $\displaystyle G^{\prime}\subset\subset[T_{1},T_{2})\times D$
is contained in an open cylinder, relatively compact in
$\displaystyle[T_{1},T_{2})\times D$. Similarly for
$\displaystyle(T_{1},T_{2})\times D$. ∎

We continue with several examples of caloric functions.

######  Example 5.8.

For every fixed $\displaystyle x\in\mathbb{R}^{d}$, the function
$\displaystyle(t,y)\mapsto p_{t}^{D}(x,y)$ satisfies the mean value property
on every $\displaystyle(\varepsilon,T)\times D$ for $\displaystyle
0<\varepsilon<T<\infty$, hence it is caloric on $\displaystyle(0,\infty)\times
D$.

######  Example 5.9.

If we let

(5.9)
$\displaystyle\displaystyle\eta_{t,Q}(x):=0,\quad(t,x)\in(-\infty,0]\times\mathbb{R}^{d}\cup(0,\infty)\times
D^{c},\ Q\in\partial D,$

then for every fixed $\displaystyle Q\in\partial D$, the function
$\displaystyle(t,x)\mapsto\eta_{t,Q}(x)$ is caloric in
$\displaystyle(-\infty,\infty)\times D$. Indeed, the mean value property in
$\displaystyle(\varepsilon,T)\times D$, with $\displaystyle
0<\varepsilon<T<\infty$ is a consequence of (1.8). Then, by Lemma 3.5,

$\displaystyle\displaystyle\eta_{t,Q}(x)$
$\displaystyle\displaystyle=\int_{0}^{t}\int_{U^{c}}J^{D}(t,x,s,z)\eta_{s,Q}(z)\,dz\,ds$
$\displaystyle\displaystyle=\int_{-R}^{t}\int_{U^{c}}J^{D}(t,x,s,z)\eta_{s,Q}(z)\,dz\,ds,$

for any $\displaystyle R\geq 0$.

######  Example 5.10.

If $\displaystyle f\colon\mathbb{R}^{d}\to[0,\infty)$ is a nonnegative
measurable function and $\displaystyle P_{1}^{D}f(x)$ is finite for all
$\displaystyle x\in D$, then $\displaystyle(t,x)\mapsto P_{t}^{D}f(x)$ is
caloric in $\displaystyle[0,\infty)\times D$, with the usual convention
$\displaystyle P_{0}^{D}f:=f$.

The following class of functions is of particular interest for us. We will
show in the next section that it coincides with the class of all singular
caloric functions.

######  Lemma 5.11.

If $\displaystyle\mu(dQ\,ds)$ is a locally finite nonnegative Borel measure on
$\displaystyle\partial D\times[0,\infty)$, then

(5.10) $\displaystyle\displaystyle
h(t,x):=\begin{cases}\int_{[0,t)}\int_{\partial
D}\eta_{t-\tau,Q}(x)\mu(dQ\,d\tau),\quad&t>0,\ x\in D,\\\ 0,\quad&\mbox{
elsewhere},\end{cases}$

is singular caloric in $\displaystyle[0,\infty)\times D$.

###### Proof.

By Lemma 3.5, $\displaystyle h$ is finite for all $\displaystyle t>0$ and
$\displaystyle x\in D$, and by (5.9), we have

$\displaystyle\displaystyle\int_{[0,t)}\int_{\partial
D}\eta_{t-\tau,Q}(x)\mu(dQ\,d\tau)=\int_{[0,\infty)}\int_{\partial
D}\eta_{t-\tau,Q}(x)\mu(dQ\,d\tau),\quad t\geq 0,\ x\in D.$

Therefore, the mean value property for $\displaystyle h$ follows from
Fubini–Tonelli and caloricity of $\displaystyle\eta$. ∎

######  Remark 5.12.

We note that the viscosity solution considered in [28, Section 2.4.1],
although non-differentiable, is Lipschitz in time. The function $\displaystyle
n_{t,Q}$ is not even Lipschitz in $\displaystyle t$ because for $\displaystyle
t\in(0,1)$ and fixed $\displaystyle x\in D$,

$\displaystyle\displaystyle\frac{\eta_{t,Q}(x)}{t}$
$\displaystyle\displaystyle=\frac{1}{t}\lim\limits_{y\to
Q}\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>t)}\frac{\mathbb{P}^{y}(\tau_{D}>t)}{\mathbb{P}^{y}(\tau_{D}>1)}\gtrsim\frac{p_{t}(x,Q)}{t}\lim_{y\to
Q}\frac{\mathbb{P}^{y}(\tau_{D}>t)}{\mathbb{P}^{y}(\tau_{D}>1)}\gtrsim|x-Q|^{-d-\alpha}\lim_{y\to
Q}\frac{\mathbb{P}^{y}(\tau_{D}>t)}{\mathbb{P}^{y}(\tau_{D}>1)}.$

We see, indeed, that the last limit is comparable to $\displaystyle t^{-1/2}$
if $\displaystyle D$ is $\displaystyle C^{1,1}$ by (2.11). Furthermore, for
Lipschitz $\displaystyle D$ it also explodes as $\displaystyle t\to 0^{+}$
because of the proof of Lemma B.2 and [11, Lemma 3].

######  Lemma 5.13.

If $\displaystyle u$ is caloric in $\displaystyle(T_{1},T_{2})\times D$ for
some $\displaystyle T_{1}<T_{2}$, then $\displaystyle u\in L^{1}_{\rm
loc}((T_{1},T_{2})\times\mathbb{R}^{d})$.

###### Proof.

The proof is similar to the one of [17, Lemma 4.5]. First note that for any
fixed $\displaystyle x\in D$, $\displaystyle r>0$, and $\displaystyle
B=B(x,r)$, by (2.10) we have

$\displaystyle\displaystyle
J^{B}(t,x,s,z)=\int_{B}p_{t-s}^{B}(x,y)\nu(y,z)\,dy$
$\displaystyle\displaystyle\approx\int_{B}p_{t-s}(x,y)\mathbb{P}^{y}(\tau_{B}>t-s)\nu(y,z)\,dy$
$\displaystyle\displaystyle\geq c\int_{B(x,r/2)}p_{t-s}(x,y)\,dy\geq C>0,$

with $\displaystyle C$ depending only on $\displaystyle r$ and $\displaystyle
R$, where $\displaystyle\delta_{B}(z),t-s\leq R$. Thus, $\displaystyle
J^{B}(t,x,\cdot,\cdot)$ is locally bounded from below. Now, take two disjoint
balls $\displaystyle B_{1},B_{2}\subseteq D$, centered at some points
$\displaystyle x_{1},x_{2}\in D$ respectively, and let $\displaystyle
T_{1}<t_{0}<t<T_{2}$ and $\displaystyle R>0$. Since $\displaystyle u$ is
nonnegative and caloric, for $\displaystyle i=1,2$ we get

$\displaystyle\displaystyle\infty>u(t,x)\geq\int_{t_{0}}^{t}\int_{B_{i}^{c}}u(s,z)J^{B_{i}}(t,x,s,z)\,dz\,ds\geq
C\int_{t_{0}}^{t}\int_{B(0,R)\setminus B_{i}}u(s,z)\,dz\,ds.$

Therefore $\displaystyle u\in L^{1}((t_{0},t)\times(B(0,R)\setminus B_{i}))$
for $\displaystyle i=1,2$. But $\displaystyle B_{1}\cap B_{2}=\emptyset$, so
$\displaystyle u\in L^{1}((t_{0},t)\times B(0,R))$. Since $\displaystyle R$
can be chosen arbitrarily large, the proof is complete.

∎

The following result shows that the so-called ancient solutions, i.e.,
functions caloric in a time interval of the form $\displaystyle(-\infty,T)$,
can be conveniently studied by considering only the lateral Poisson integrals.

######  Lemma 5.14.

If $\displaystyle u$ is caloric in $\displaystyle(-\infty,T)\times D$ for some
$\displaystyle T\in\mathbb{R}$, then for all $\displaystyle x\in
U\subset\subset D$ and $\displaystyle t<T$ we have

(5.11) $u(t,x)=\mathbb{E}^{(t,x)}[u(\tau_{(-\infty,t)\times
U},X_{\tau_{(-\infty,t)\times
U}})]=\int_{-\infty}^{t}\int_{U^{c}}J^{U}(t,x,s,z)u(s,z)\,dz\,ds.$

In particular, the integral on the right-hand side of (5.11) is finite.

###### Proof.

Let $\displaystyle t,x,U$ be as in the statement. By the definition of
caloricity, for $\displaystyle v<t$ we have

$\displaystyle\displaystyle
u(t,x)=\int_{v}^{t}\int_{U^{c}}J^{U}(t,x,s,z)u(s,z)\,dz\,ds+\int_{U}p_{t-v}^{U}(x,y)u(v,y)\,dy.$

The first integral on the right-hand side increases to the right-hand side of
(5.11) by the monotone convergence theorem and the second integral decreases.
It suffices to prove that

(5.12) $a:=\lim\limits_{v\to-\infty}\int_{U}p_{t-v}^{U}(x,y)u(v,y)\,dy=0.$

To this end note that for every $\displaystyle v<t$,

(5.13) $\displaystyle\displaystyle\int_{U}p_{t-v}^{U}(x,y)u(v,y)\,dy\geq a.$

Let $\displaystyle n>0$ be so large that $\displaystyle U\subset\subset D_{n}$
(see (2.1)). Recall that $\displaystyle\lambda_{1}(V)$ is the first eigenvalue
of the Dirichlet fractional Laplacian for an open set $\displaystyle V$. We
claim that

(5.14) $\displaystyle\displaystyle\lambda_{1}(D_{n})<\lambda_{1}(U).$

A weak inequality is well known as the domain monotonicity. In order to prove
the strict inequality, assume without loss of generality that $\displaystyle
0\in U$. Then there exists $\displaystyle q>1$ such that $\displaystyle
qU\subset\subset D_{n}$, so, by domain monotonicity,
$\displaystyle\lambda_{1}(D_{n})\leq\lambda_{1}(qU)=q^{-\alpha}\lambda_{1}(U)$,
which yields (5.14).

By (2.13), (2.12), and the fact that each eigenfunction is bounded from above
and bounded from below away from the boundary, for $\displaystyle s<t$,
$\displaystyle s\to-\infty$, we get

$\displaystyle\displaystyle\infty>u(t,x)$
$\displaystyle\displaystyle\geq\int_{D_{n}}u(s,y)p_{t-s}^{D_{n}}(x,y)\,dy\geq\int_{U}u(s,y)p_{t-s}^{D_{n}}(x,y)\,dy\approx\int_{U}u(s,y)e^{-\lambda_{1}(D_{n})(t-s)}\,dy$
$\displaystyle\displaystyle=e^{(-\lambda_{1}(D_{n})+\lambda_{1}(U))(t-s)}\int_{U}u(s,y)e^{-\lambda_{1}(U)(t-s)}\,dy$
$\displaystyle\displaystyle\gtrsim
e^{(-\lambda_{1}(D_{n})+\lambda_{1}(U))(t-s)}\int_{U}u(s,y)p_{t-s}^{U}(x,y)\,dy.$

By (5.14), we must have $\displaystyle a=0$. ∎

###  5.3. Caloric functions are continuous

This subsection is devoted to proving that caloric functions are continuous,
hence locally bounded. The proof is based on certain estimates for the kernel
$\displaystyle J^{D}$, which may be of independent interest. Let us note in
passing that bounded caloric functions are known to be locally Hölder
continuous [30, Theorem 4.14].

######  Proposition 5.15.

Assume that $\displaystyle u$ is a nonnegative caloric function in
$\displaystyle(T_{0},T_{1})\times D$ for some $\displaystyle T_{0}<T_{1}$.
Then, $\displaystyle u$ is continuous and locally bounded therein.

We fix arbitrary $\displaystyle(t_{0},x_{0})\in(T_{0},T_{1})\times D$,
$\displaystyle r\in(0,\delta_{D}(x_{0})/2)$, and let $\displaystyle
B_{\rho}=B(x_{0},\rho)$ for $\displaystyle\rho>0$. We first establish some
basic facts about the lateral Poisson kernel. With a slight conflict of
notation, we introduce the Euclidean distance between sets $\displaystyle
A,B\in\mathbb{R}^{d}$,

$\displaystyle d(A,B):=\inf\\{|b-a|:\;a\in A,b\in B\\}.$

######  Lemma 5.16.

Let $\displaystyle D$ be a Lipschitz open set, $\displaystyle U\subset\subset
D$, and $\displaystyle 0<T<\infty$. Then,

(5.15) $\displaystyle\displaystyle J^{D}(t,x,s,z)\approx
J^{D}(t,x_{0},s,z),\quad x\in U,\ z\in D^{c},\ 0<t-s<T,$

and

(5.16) $\displaystyle\displaystyle J^{D}(t,x,s,z)\lesssim
J^{D}(t^{\prime},x,s,z),\quad x\in U,\ z\in D^{c},\ 0<t-s\leq t^{\prime}-s<T,$

with the comparability constants depending only on $\displaystyle
d,\alpha,\underline{D},d(U,D^{c})$, and $\displaystyle T$.

###### Proof.

Let $\displaystyle U^{\prime}$ be such that $\displaystyle U\subset\subset
U^{\prime}\subset\subset D$. We pick $\displaystyle U^{\prime}$ so that the
constants below depend only on $\displaystyle D$ and $\displaystyle U$, e.g.,
by assuming $\displaystyle d(U,D^{c})/2\geq d(U^{\prime},D^{c})\geq
d(U,D^{c})/3$. We first prove (5.15). By (2.10),

$\displaystyle\displaystyle J^{D}(t,x,s,z)$
$\displaystyle\displaystyle=\int_{D}p_{t-s}^{D}(x,y)\nu(y,z)\,dy$
$\displaystyle\displaystyle\approx\mathbb{P}^{x}(\tau_{D}>t-s)\int_{D}p_{t-s}(x,y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy$
(5.17)
$\displaystyle\displaystyle\approx\mathbb{P}^{x_{0}}(\tau_{D}>t-s)\bigg{(}\int_{D\setminus
U^{\prime}}+\int_{U^{\prime}}\bigg{)}p_{t-s}(x,y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy,$

with constants depending on $\displaystyle d,\alpha,\underline{D},d(U,D^{c})$,
and $\displaystyle T$. For $\displaystyle y\in D\setminus U^{\prime}$,
$\displaystyle|x-y|\approx|x_{0}-y|$, so by (2.7),

$\displaystyle\displaystyle\int_{D\setminus
U^{\prime}}p_{t-s}(x,y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy\approx\int_{D\setminus
U^{\prime}}p_{t-s}(x_{0},y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy.$

For $\displaystyle y\in U^{\prime}$,
$\displaystyle\mathbb{P}^{y}(\tau_{D}>t-s)\approx 1$ and
$\displaystyle\nu(y,z)\approx\nu(x_{0},z)$. Using this and the fact that
$\displaystyle U\subset\subset U^{\prime}$, we find that

$\displaystyle\displaystyle\int_{U^{\prime}}p_{t-s}(x,y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy$
$\displaystyle\displaystyle\approx\nu(x_{0},z)\int_{U^{\prime}}p_{t-s}(x,y)\,dy\approx\nu(x_{0},z)\int_{U^{\prime}}p_{t-s}(x_{0},y)\,dy$
$\displaystyle\displaystyle\approx\int_{U^{\prime}}p_{t-s}(x_{0},y)\mathbb{P}^{y}(\tau_{D}>t-s)\nu(y,z)\,dy.$

Coming back to (5.17), we obtain (5.15). We now proceed to proving (5.16). We
split in a similar way:

$\displaystyle\displaystyle
J^{D}(t,x,s,z)=\bigg{(}\int_{U^{\prime}}+\int_{D\setminus
U^{\prime}}\bigg{)}p_{t-s}^{D}(x,y)\nu(y,z)\,dy.$

By Lemma B.1,

$\displaystyle\displaystyle\int_{D\setminus
U^{\prime}}p_{t-s}^{D}(x,y)\nu(y,z)\,dy\lesssim\int_{D\setminus
U^{\prime}}p_{t^{\prime}-s}^{D}(x,y)\nu(y,z)\,dy$

For the integral over $\displaystyle U^{\prime}$ we use:(2.10)

$\displaystyle\displaystyle\int_{U^{\prime}}p_{t-s}^{D}(x,y)\nu(y,z)\,dy\approx\nu(x_{0},z)\int_{U^{\prime}}p_{t-s}^{D}(x,y)\,dy\approx\nu(x_{0},z)\int_{U^{\prime}}p_{t-s}(x,y)\mathbb{P}^{x}(\tau_{D}>t-s)\mathbb{P}^{y}(\tau_{D}>t-s)\,dy.$

For $\displaystyle w\in U^{\prime}$ and $\displaystyle 0<t-s<T$, we have
$\displaystyle\mathbb{P}^{w}(\tau_{D}>t-s)\approx 1$ and by (2.7),
$\displaystyle\int_{U^{\prime}}p_{t-s}(x,y)\,dy\approx 1$, with comparability
constants depending only on $\displaystyle T,U^{\prime}$, and
$\displaystyle\underline{D}$. It follows that

$\displaystyle\displaystyle\nu(x_{0},z)\int_{U^{\prime}}p_{t-s}(x,y)\mathbb{P}^{x}(\tau_{D}>t-s)\mathbb{P}^{y}(\tau_{D}>t-s)\,dy$
$\displaystyle\displaystyle\approx$
$\displaystyle\displaystyle\,\nu(x_{0},z)\int_{U^{\prime}}p_{t^{\prime}-s}(x,y)\mathbb{P}^{x}(\tau_{D}>t^{\prime}-s)\mathbb{P}^{y}(\tau_{D}>t^{\prime}-s)\,dy$
$\displaystyle\displaystyle\approx$
$\displaystyle\displaystyle\int_{U^{\prime}}p_{t^{\prime}-s}^{D}(x,y)\nu(y,z)\,dy,$

which ends the proof. ∎

###### Proof of Proposition 5.15.

We will show continuity at the fixed point $\displaystyle(t_{0},x_{0})$. Let
$\displaystyle x\in B_{r/2}$, $\displaystyle t_{1}\in(T_{0},t_{0})$ and
$\displaystyle t\in(t_{1},T_{1})$, so that $\displaystyle
T_{1}<t_{1}<t<T_{0}$. We have

$\displaystyle\displaystyle u(t,x)$
$\displaystyle\displaystyle=\int_{B_{r}}u(t_{1},y)p_{t-t_{1}}^{B_{r}}(x,y)\,dy+\int_{t_{1}}^{t}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t,x,\tau,z)\,dz\,d\tau,$
$\displaystyle\displaystyle u(t_{0},x_{0})$
$\displaystyle\displaystyle=\int_{B_{r}}u(t_{1},y)p_{t_{0}-t_{1}}^{B_{r}}(x,y)\,dy+\int_{t_{1}}^{t_{0}}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t_{0},x,\tau,z)\,dz\,d\tau.$

Since $\displaystyle u$ is nonnegative and caloric, all integrals above are
finite. For $\displaystyle(t,x)$ sufficiently close to
$\displaystyle(t_{0},x_{0})$, we have $\displaystyle
p_{t-t_{1}}^{B_{r}}(x,y)\approx p_{t_{0}-t_{1}}^{B_{r}}(x_{0},y)$ uniformly in
$\displaystyle y$. Therefore, by the dominated convergence theorem,

$\displaystyle\displaystyle\int_{B_{r}}u(t_{1},y)p_{t-t_{1}}^{B_{r}}(x,y)\,dy\mathop{\longrightarrow}\limits_{(t,x)\to(t_{0},x_{0})}\int_{B_{r}}u(t_{1},y)p_{t_{0}-t_{1}}^{B_{r}}(x_{0},y)\,dy.$

Therefore it remains to show that

(5.18)
$\displaystyle\displaystyle\int_{t_{1}}^{t}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t,x,\tau,z)\,dz\,d\tau\mathop{\longrightarrow}\limits_{(t,x)\to(t_{0},x_{0})}\int_{t_{1}}^{t_{0}}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t_{0},x_{0},\tau,z)\,dz\,d\tau.$

Assume that $\displaystyle t>t_{0}$ (we skip the other case, as it is
similar). Then,

$\displaystyle\displaystyle\bigg{|}\int_{t_{1}}^{t}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t,x,\tau,z)\,dz\,d\tau-\int_{t_{1}}^{t_{0}}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t_{0},x_{0},\tau,z)\,dz\,d\tau\bigg{|}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\int_{t_{1}}^{t_{0}}\int_{B_{r}}u(\tau,z)|J^{B_{r}}(t,x,\tau,z)-J^{B_{r}}(t_{0},x_{0},\tau,z)|\,dz\,d\tau+\int_{t_{0}}^{t}\int_{B_{r}}u(\tau,z)J^{B_{r}}(t,x,\tau,z)\,dz\,d\tau=:I_{1}+I_{2}.$

By Lemma 5.16, we have $\displaystyle J^{B_{r}}(t,x,\tau,z)\lesssim
J^{B_{r}}(t_{0}+\varepsilon,x_{0},\tau,z)$ for $\displaystyle
t_{1}\leq\tau\leq t\leq t_{0}+\varepsilon$, $\displaystyle x\in B_{r/2}$, and
$\displaystyle z\in B_{r}^{c}$. Therefore by the dominated convergence
theorem, $\displaystyle I_{2}\to 0$. Furthermore, by the properties of
$\displaystyle p_{t}^{B_{r}}$ and the dominated convergence theorem, it is
easy to see that $\displaystyle J^{B_{r}}(\cdot,\cdot,\tau,z)$ is continuous
on $\displaystyle(\tau,\infty)\times B_{r}$ for all
$\displaystyle\tau\in\mathbb{R}$ and $\displaystyle z\in D^{c}$. Therefore,
using the bounds of Lemma 5.16 and the dominated convergence theorem once
again, we find that $\displaystyle I_{1}\to 0$ as well. This ends the proof. ∎

##  6\. Representation of caloric functions in Lipschitz open sets

We first discuss the representation for functions caloric on
$\displaystyle[0,T)\times D$, where the meaning of the initial condition is
clearer. We then use this case to resolve the situation of functions caloric
in $\displaystyle(0,T)\times D$.

###  6.1. Functions caloric up to time $\displaystyle 0$

######  Lemma 6.1.

Assume that $\displaystyle u$ is a nonnegative caloric function in
$\displaystyle\dot{D}:=[0,T)\times D$. Then there exists a unique
decomposition $\displaystyle u=r+s$, where $\displaystyle r$ is regular
caloric in $\displaystyle\dot{D}$ and $\displaystyle s$ in singular caloric in
$\displaystyle\dot{D}$.

###### Proof.

Let $\displaystyle t<T$. Since $\displaystyle u$ has the mean value property
in every $\displaystyle\dot{D}_{n}=(0,t)\times D_{n}$ (see (2.1)), we have

$\displaystyle\displaystyle
u(t,x)=\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{\dot{D}_{n}}})=:i_{n}(t,x)+l_{n}(t,x)+s_{n}(t,x),$

where

$\displaystyle\displaystyle
i_{n}(t,x)=\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}_{n}}})\,;\,\tau_{D_{n}}>t],$
$\displaystyle\displaystyle
l_{n}(t,x)=\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}_{n}}})\,;\,\tau_{D_{n}}<t,\,\tau_{D_{n}}=\tau_{D}],$
$\displaystyle\displaystyle
s_{n}(t,x)=\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}_{n}}})\,;\,\tau_{D_{n}}<t,\,\tau_{D_{n}}<\tau_{D}].$

We let $\displaystyle n\to\infty$. By the monotone convergence, we get

$i_{n}(t,x)=\mathbb{E}^{(t,x)}[u(\dot{X}_{t})\,;\,\tau_{D_{n}}>t]\nearrow\mathbb{E}^{(t,x)}[u(\dot{X}_{t})\,;\,\tau_{D}>t]=:i(t,x),$

and by [11, (5.40)],

$l_{n}(t,x)=\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}}})\,;\,\tau_{D}<t,\,\tau_{D_{n}}=\tau_{D}]\nearrow\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}}})\,;\,\tau_{D}<t]=:l(t,x),$

the limits being finite because all $\displaystyle i_{n}$, $\displaystyle
l_{n}$, and $\displaystyle s_{n}$ are nonnegative. So, $\displaystyle
s_{n}(t,x)$ converges to some $\displaystyle s(t,x)$. Since $\displaystyle
r(t,x):=i(t,x)+l(t,x)=\mathbb{E}^{(t,x)}u(\dot{X}_{\tau_{\dot{D}}})$,
$\displaystyle r$ is regular caloric. By inspecting the definition of
$\displaystyle s_{n}$, we find that $\displaystyle s$ is singular caloric:
indeed, if $\displaystyle X_{t}$ starts from $\displaystyle x\in D^{c}$, then
the event $\displaystyle\tau_{D_{n}}<\tau_{D}$ has probability $\displaystyle
0$, so $\displaystyle s_{n}(t,x)=0$ for $\displaystyle x\in D^{c}$, and if
$\displaystyle\dot{X}$ starts from $\displaystyle(0,x)$, $\displaystyle x\in
D$, then $\displaystyle s_{n}(0,x)=0$ because $\displaystyle\tau_{D_{n}}\geq
0$.

Assume that there is another decomposition $\displaystyle
u=r^{\prime}+s^{\prime}$. Since $\displaystyle s^{\prime}=s=0$ on
$\displaystyle D^{p}$, we have that $\displaystyle r-r^{\prime}=0$ on
$\displaystyle D^{p}$ as well and therefore $\displaystyle r-r^{\prime}=0$ in
$\displaystyle\dot{D}$, because $\displaystyle r-r^{\prime}$ is regular
caloric on $\displaystyle\dot{D}$. ∎

We next give an integral representation for the singular caloric part, with
the use of the parabolic Martin kernel. We first prove the following technical
result.

######  Lemma 6.2.

Let $\displaystyle x\in D$ and $\displaystyle 0<\varepsilon<T$ be fixed. Then
there exists a modulus of continuity $\displaystyle\omega$, independent of
$\displaystyle y$ and $\displaystyle t\in[\varepsilon,T]$, such that for
$\displaystyle n$ large we have

(6.1)
$\displaystyle\displaystyle\bigg{|}\frac{p_{t}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}-\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)}\bigg{|}\leq\omega\bigg{(}\frac{1}{n}\bigg{)},\quad
y\in D_{n},\ t\in[\varepsilon,T].$

###### Proof.

First note that the expression on the right-hand side of (6.1) converges to 0
as $\displaystyle n\to\infty$ for every fixed $\displaystyle y\in D$ (the
expression is considered only when $\displaystyle 1/n<\delta_{D}(y)$). In
order to get (6.1) we will show that the convergence in uniform by using the
Arzelà–Ascoli theorem. Indeed, by Theorem 1.2, we find that
$\displaystyle\overline{D_{n}}\ni y\mapsto
p_{t}^{D_{n}}(x,y)/\mathbb{P}^{y}(\tau_{D_{n}}>1)$ are uniformly Hölder
continuous for $\displaystyle n$ large and $\displaystyle
t\in[\varepsilon,T]$. Furthermore, it is well-known that a Hölder continuous
function in $\displaystyle\overline{D_{n}}$ can be extended to a function on
$\displaystyle\overline{D}$ with the same Hölder regularity, see, e.g., Banach
[4, IV (7.5)]. If we denote the corresponding extensions by $\displaystyle
f_{n}$ then by the Arzelà–Ascoli theorem, we find that

$\displaystyle\displaystyle\bigg{|}f_{n}(t,y)-\frac{p_{t}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)}\bigg{|}\leq\omega\bigg{(}\frac{1}{n}\bigg{)},\quad
y\in D,\ t\in[\varepsilon,T].$

In particular, (6.1) follows. ∎

######  Theorem 6.3.

Assume that $\displaystyle u$ is singular caloric in $\displaystyle[0,T)\times
D$. Then there exists a nonnegative Borel measure $\displaystyle\mu$ on
$\displaystyle\partial D\times[0,T)$ such that

(6.2) $u(t,x)=\int_{[0,t)}\int_{\partial D}\eta_{t-s,Q}(x)\mu(dQ\,ds),\quad
x\in D,\ t\in(0,T).$

###### Proof.

Let $\displaystyle D_{n}$ be as in Lemma 6.1 and let $\displaystyle N$ be
large enough, so that $\displaystyle x,x_{0}\in D_{N}$. Since $\displaystyle
u$ is singular caloric, for natural $\displaystyle n>N$ we have

$\displaystyle\displaystyle u(t,x)$
$\displaystyle\displaystyle=\mathbb{E}^{(t,x)}[u(\dot{X}_{\tau_{\dot{D}_{n}}})\,;\,\tau_{D_{n}}<t,X_{\tau_{D_{n}}}\in
D\setminus D_{n}]$ $\displaystyle\displaystyle=\int_{0}^{t}\int_{D\setminus
D_{n}}u(s,z)\int_{D}p_{t-s}^{D_{n}}(x,y)\nu(y,z)\,dy\,dz\,ds$
$\displaystyle\displaystyle=\int_{0}^{t}\int_{D}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\int_{D\setminus
D_{n}}\mathbb{P}^{y}(\tau_{D_{n}}>1)u(s,z)\nu(y,z)\,dz\,dy\,ds.$

We define

$\displaystyle\mu_{n}(dy\,ds)=\int_{D\setminus
D_{n}}\mathbb{P}^{y}(\tau_{D_{n}}>1)u(s,z)\nu(y,z)\,dz\,dy\,ds.$

Note that by (2.10), if we fix $\displaystyle\theta>0$, then we have
$\displaystyle\mathbb{P}^{y}(\tau_{D_{n}}>1)\lesssim
p_{s+\theta}^{D_{n}}(x_{0},y)$ uniformly in $\displaystyle s\in(0,t)$.
Therefore, since $\displaystyle u$ is caloric, for $\displaystyle\theta$
sufficiently small we have

$\displaystyle\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}\mu_{n}(dy\,ds)\lesssim\int_{0}^{t}\int_{\mathbb{R}^{d}}\
\int_{D\setminus
D_{n}}p_{t+\theta-s}^{D_{n}}(x_{0},y)u(s,z)\nu(y,z)\,dz\,dy\,ds\leq
u(x_{0},t+\theta),$

which means that the masses of $\displaystyle\mu_{n}$ are uniformly bounded.
With this notation we have

$\displaystyle\displaystyle
u(t,x)=\int_{0}^{t}\int_{D}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\,\mu_{n}(dy\,ds).$

The goal is then to show that the right-hand side converges to the right-hand
side of (6.2). To this end we will isolate small times and look separately at
$\displaystyle D_{N}$ and $\displaystyle D\setminus D_{N}$.

Note that all $\displaystyle\mu_{n}$ are supported in $\displaystyle
D\times[0,T]$, so the sequence $\displaystyle(\mu_{n})$ is tight and we can
extract a subsequence $\displaystyle\mu_{n_{k}}$ converging weakly to
$\displaystyle\mu$. Furthermore, for every $\displaystyle U\subset\subset D$
and $\displaystyle 0<t<T$, we have that $\displaystyle\mu_{n}(U\times[0,t])\to
0$ as $\displaystyle n\to\infty$, so
$\displaystyle\mu|_{\overline{D}\times[0,T)}$ must be concentrated on
$\displaystyle\partial D\times[0,T)$.

Since for $\displaystyle y\in D_{N}$ we have $\displaystyle
p_{t-s}^{D_{n}}(x,y)\approx p_{t-s}^{D}(x,y)$ for $\displaystyle n>N+1$, we
find that

(6.3)
$\lim\limits_{n\to\infty}\int_{0}^{t}\int_{D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\mu_{n}(dy\,ds)\lesssim\lim\limits_{n\to\infty}\int_{0}^{t}\int_{D\setminus
D_{n}}u(s,z)\int_{D_{N}}p_{t-s}^{D}(x,y)\nu(y,z)\,dy\,dz\,ds=0.$

We will now show that there exists a modulus of continuity
$\displaystyle\omega$ independent of $\displaystyle n$ such that

(6.4)
$\int_{t-\epsilon}^{t}\int_{D}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\mu_{n}(dy\,ds)<\omega(\epsilon).$

To this end we will show that the left-hand side converges to 0 as
$\displaystyle\epsilon\to 0^{+}$ for each $\displaystyle n>N$, and that it is
nonincreasing with respect to $\displaystyle n$ for each (small)
$\displaystyle\epsilon$. By the definition of $\displaystyle\mu_{n}$ and the
fact that $\displaystyle u$ is caloric,

(6.5)
$\displaystyle\displaystyle\int_{t-\epsilon}^{t}\int_{D}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\mu_{n}(dy\,ds)$
$\displaystyle\displaystyle=\int_{t-\epsilon}^{t}\int_{D\setminus
D_{n}}J^{D_{n}}(t,x,s,z)u(s,z)\,dz\,ds$
$\displaystyle\displaystyle=u(t,x)-\int_{D_{n}}p_{\epsilon}^{D_{n}}(x,y)u(t-\epsilon,y)\,dy.$

The last expression converges to 0 for $\displaystyle\varepsilon\to 0^{+}$ for
all fixed $\displaystyle n$, because $\displaystyle u$ is continuous in both
variables, and it is nonincreasing with respect to $\displaystyle n$ because
of the domain monotonicity. This proves (6.4).

Note also that the right-hand side of (6.2) is finite because
$\displaystyle\mu$ is a finite measure and $\displaystyle\eta_{s,Q}(x)$ is
bounded in $\displaystyle s$ and $\displaystyle Q$ for fixed $\displaystyle
x$. Therefore,

(6.6) $\displaystyle\displaystyle\lim\limits_{\epsilon\to
0^{+}}\int_{[t-\epsilon,t)}\int_{\partial D}\eta_{t-s,Q}(x)\mu(dQ\,ds)=0.$

By (6.3),(6.4), and (6.6), for any $\displaystyle\delta>0$ there exist
$\displaystyle\epsilon$ (small) and $\displaystyle N_{0}$ (large) such that
for $\displaystyle n>N_{0}$,

$\displaystyle\displaystyle\bigg{|}\int_{0}^{t}\int_{D}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\,\mu_{n}(dy\,ds)-\int_{[0,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\,\bigg{|}\int_{[t-\epsilon,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}+\bigg{|}\int_{t-\epsilon}^{t}\int_{D\setminus
D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\mu_{n}(dy\,ds)\bigg{|}+\bigg{|}\int_{0}^{t}\int_{D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\mu_{n}(dy\,ds)\bigg{|}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\,\bigg{|}\int_{0}^{t-\epsilon}\int_{D\setminus
D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\,\mu_{n}(dy\,ds)-\int_{[0,t-\epsilon)}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\,3\delta+\bigg{|}\int_{0}^{t-\epsilon}\int_{D\setminus
D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\,\mu_{n}(dy\,ds)-\int_{[0,t-\epsilon)}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}.$

Furthermore, if $\displaystyle N_{0}$ is large enough, then by Lemma 6.2,

$\displaystyle\displaystyle\bigg{|}\int_{0}^{t-\epsilon}\int_{D\setminus
D_{N}}\frac{p_{t-s}^{D_{n}}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}\,\mu_{n}(dy\,ds)-\int_{[0,t-\epsilon]}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\,\delta+\bigg{|}\int_{0}^{t-\epsilon}\int_{D\setminus
D_{N}}\frac{p_{t-s}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D}>1)}\,\mu_{n}(dy\,ds)-\int_{[0,t-\epsilon]}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds)\bigg{|}.$

By Lemma C.1,
$\displaystyle\mu_{n}\cdot\textbf{1}_{D\times[0,t-\varepsilon]}\to\mu\textbf{1}_{D\times[0,t-\varepsilon)}$
weakly. By Corollary 3.4,
$\displaystyle(s,y)\mapsto\frac{p_{t-s}^{D}(x,y)}{\mathbb{P}^{y}(\tau_{D_{n}}>1)}$
is in $\displaystyle C([0,t-\epsilon]\times\overline{D})$. So, the last
expression is smaller than $\displaystyle 2\delta$ for $\displaystyle n$ large
enough, which ends the proof. ∎

######  Theorem 6.4.

The measure $\displaystyle\mu$ obtained in Theorem 6.3 is unique.

###### Proof.

Following [11, 22], we start by showing that the measures
$\displaystyle\mu_{n}^{Q}$ corresponding to $\displaystyle\eta_{t,Q}$ converge
to $\displaystyle\delta_{Q}\otimes\delta_{0}$ for $\displaystyle t>0$,
$\displaystyle Q\in\partial D$. To this end, fix $\displaystyle Q\in\partial
D$ and let

$\displaystyle\displaystyle\mu_{n}^{Q}(y,s)=\mathbb{P}^{y}(\tau_{D_{n}}>1)\int_{D\setminus
D_{n}}\eta_{s,Q}(z)\nu(y,z)\,dz,\quad s>0,\ y\in\mathbb{R}^{d}.$

By Lemma 3.5,
$\displaystyle\mu_{n}^{Q}((B(Q,\varepsilon)\times[0,\varepsilon))^{c})\to 0$
as $\displaystyle n\to\infty$, for any $\displaystyle\varepsilon>0$. So,
$\displaystyle\mu_{n}$ converges weakly to
$\displaystyle\delta_{Q}\otimes\delta_{0}$.

Now, let $\displaystyle u$ be a singular caloric function and assume that

$\displaystyle\displaystyle u(t,x)=\int_{[0,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\mu(dQ\,ds).$

Let $\displaystyle\mu_{n}(y,s)=\int_{D\setminus
D_{n}}\mathbb{P}^{y}(\tau_{D}>1)u(s,z)\nu(y,z)\,dz$. By Fubini–Tonelli,

$\displaystyle\displaystyle\mu_{n}(y,s)$
$\displaystyle\displaystyle=\int_{D\setminus
D_{n}}\mathbb{P}^{y}(\tau_{D}>1)\nu(y,z)\int_{[0,s)}\int_{\partial
D}\eta_{s-\tau,Q}(z)\mu(dQ\,d\tau)\,dz$
$\displaystyle\displaystyle=\int_{[0,s)}\int_{\partial
D}\mu_{n}^{Q}(y,s-\tau)\mu(dQ\,d\tau).$

Let $\displaystyle f\in C_{b}(\overline{D}\times[0,T])$. Then,

$\displaystyle\displaystyle\int_{0}^{t}\int_{D}f(y,s)\mu_{n}(y,s)\,dy\,ds$
$\displaystyle\displaystyle=\int_{0}^{t}\int_{D}f(y,s)\int_{[0,s)}\int_{\partial
D}\mu_{n}^{Q}(y,s-\tau)\mu(dQ\,d\tau)\,dy\,ds$
$\displaystyle\displaystyle=\int_{[0,t)}\int_{\partial
D}\int_{0}^{t-\tau}\int_{D}f(y,s+\tau)\mu_{n}^{Q}(y,s)\,dy\,ds\,\mu(dQ\,d\tau).$

Since $\displaystyle\mu_{n}^{Q}\implies\delta_{Q}\otimes\delta_{0}$, the above
integral with respect to $\displaystyle dy\,ds$ converges to $\displaystyle
f(Q,\tau)$. Therefore, by the dominated convergence theorem,

$\displaystyle\displaystyle\int_{0}^{t}\int_{D}f(y,s)\mu_{n}(y,s)\,dy\,ds\mathop{\longrightarrow}_{n\to\infty}\int_{[0,t)}\int_{\partial
D}f(Q,s)\mu(dQ\,ds),$

which means that
$\displaystyle\mu_{n}\implies\mu\cdot\textbf{1}_{\overline{D}\times[0,t)}$.
Thus, $\displaystyle\mu$ is uniquely determined by $\displaystyle u$. ∎

###  6.2. Functions caloric on $\displaystyle(0,T)\times D$

######  Theorem 6.5.

Assume that $\displaystyle u$ is caloric on $\displaystyle(0,T)\times D$ and
let $\displaystyle g=u|_{D^{c}}$. Then there exist unique bounded nonnegative
measures $\displaystyle\mu$ on $\displaystyle[0,T)\times\partial D$ and
$\displaystyle\mu_{0}$ on $\displaystyle D$ such that for all $\displaystyle
0<t<T$ and $\displaystyle x\in D$,

$\displaystyle\displaystyle
u(t,x)=P_{t}^{D}\mu_{0}(x)+\int_{[0,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\,\mu(dQ\,ds)+\int_{0}^{t}\int_{D^{c}}g(s,z)J^{D}(t,x,s,z)\,dz\,ds.$

###### Proof.

By the results of the previous subsection, there is a nonnegative measure
$\displaystyle\mu$ on $\displaystyle(0,T)\times\partial D$ such that for all
$\displaystyle 0<\varepsilon<t<T$ and $\displaystyle x\in D$,

$\displaystyle\displaystyle
u(t,x)=P_{t-\varepsilon}^{D}u(\varepsilon,\cdot)(x)+\int_{[\varepsilon,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\,\mu(dQ\,ds)+\int_{\varepsilon}^{t}\int_{D^{c}}g(s,z)J^{D}(t,x,s,z)\,dz\,ds.$

By nonnegativity, and the monotone convergence theorem, the last two integrals
increase and converge as $\displaystyle\varepsilon\to 0^{+}$, so that

$\displaystyle\displaystyle u(t,x)=\lim\limits_{\varepsilon\to
0^{+}}P_{t-\varepsilon}^{D}u(\varepsilon,\cdot)(x)+\int_{(0,t)}\int_{\partial
D}\eta_{t-s,Q}(x)\,\mu(dQ\,ds)+\int_{0}^{t}\int_{D^{c}}g(s,z)J^{D}(t,x,s,z)\,dz\,ds,$

where the remaining limit exists and the expression under it decreases. Since
$\displaystyle p_{t-\varepsilon}^{D}(x,y)\approx p_{t}^{D}(x,y)$ and
$\displaystyle p_{t}^{D}(x,\cdot)\approx 1$ for any $\displaystyle
U\subset\subset D$ we find that $\displaystyle u(\varepsilon,\cdot)$ have
bounded integral on $\displaystyle U$. Therefore, by the Prokhorov theorem,