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

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 $\alpha$-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.

1 Introduction

Let $0< \alpha < 2$ and $d \ge 2$. For $u\in C^2_b(\mathbb {R}^d)$, define

$$\begin{aligned} (-\Delta )^{\alpha /2} u(x) {:}{=} \lim \limits _{\varepsilon \rightarrow 0^+} \int _{|x-y|>\varepsilon } (u(x) - u(y))\nu (x,y)\, dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

where $\nu (x,y) = c_{d,\alpha }|x-y|^{-d-\alpha }$, and denote $\Delta ^{\alpha /2} {:}{=} -(-\Delta )^{\alpha /2}$. Let $D \subset \mathbb {R}^{d}$ be a nonempty bounded open Lipschitz set with localization radius $r_0\in (0,\infty )$ and Lipschitz constant $\lambda \in (0,\infty )$. One of our goals is to investigate the structure of nonnegative solutions to the initial-boundary value problem for the fractional heat equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu(t,x) = \Delta ^{\alpha /2}u(t,x),\quad &{}t\in (0,T),\ x\in D,\\ u(t,x) = g(t,x),\quad &{}t\in (0,T),\ x\in D^c,\\ u(0,x) = u_0(x),\quad &{}x\in D. \end{array}\right. } \end{aligned}$$

(1.1)

Solutions to (1.1) are called caloric functions. They are defined in terms of the mean value property for the space-time $\alpha$-stable Lévy process; we refer to Sect. 5 for details and connections with the classical notion of solution to (1.1). As shown by Bogdan [13] (see also Abatangelo [1] and Bogdan, Kulczycki, and Kwaśnicki [23]), nonnegative harmonic functions for the fractional Laplacian on D can be decomposed into a regular part, which can be recovered from the exterior values, and a singular part, vanishing outside of D and represented as an integral with respect to a finite measure on $\partial D$ of the (elliptic) Martin kernel for D and the fractional Laplacian. Our ultimate goal, which we complete in Sect. 6, is to give a counterpart of this decomposition for nonnegative caloric functions. In particular, in Theorems 6.3 and 6.4, we show that every nonnegative singular caloric function, i.e., such that $u_0=g=0$, can be expressed as integral with respect to the parabolic Martin kernel $\eta _{t,Q}(x)$:

$$\begin{aligned} u(t,x) = \int _{[0,t)} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds),\quad x\in D,\ t\in (0,T), \end{aligned}$$

(1.2)

with a unique finite Borel measure $\mu$ on $\partial D\times [0,T)$.

Singular caloric functions were recently represented by Chan, Gómez-Castro, and Vázquez [28] for domains more regular than Lipschitz, such as $C^{1,1}$ domains. While the authors of [28] 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 [48]), the results of [28] are not applicable in our setting. Another difference between [28] 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. Note that in our paper, the boundary data may be a measure; for example $\mu = \delta _{Q_0} \otimes \delta _0$ represents a fixed parabolic Martin kernel $\eta _{t,Q_0}(x)$. Furthermore, in Theorem 6.5, we demonstrate that even without a prescribed initial condition, $u(\varepsilon ,\cdot )$ converges to a measure $\mu _0$ on D as $\varepsilon \rightarrow 0^+$. This measure finitely integrates the function $x\mapsto \mathbb {P}^x(\tau _D>1)$ on D (see below), similar to the condition used in [28]. As a consequence, for general nonnegative caloric functions, we get the following representation:

$$\begin{aligned} u(t,x)&= P_t^D \mu _0(x) + \int _{[0,t)}\int _{\partial D} \eta _{t-s,Q}(x)\, \mu (dQ\, ds) \nonumber \\&\quad + \int _0^t\int _{D^c} g(s,z)J^D(t,x,s,z)\, dz\, ds, \end{aligned}$$

(1.3)

which is our first main result. Here, $P_t^D$ is the Dirichlet heat semigroup of D for $\Delta ^{\alpha /2}$, and $J^D$ is the so-called lateral Poisson kernel, see below for details. To obtain the representation (1.3), we prove several new boundary regularity results for the fractional Laplacian in Lipschitz sets, namely Theorems 1.2 and 1.4 and Corollaries 1.3 and 1.5. They are of independent interest and may be considered the second main contribution of the paper.

To prove the results, we utilize some basic probabilistic potential theory. Let $X = ( X_{t} )_{t \ge 0}$ be the isotropic $\alpha$-stable Lévy process in $\mathbb {R}^{d}$, see, e.g., Sato [60]. For $x \in \mathbb {R}^{d}$, we denote by $\mathbb {P}^{x}$ and $\mathbb {E}^{x}$ the probability and the expectation of the process starting from x, and $\mathbb {P}{:}{=}\mathbb {P}^{0}$, $\mathbb {E}{:}{=}\mathbb {E}^{0}$. We then consider

$$\begin{aligned} \tau _{D} {:}{=} \inf \{ s > 0 \, \ X_{s} \notin D\}, \end{aligned}$$

(1.4)

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

$$\begin{aligned} \mathbb {P}^x(\tau _D > t) = \int _D p_t^D(x,y)\, dy, \end{aligned}$$

where $p_t^D$ is the Dirichlet heat kernel of $\Delta ^{\alpha /2}$ in D (for details see Sect. 2). Furthermore, let $G_D$ be the (elliptic) Green function of $\Delta ^{\alpha /2}$ in D. We fix arbitrary $t_0\in (0,\infty )$ and $x_0\in D$, 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 $p_t^D$ by constructing a ratio that converges to a nontrivial limit at the boundary of D. Each of the following expressions will be called a parabolic Martin kernel:

$$\begin{aligned}&\eta _{t, Q}(x) {:}{=} \lim _{D\ni y \rightarrow Q} \frac{p_{t}^{D}(x,y)}{\mathbb {P}^{y} ( \tau _{D} > 1 )}, \end{aligned}$$

(1.5)

$$\begin{aligned}&\eta ^{x_0}_{t,Q}(x) {:}{=}\lim \limits _{D\ni y \rightarrow Q}\frac{p_t^D(x,y)}{G_D(x_0,y)}, \end{aligned}$$

(1.6)

$$\begin{aligned}&\widetilde{\eta }_{t,Q}(x) {:}{=} \lim \limits _{D\ni y\rightarrow Q} \frac{p_{t}^D(x,y)}{p_{t_0}^D(x_0,y)}. \end{aligned}$$

(1.7)

Here, $t>0$, $x\in D$, and $Q\in \partial D$. We recall that the heat kernel plays the role of the Green function for the heat equation, see, e.g., Doob [36], Watson [63], or Bogdan and Hansen [21, Subsection 9.4]. This might indicate that $\widetilde{\eta }$ is the canonical parabolic Martin kernel, however $\eta$ and $\eta ^{x_0}$ offer a more explicit description of the boundary behavior of $p_t^D$ and are more convenient to handle via the existing elliptic theory. If D is $C^{1,1}$, then one can also normalize $p^D_t$ by using $\delta _D(y)^{\alpha /2}$ with

$$\begin{aligned} \delta _D(y){:}{=} \inf \{|x-y|: x\in \partial D\}, \end{aligned}$$

see Chen, Kim, and Song [30]; see also [28]. The next result may be considered as a consequence and a follow-up of the approximate factorization (2.6) of $p_t^D$ by Bogdan, Grzywny, and Ryznar [19].

Theorem 1.1

Recall that $D\subset \mathbb {R}^d$ is open, bounded, and Lipschitz with localization radius $r_0$, Lipschitz constant $\lambda$, and reference point $x_0$ and time $t_0$. Then, the limits in (1.5), (1.6), and (1.7) exist for all $t>0$, $x\in D$, and $Q\in \partial D$. Furthermore, they are finite, strictly positive, continuous in t and x, and

$$\begin{aligned} \eta _{1, Q}(x)&\approx \mathbb {P}^{x} ( \tau _{D} > 1 ), \quad x\in D, \end{aligned}$$

(1.8)

$$\begin{aligned} \eta _{t + s, Q}(x)&= \int _{D} \eta _{t, Q}(z) p_{s}^{D}(z,x)\, dz,\quad 0<s,t<\infty ,\quad x\in D. \end{aligned}$$

(1.9)

The formula (1.8) is a sample of the more general estimates for $\eta$ 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 regularity¹ result. In the elliptic setting, for $G_D$, it is usually proved using the boundary Harnack principle. For solutions of parabolic equations like (1.1), we may utilize the elliptic results after expressing the numerators and denominators in (1.5), (1.6), and (1.7) as Green potentials. This is precisely our approach—it was used before by Bogdan, Palmowski, and Wang [24] for Lipschitz cones at the vertex. We further remark that an early version of proof of Theorem 1.1 for (1.5) has appeared in the PhD thesis of the first-named author [4].

To obtain the representation of nonnegative caloric functions, we refine Theorem 1.1 to ensure a uniform rate of convergence in (1.5). To this end, we extend the spatial domain of the functions in (1.5), (1.6), (1.7), by additionally defining, for $t>0$, $x\in D$, and $y\in D$,

$$\begin{aligned} \eta ^{x_0}_{t,y}(x) {:}{=}\frac{p_t^D(x,y)}{G_D(x_0,y)},\quad \eta _{t,y}(x) {:}{=} \frac{p_t^D(x,y)}{\mathbb {P}^y(\tau _D>1)},\quad \widetilde{\eta }_{t,y}(x) {:}{=} \frac{p_t^D(x,y)}{p_{t_0}^D(x_0,y)}. \end{aligned}$$

Theorem 1.2

Recall that $D\subset \mathbb {R}^d$ is open, bounded, and Lipschitz with localization radius $r_0$, Lipschitz constant $\lambda$, and reference point $x_0$ and time $t_0$. Fix $r_1\in (0,\infty )$ and $0<T_1<T_2<\infty$. For $x\in D$ and $t\in [T_1,T_2]$, $\eta$, $\eta ^{x_0}$, and $\widetilde{\eta }$ are Hölder continuous in y on $\overline{D}$, $\overline{D}$, and $\overline{D}{\setminus } B(x_0,r_1)$, respectively. The Hölder exponents and constants depend only on $d,\alpha ,\underline{D},T_1,T_2$ (for $\eta ^{x_0}$ also on $x_0,r_1$; for $\widetilde{\eta }$ also on $t_0,x_0$).

Here and below, we say constants depend on $\underline{D}$ if they depend only on $r_0,\lambda$, and an upper bound for $\textrm{diam}(D)$. Theorem 1.2 yields the following boundary regularity for the Dirichlet heat semigroup

$$\begin{aligned} P_t^Df(y){:}{=}\int _D p_t^D(x,y)f(x)\, dx. \end{aligned}$$

Corollary 1.3

Fix $r_1\in (0,\infty )$. Let $u_0\in L^1(D)$, $0<T_1<T_2<\infty$, and $t\in [T_1,T_2]$. Then, the functions

$$\begin{aligned} \frac{P_t^D u_0(y)}{G_D(x_0,y)},\quad \frac{P_t^D u_0(y)}{\mathbb {P}^y(\tau _D>1)},\quad \frac{P_t^Du_0(y)}{p_{t_0}^D(x_0,y)} \end{aligned}$$

are Hölder continuous in y on $\overline{D}{\setminus } B(x_0,r_1)$, $\overline{D}$, and $\overline{D}$ respectively. The Hölder exponents and constants depend only on $d,\alpha ,\underline{D},T_1,T_2$ (and $t_0,x_0$, $r_1$, where relevant).

Theorem 1.2 and Corollary 1.3 can be viewed as analogues of the boundary regularity result for $C^{1,1}$ open sets by Fernández-Real and Ros-Oton [39, Theorem 1.1 (b)], see also [40]. However, such regularity results for nonlocal equations are quite scarce for Lipschitz and less regular domains. That is, much is known about harmonic functions [12, 23, 48], but the first result for the Poisson equation ($\Delta ^{\alpha /2}u = -f$) appeared only recently but in the PDE literature the first results for the Dirichlet problem for the Poisson equation ($\Delta ^{\alpha /2}u = -f$) appeared only recently in the paper of Lian, Zhang, Li, and Hong [64, Theorem 3.11]; similar results were implicit in the probability literature, at least for bounded f, see [48, Theorem 2, Lemma 17] and [12, Lemma 3]. Other related works are by Ding and Zhang [65] and Borthagaray and Nochetto [27], but we note that [27, 64, 65] do not treat the $\textit{relative}$ boundary regularity, which is a stronger property. For regularity results in $C^{1,\gamma }$ domains with $\gamma \in (0,1)$, see, e.g., Abels and Grubb [2] or Dong and Ryu [35] 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 $D\subset \mathbb {R}^d$ is open, bounded, and Lipschitz with localization radius $r_0$, Lipschitz constant $\lambda$, and reference point $x_0$. Let $r>0$. There exists $p_0 = p_0(d,\alpha ,\underline{D},r)>1$ and constants $C\in (0,\infty )$ and $\sigma \in (0,1]$ depending only on $d,\alpha ,\underline{D},p,r$, such that for all $p\in [1,p_0)$,

$$\begin{aligned} \bigg \Vert \frac{G_D(y,\cdot )}{G_D(x_0,y)} - \frac{G_D(y',\cdot )}{G_D(x_0,y')}\bigg \Vert _{L^p(D)} \le C|y-y'|^{\sigma },\quad y,y'\in \overline{D}{\setminus }B(x_0,r). \end{aligned}$$

Recall that Green potentials $v(x) = G_Df(x){:}{=}\int _D G_D(x,y)f(y)dy$ solve the Dirichlet problem for the Poisson equation:

$$\begin{aligned} {\left\{ \begin{array}{ll}(-\Delta )^{\alpha /2} v(x) = f(x),\quad &{}x\in D,\\ \hspace{41pt} v(x) = 0,\quad &{}x\in D^c, \end{array}\right. } \end{aligned}$$

see [15]. Theorem 1.4 yields a boundary, or relative, Hölder estimate, as follows.

Corollary 1.5

Let $p>p_0/(p_0-1)$ and let $f\in L^p(D)$. Then, $G_D f(y)/G_D(x_0,y)$ is Hölder continuous in $\in D{\setminus } B(x_0,r)$ with Hölder constant and exponent depending only on $d,\alpha ,\underline{D},p,r$ and $\Vert f\Vert _{L^p(D)}$.

A similar result for $C^{1,1}$ domains was obtained by Ros-Oton and Serra [57] with explicit and sharp Hölder exponents. Our regularity results are far from being sharp in terms of $p_0$ and $\sigma$, but this is to be expected for Lipschitz sets—some insight about precise boundary behavior can be gained from the results on cones [6, 33, 55] or numerical considerations [38], but we do not pursue this point here. Note that Corollary 1.5 implies that the Green potentials have the same decay rate at the boundary as harmonic functions, without restrictions on the Lipschitz constant of D. This stands in sharp contrast to the case of local operators, where such comparability is known to be false if the Lipschitz constant of D is too large, see, e.g., [3, 59].

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 [31]. Here we focus on the mean-value property in cylinders, which seems adequate for the initial-exterior problem (1.1). 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 u satisfies the mean-value property $(0,T)\times D$, then u(t, x) can be recovered from the space-time isotropic $\alpha$-stable process $s\mapsto (t-s,X_s+x)$, which starts from (t, x) at time $s=0$, by computing the expectation of $u(t-s,X_s+x)$ at the place of the first exit of the process from $(0,T)\times D$. The exit can occur when $x+X_s$ leaves D before time t—in which case the exterior conditions affect the expectation—or when the time coordinate $t-s$ reaches 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 $(0,T)\times U$ for all open (relatively compact sets) $U\subset \subset D$. We refer to the book of Freidlin [41, 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 (1.5) indicates how the isotropic $\alpha$-stable process in D, conditioned on surviving at least time 1, behaves near the boundary of D. More precisely, it implies the existence of a “Yaglom limit”, see Theorem 3.7 below. Thanks to (1.9), $\eta _{t,Q}(y)$ may be understood as the entrance law for the killed process from Q into D, see Blumenthal [9]. This was used in [45, 53] 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 $G_D(x_0,\cdot )$ at the boundary. Then we approximate $p_t^D$ by Green potentials and obtain the limit in (1.5) 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 $p_t^D$ as a Green potential and we apply Theorem 1.4. We make use of the spectral theory to show that $p_t^D$ has regularity necessary for the proof; some ideas here were inspired by [28]. The boundary measure in the representation of singular caloric functions is obtained from an approximating sequence constructed via the lateral Poisson kernel. Our construction is quite different than the one in [28], in particular it does not use the inhomogeneous fractional heat equation. Needless to say, our results point out directions of development for other nonlocal operators and various classes of open sets.

The structure of the rest of the paper is as follows. Section 2 contains basic definitions and facts. In Sect. 3, we prove Theorem 1.1 and its consequences. In Sect. 4, we prove Theorems 1.4 and 1.2. In Sect. 5, we introduce the caloric functions and the parabolic Poisson kernel and study their properties. Then in Sect. 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 f and g, we write $f(x)\lesssim g(x)$, $x\in A$, if there is a number $C\in (0,\infty )$, referred to as constant, such that $f(x)\le C g(x)$, $x\in A$. We write $C=C(d,\alpha ,\ldots )$ if C is a constant depending only on $d,\alpha ,\ldots$, that is, C may be considered as a function of the parameters $d,\alpha ,\ldots$, but not of $x\in A$. We say that f and g are comparable and write $f\approx g$ if $f\lesssim g$ and $g\lesssim f$ (this notation was used in Sect. 1). We often use ${:}{=}$ and occasionally employ cursive for definitions.

2.1 Geometry

Let $B(x,r) {:}{=} \{y\in \mathbb {R}^d: |y-x| < r\}$. Recall that D is a Lipschitz open set with constant $\lambda \in (0,\infty )$ and localization radius $r_0\in (0,\infty )$. This means that for every $Q\in \partial D$ there is a rigid motion $R_Q$ and a Lipschitz function $f_Q:\mathbb {R}^{d-1}\rightarrow \mathbb {R}$ with Lipschitz constant $\lambda$, such that $R_Q(Q) = 0$ and $D\cap B(Q,r_0)=R_Q^{-1}(B(0,r_0)\cap \{y_d > f_Q(y_1,\ldots ,y_{d-1})\})$. For $r>0$, we let

$$\begin{aligned} D_{r} {:}{=} \{x\in D: \delta _D(x) > 1/r\}. \end{aligned}$$

(2.1)

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

$$\begin{aligned} \mathcal {A}_r(y) {:}{=}{\left\{ \begin{array}{ll} \{A\in D: B(A,\kappa r)\subseteq D\cap B(y,r)\},\quad &{}r\le r_0/2,\\ \{x_0\},\quad &{}r>r_0/2.\end{array}\right. } \end{aligned}$$

Lemma 2.1

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

Proof

Obviously, it suffices to consider $r\le r_0/2$. For $y\in \partial D$ the statement is true even with $\kappa$ replaced by $2\kappa = 1/(2\sqrt{1+\lambda ^2})$. Indeed, if we consider the interior right-circular cone with angle $\textrm{arccot}(\lambda )$ and vertex at y, then the point $A\in D$ on the axis of the cone such that $|A-y|=r$ satisfies $B(A,r/(2\sqrt{1+\lambda ^2})) \subseteq D\cap B(y,r)$. If $y\in D$ and $y\notin \mathcal {A}_r(y)$, then there is $Q\in \partial D$ with $|y-Q| = \delta _D(y) < \kappa r$ and $A\in D$ with

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

$\square$

Thus, by definition (see, e.g., [19]), D is $\kappa$-fat at each scale $r\in (0,r_0/2)$. We will denote by $A_r(y)$ an arbitrary point in $\mathcal {A}_r(y)$. The actual choice is unimportant in the sense that if $A_1,A_2\in \mathcal {A}_r(y)$ and $u\ge 0$ is harmonic in $B(A_1,\kappa r)$ and $B(A_2,\kappa r)$—see Definition 2.4 below—then we have the comparability $C^{-1}u(A_1)\le u(A_2)\le C u(A_1)$, where $C=C(d,\alpha )$; see the Harnack inequality in [14, Lemma 1], see also [15, Lemma 4.4].

For $x,y\in D$, let $r_{x,y} {:}{=} |x-y|\vee \delta _D(x)\vee \delta _D(y)$. Let $\mathcal {A}_{x,y} {:}{=} \{x_0\}$ if $r_{x,y}>r_0/32$, and otherwise let

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

Then, $\mathcal A_{x,y}$ is nonempty, see [48]. We denote by $A_{x,y}$ any point in $\mathcal {A}_{x,y}$. The actual choice is unimportant in the sense that under suitable assumptions on functions $u\ge 0$, there exists $C = C(d,\alpha ,\underline{D})$ such that for all $A_1,A_2\in \mathcal {A}_{x,y}$, $C^{-1}u(A_1)\le u(A_2)\le C u(A_1)$. See Remark 2.2, following (2.10).

2.2 Potential theory

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

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

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

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

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

$$\begin{aligned} \int _{\mathbb {R}^{d}} p_{t}(x) e^{i \langle x, \xi \rangle } \, dx = e^{-t |\xi |^{\alpha }}, \quad \xi \in \mathbb {R}^{d}. \end{aligned}$$

(2.2)

The associated semigroup of operators acts on, e.g., $u\in C_0(\mathbb {R}^d)$ as follows:

$$\begin{aligned} P_t u(x) {:}{=} \int _{\mathbb {R}^d} u(y) p_t(x,y) \, dy, \quad x\in \mathbb {R}^d,\ t\ge 0. \end{aligned}$$

We have the following scaling property as a consequence of (2.2):

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

(2.3)

Furthermore, there exists a constant c such that

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

see, e.g., [11, 26]. Thus, in short,

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

(2.4)

Recall that $\tau _D$ is the first exit time from D defined in (1.4). Since D is bounded, then $\tau _{D} < \infty$ almost surely, see, e.g., Pruitt [56]. The Dirichlet heat kernel $p_{t}^{D}(x,y)$ of D is defined by Hunt’s formula:

$$\begin{aligned} p_{t}^{D} (x,y) = p_{t}(x,y) - \mathbb {E}^{x} \big [p_{t - \tau _{D}} ( X_{\tau _{D}}, y)\,; \, \tau _{D} < t \big ], \end{aligned}$$

(2.5)

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

$$\begin{aligned} \mathbb {E}^{x} \big [p_{t - \tau _{D}} ( X_{\tau _{D}}, y)\,; \, \tau _{D}< t \big ] {:}{=} \int _{\{ \tau _{D} < t \}} p_{t - \tau _{D}} ( X_{\tau _{D}}, y )\, d\mathbb {P}^{x}. \end{aligned}$$

It is well known that $p_t^D(x,y)$ is jointly continuous, see Appendix A for more regularity properties. Since D is Lipschitz, it satisfies the exterior cone condition. Therefore, $\mathbb {P}^{x} ( \tau _{D} = 0) = 1$ for all $x \in D^{c}$ by Blumenthal’s zero–one law. In particular $p_{t}^{D}(x,y) = 0$ when x or y are outside of D. For bounded or nonnegative functions f we have

$$\begin{aligned} P_{t}^{D} f (x) = \int _{\mathbb {R}^{d}} f(y) p_{t}^{D} (x,y)\, dy = \mathbb {E}^{x} \big [f ( X_{t})\,; \, \tau _D>t \big ], \end{aligned}$$

see [32, Section 2]. We also note that

$$\begin{aligned} 0 \le p_{t}^{D} (x,y) = p_{t}^{D} (y,x) \le p_{t}(y - x) \end{aligned}$$

and $p_t^D$ satisfies the Chapman–Kolmogorov equations:

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

see [17, 30]. The following scaling property follows from (2.3),

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

By [19, Theorem 1], for every $T>0$ we have the approximate factorization:

$$\begin{aligned} p_{t}^{D} (x,y) \approx \mathbb {P}^{x} (\tau _{D}> t ) p_{t}(x,y)\mathbb {P}^{y} ( \tau _{D} > t ), \quad x,y \in D,\ t\in (0,T). \end{aligned}$$

(2.6)

If D is (open, bounded, and) $C^{1,1}$, then the (2.6) takes on a more explicit form [30]:

$$\begin{aligned} p_t^D(x,y) \approx \bigg (1\wedge \frac{\delta _D(x)^{\alpha /2}}{\sqrt{t}}\bigg )p_t(x,y)\bigg (1\wedge \frac{\delta _D(y)^{\alpha /2}}{\sqrt{t}}\bigg ),\quad x,y \in D,\ t\in (0,T).\nonumber \\ \end{aligned}$$

(2.7)

We also recall the large time estimates. Let $\lambda _1 = \lambda _1(D) >0$ be the first eigenvalue and $\varphi _1$ the first eigenfunction of the Dirichlet fractional Laplacian on D, see Sect. 2.3 below for more details. By the intrinsic ultracontractivity due to Kulczycki [49], for every $T>0$ we have

$$\begin{aligned} p_t^D(x,y) \approx e^{-\lambda _1 t}\varphi _1(x)\varphi _1(y),\quad x,y\in D,\ t\in (T,\infty ). \end{aligned}$$

(2.8)

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

$$\begin{aligned} p_t^D(x,y) \approx e^{-\lambda _1 t}\delta _D(x)^{\alpha /2}\delta _D(y)^{\alpha /2},\quad x,y\in D,\ t\in (T,\infty ), \end{aligned}$$

(2.9)

see [30, Theorem 1.1 (ii)]. We define the killing intensity of X on D as

$$\begin{aligned} \kappa _{D}(z){:}{=} \int _{D^{c}} \nu (z - y)\, dy,\quad z\in D. \end{aligned}$$

By [60, Theorem 31.5], $\Delta ^{\alpha /2}$ coincides with the generator of $X_t$ for the class $C^2_c(\mathbb {R}^d)$ of real-valued twice continuously differentiable functions compactly supported in $\mathbb {R}^d$.

The Green function of D is given by the formula:

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

In particular, $G_D(x,y) = 0$ if either $x \in D^{c}$ or $y\in D^c$. We note that $G_D$ is finite for all $x\ne y$ and by (2.5), $G_D(x,y)\le G_{\mathbb {R}^d}(x,y)= c|x-y|^{\alpha -d}$. For further reference, we recall the Green function estimates of Jakubowski [48, Theorem 1]: If we let

$$\begin{aligned} \Phi (x) {:}{=} G_D(x_0,x)\wedge 1, \end{aligned}$$

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

$$\begin{aligned} C^{-1}|x-y|^{\alpha -d}\frac{\Phi (x)\Phi (y)}{\Phi (A_{x,y})^2}\le G_D(x,y)\le C|x-y|^{\alpha -d}\frac{\Phi (x)\Phi (y)}{\Phi (A_{x,y})^2}\,,\qquad x,y\in D, \end{aligned}$$

(2.10)

see Sect. 2.1 for notation and the following remark.

Remark 2.2

We note that if $A_1,A_2\in \mathcal A_{x,y}$, then $\Phi (A_1)\approx \Phi (A_2)$; see [48, Lemma 13]. We also note that [48] uses an extra reference point $x_1$ to define $A_{x,y}$ for $r_{x,y}\ge r_0/32$, but the resulting values of $\Phi (A_{x,y})$ are trivially comparable in both settings. In particular, (2.10) remains true in the present (simplified) setting.

Remark 2.3

It is implicit in (2.6) and (2.8) that $\varphi _1(y)\approx \mathbb {P}^y(\tau _D>1)$, $y\in D$. Furthermore, by [19, Theorem 2], $\mathbb {P}^y(\tau _D>1)\approx \mathbb {E}^y \tau _D$, $y\in D$, and, by [48, Lemma 17], $\mathbb {E}^y \tau _D\approx \Phi (y)$, $y\in D$. Therefore,

$$\begin{aligned} \varphi _1(y)\approx \mathbb {P}^y(\tau _D>1)\approx \mathbb {E}^y \tau _D\approx \Phi (y),\quad y\in D. \end{aligned}$$

(2.11)

In our proofs, we mostly use the survival probability and $\Phi$, but we also refer to results stated in terms of $\varphi _1$ and the expected exit time.

We define the Green operator (or Green potential)

$$\begin{aligned} ( G_{D} f ) (x) {:}{=} \int _{D} G_{D}(x,y) f(y)\, dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

for integrable or nonnegative functions f. For $f\in L^1(D)$, the function $u{:}{=}G_Df$ is a distributional solution of $(-\Delta )^{\alpha /2} u = f$ in D, see [15, Proposition 3.13].

Definition 2.4

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

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

  $$\begin{aligned} u(x) = \mathbb {E}^{x} u ( X_{\tau _{B}} ) < \infty , \quad x \in B. \end{aligned}$$

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

  $$\begin{aligned} u(x) = \mathbb {E}^{x} u ( X_{\tau _{D}} ) < \infty , \quad x \in D. \end{aligned}$$

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

We will often write ‘harmonic’ instead of ‘$\alpha$-harmonic’. Since $\tau _{B} \le \tau _{D}$ for $B \subset D$, by the strong Markov property it follows that regular harmonic functions are harmonic. Also by the strong Markov property, $G_D(\cdot , y)$ is harmonic in $D\setminus \{y\}$, see [32, Theorem 2.5] or [50, (2.1)].

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

$$\begin{aligned} u(x) = \int _{D^{c}} u(z)\, \omega _{D}^{x} (dz), \quad x \in D. \end{aligned}$$

The $\alpha$-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:

$$\begin{aligned} P_D(x,z) {:}{=} \int _D G_D(x,y)\nu (y,z)\, dy,\quad x\in D,\, z\in D^c, \end{aligned}$$

(2.12)

see [12, Lemma 6]. Therefore, for every regular harmonic u we have the representation

$$\begin{aligned} u(x) = \int _{D^{c}} P_{D}(x,z) u(z)\, dz, \quad x \in D. \end{aligned}$$

We also recall the Ikeda–Watanabe formula from [47]:

$$\begin{aligned} \mathbb {P}^{x} \big [ \tau _{D} \in I, X_{\tau _{D}-} \in A, X_{\tau _{D}} \in B \big ] = \int _{I} \int _{B} \int _{A} \nu (y,z)p_{u}^{D} (x,dy) \, dz\, du, \end{aligned}$$

(2.13)

where $I \subset (0,\infty )$, $A \subset D$, and $B \subset ( \overline{D} )^{c}$. See also [7, Lemma 1], [12, 25, (4.13)], or [62, Theorem 2.4].

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

$$\begin{aligned} M_{D}^{x_{0}} (y,Q) = \lim _{D\ni x \rightarrow Q} \frac{G_{D}(x,y)}{G_{D}(x, x_{0})}. \end{aligned}$$

(2.14)

In [13, Lemma 6] it is shown that the Martin kernel exists, the mapping $(y, Q) \mapsto M_{D}^{x_{0}}(y,Q)$ is continuous on $D \times \partial D$, and for every $Q \in \partial D$ the function $M_{D}^{x_{0}}(\cdot , Q)$ is singular $\alpha$-harmonic in D.

2.3 Auxiliary results on $P_t^D$ and its spectral decomposition

We recall that the operators $P_t^D$ are compact on $L^2(D)$, see, e.g., [16, Chapter 4]. Therefore there exist a nondecreasing sequence of nonnegative numbers $\lambda _n$ diverging to infinity and an orthonormal sequence of functions $\varphi _n\in C_0(D)$ such that for every $\phi \in L^2(D)$, we have

$$\begin{aligned} P_t^D\phi (x) = \sum \limits _{n=1}^\infty e^{-\lambda _n t} \langle \phi ,\varphi _n\rangle \varphi _n(x) \end{aligned}$$

(2.15)

and

$$\begin{aligned} p_t^D(x,y) = \sum \limits _{n=1}^\infty e^{-\lambda _n t} \varphi _n(x)\varphi _n(y),\quad x,y\in D,\; t>0. \end{aligned}$$

(2.16)

The fractional Weyl bounds [10, 42] read

$$\begin{aligned} \lambda _n \approx n^{\alpha /d}. \end{aligned}$$

(2.17)

Note that $P_t^D\varphi _n(x) = e^{-\lambda _n t} \varphi _n(x)$ for all $x\in D$. Therefore,

$$\begin{aligned} G_D \varphi _n (x)= \int _0^\infty P_t^D \varphi _n (x) \, dt = \lambda _n^{-1} \varphi _n (x),\quad x\in D. \end{aligned}$$

(2.18)

By iterating (2.18) and using the regularity results for the fractional Laplacian [44, 58], we find that $\varphi _n$ are smooth in D. Furthermore, by [39, Proposition 3.1], there exist $C>0$ and $w\ge 1$, such that

$$\begin{aligned} \Vert \varphi _n\Vert _{\infty } \le C \lambda _n^{w-1},\quad n\in \mathbb {N}. \end{aligned}$$

(2.19)

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

$$\begin{aligned} L^D \phi {:}{=} \lim \limits _{t\rightarrow 0^+} \frac{P_t^D\phi - \phi }{t}. \end{aligned}$$

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

Lemma 2.5

• (1) We have $\varphi _n\in D(L^D)$ and $L^D\varphi _n (x)= -\lambda _n \varphi _n(x)$ for all $x\in D$.

• (2) We have

  $$\begin{aligned} F {:}{=} \{\phi \in L^2(D): \sum \limits _{n=1}^\infty \lambda _n^2 |\langle \phi ,\varphi _n\rangle |^2 < \infty \} \subseteq D(L^D), \end{aligned}$$

  and for each $\phi \in F,$

  $$\begin{aligned} L^D\phi = \sum \limits _{n=1}^\infty \lambda _n \langle \phi ,\varphi _n\rangle \varphi _n. \end{aligned}$$

• (3) For every $y\in D$ and $t>0$, $p_t^D(\cdot ,y) \in F.$

• (4) For every $x,y\in D$, we have $L^D_x p_t^D(x,y) = \Delta ^{\alpha /2}_x p_t^D(x,y)$.

Proof

Statements (1) and (2) follow quite easily from (2.15) and (2.17). In order to prove (3), we first let $m\in \mathbb {N}$. Then, by (2.16) and (2.19),

$$\begin{aligned} |\langle p_t^D(\cdot ,y),\varphi _m\rangle | = |e^{-\lambda _m t}\varphi _m(x)| \le e^{-\lambda _m t} \Vert \varphi _m\Vert _{\infty } \le Ce^{-\lambda _m t} \lambda _m^{w-1}. \end{aligned}$$

Using (2.17), we get (3).

We now prove (4). Let $x,y\in D$ and note that $z\mapsto p_t^D(z,y) \in C^2(D)\cap C_c(\mathbb {R}^d)$. Let $\phi \in C^2_c(B(x,\delta _D(x)/2))$ (extended by 0 to the whole of $\mathbb {R}^d$) and $g\in C_c(\mathbb {R}^d)$ be such that $\phi ({z}) + g({z}) = p_t^D({z},y)$ for $z\in D$ and $g({z}) = 0$ for $z\in B(x,\delta _D(x)/4)$. Note that by (2.4),

$$\begin{aligned} \frac{p_t^D(x,z)}{t} \le \frac{p_t(x,z)}{t} \lesssim \nu (x,z), \end{aligned}$$

(2.20)

which for $|x-z|>\delta _D(x)/4$ is uniformly bounded. Furthermore, since by [52, (2.10)] we have $p_t(x,z)/t \rightarrow \nu (x,z)$ as $t\rightarrow 0^+$ for all $x,z\in \mathbb {R}^d$, $x\ne z$, by (2.5) we find that for fixed $x,z\in D$, $x\ne z$,

$$\begin{aligned} \lim \limits _{t\rightarrow 0^+} \frac{p_t^D(x,z)}{t} = \nu (x,y) + \lim \limits _{t\rightarrow 0^+} \frac{1}{t} \mathbb {E}^x[p_{t-\tau _D}(X_{\tau _D},z)\, ; \, \tau _D<t]. \end{aligned}$$

Since x and z are fixed we have $p_{t-\tau _D}(X_{\tau _D},z) \lesssim t$, so the limit on the right hand side is equal to 0, hence $p_t^D(x,z)/t\rightarrow \nu (x,z)$ as $t\rightarrow 0^+$ for all $x,z\in D, x\ne z$. By this, (2.20), and the dominated convergence theorem, we get $\Delta ^{\alpha /2}g(x) = L^Dg(x)$.

Let L be the $C_0(\mathbb {R}^d)$-generator of the semigroup induced by $p_t$. By Sato [60, Theorem 31.5], we have $\Delta ^{\alpha /2} \phi (x) = L\phi (x)$. Therefore,

$$\begin{aligned} L^D\phi (x) = \Delta ^{\alpha /2}\phi (x) + \lim \limits _{t\rightarrow 0^+} \frac{P_t^D\phi (x) - P_t\phi (x)}{t}. \end{aligned}$$

We will show that the last limit exists and is equal to 0. By (2.5), Fubini–Tonelli, and the fact that $X_{\tau _D}\in D^c$ almost surely,

$$\begin{aligned} \frac{|P_t^D\phi (x) - P_t\phi (x)|}{t}&\le \Vert \phi \Vert _{\infty } \frac{1}{t} \mathbb {E}^x\bigg [\int _{B(x,\delta _D(x)/2)} p_{t-\tau _D}(X_{\tau _D},z)\, dz\, ; \, \tau _D<t\bigg ]\\&\lesssim \mathbb {P}^x(\tau _D<t) \mathop {\longrightarrow }\limits ^{t\rightarrow 0^+} 0. \end{aligned}$$

By collecting the above results we find that

$$\begin{aligned} \Delta ^{\alpha /2}_x p_t^D(x,y) = \Delta ^{\alpha /2}\phi (x) + \Delta ^{\alpha /2} g(x) = L^D\phi (x) + L^D g(x) = L^D_x p_t^D(x,y), \end{aligned}$$

which ends the proof. $\square$

Corollary 2.6

For every $t>0$, $\Delta ^{\alpha /2}_x p_t^D$ is bounded in $D\times D$.

Proof

By Lemma 2.5 and (2.19), we have

$$\begin{aligned} |\Delta ^{\alpha /2}_x p_t^D(x,y)|&= |L^D_x p_t^D(x,y)| = \bigg |\sum \limits _{n=1}^\infty \lambda _n e^{-\lambda _n t} \varphi _n(x)\varphi _n(y)\bigg | \\ {}&\le \sum \limits _{n=1}^\infty C\lambda _n e^{-\lambda _n t} \lambda _n^{2w-2} \le C_0<\infty . \end{aligned}$$

$\square$

Lemma 2.7

Let $\phi \in C_c^\infty (D)$. Then,

$$\begin{aligned} P_t^D L\phi (y) = \sum \limits _{n=1}^\infty e^{-\lambda _n t}\lambda _n \langle \phi ,\varphi _n\rangle \varphi _n(y) ,\quad y\in D. \end{aligned}$$

Proof

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

$$\begin{aligned} P_t^D L\phi (y) = \sum \limits _{n=1}^\infty e^{-\lambda _n t} \langle \varphi _n,L\phi \rangle \varphi _n(y). \end{aligned}$$

By (2.18) we have $\varphi _n = G_D[\lambda _n\varphi _n]$. Therefore, by [15, Proposition 3.13],

$$\begin{aligned} \langle \varphi _n,L\phi \rangle = \langle G_D[\lambda _n\varphi _n],L\phi \rangle = \langle \lambda _n\varphi _n,\phi \rangle , \end{aligned}$$

which ends the proof of the lemma. $\square$

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

Lemma 2.8

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

$$\begin{aligned} \Vert u\Vert _{L^p(D)} \le c \bigg (\sum \limits _{n=1}^\infty |\langle u,\varphi _n\rangle |^{p'} \lambda _n^{w-1}\bigg )^{1/p'}, \end{aligned}$$

where $p' = p/(p-1)$ is the Hölder conjugate exponent of p.

Proof

Let $\phi \in L^2(D)$. By (2.19), we have $\Vert \varphi _n\Vert _{\infty }\le C\lambda _n^{w-1}$ for some $C>0$ and $w\ge 1$ independent of n. Therefore for $x\in D$,

$$\begin{aligned} \Vert \phi \Vert _{\infty } \le \sum \limits _{n=1}^\infty |\langle \phi ,\varphi _n\rangle | \Vert \varphi _n\Vert _{\infty } \le C \sum \limits _{n=1}^\infty |\langle \phi ,\varphi _n\rangle | \lambda _n^{w-1}. \end{aligned}$$

If we let $\hat{\phi } = (\langle \phi ,\varphi _1\rangle ,\langle \phi ,\varphi _2\rangle ,\ldots )$ and denote by $l^p_\lambda$ the space of sequences with the p-th powers summable with the weight $(\lambda _1^{w-1},\lambda _2^{w-1},\ldots )$, then the above means that $\hat{\phi }\mapsto \phi$ is bounded from $l^1_\lambda$ to $L^\infty (D)$. By Parseval’s identity, this map is also bounded from $l^2$ to $L^2(D)$, hence also from $l^2_\lambda$ to $L^2(D)$. The statement of the lemma follows from the Riesz–Thorin theorem. $\square$

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 D. This extends what is already known about the asymptotics of Green potentials at the vertex of cone [24, Lemma 3.5]; we also propose a different proof.

Lemma 3.1

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

$$\begin{aligned} \lim _{x \rightarrow Q} \int _{D} \frac{G_{D}(x,y)}{G_{D} (x, x_{0})} f(y)\, dy = \int _{D} \lim _{x \rightarrow Q} \frac{G_{D}(x,y)}{G_{D} (x, x_{0})} f(y)\, dy < \infty , \quad x \in D. \end{aligned}$$

Proof

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

$$\begin{aligned} \rho = ( \delta _D(x_1) \wedge \delta _D(x_2) \wedge |x_1-x_2| )/3, \end{aligned}$$

so that $B(x_{1}, \rho ), B(x_{2}, \rho ) \subset D$ and $B(x_{1}, \rho ) \cap B(x_{2}, \rho ) = \emptyset$. We know that $M_{D}^{x_{0}}( \cdot , Q )$ given by (2.14) is regular $\alpha$-harmonic on $B(x_{1}, \rho )$ and $B(x_{2}, \rho )$, and for x sufficiently close to $\partial D$ so is $G_{D} (x,\cdot )$. Therefore, for $i=1,2$,

$$\begin{aligned} \int _{B(x_{i}, \rho )^{c}} \lim _{x \rightarrow Q} \frac{G_{D}(x,y)}{G_{D} (x, x_{0})} \,\omega _{B(x_{i}, \rho )}^{x_{i}}(dy)&= \int _{B(z_{i}, \rho )^{c}} M_{D}^{x_{0}}(y, Q)\, \omega _{B(z_{i}, \rho )}^{z_{i}}(dy) \\&= M_{D}^{x_{0}}(x_{i}, Q) \\&= \lim _{x \rightarrow Q} \frac{G_{D}(x, x_{i})}{G_{D}(x, x_{0})} \\&= \lim _{x \rightarrow Q} \frac{\int _{B(x_{i}, \rho )^{c}} G_{D}(x,y) \,\omega _{B(x_{i}, \rho )}^{x_{i}}(dy)}{G_{D}(x, x_{0})} \\&= \lim _{x \rightarrow Q} \int _{B(x_{i}, \rho )^{c}} \frac{G_{D}(x,y)}{G_{D} (x, x_{0})}\, \omega _{B(x_{i}, \rho )}^{x_{i}}(dy). \end{aligned}$$

The $\alpha$-harmonic measures $\omega ^{x_i}_{B(x_i,\rho )}(dy)$ are absolutely continuous and have radially decreasing densities $g_i$, see, e.g., [13]. Therefore there exists $C>0$ such that $\omega ^{x_i}_{B(x_i,\rho )}(dy) = g_i(y)\, dy$ and $g_i(y)\ge C$ for $y\in D\cap (B(x_i,\rho )^c)$. Let $g = g_1 + g_2$. Vitali’s theorem [61, Theorem 16.6 (i) and (iii)] yields the following $L^1$ convergence:

$$\begin{aligned} \lim \limits _{x\rightarrow Q}\int _D \bigg |\frac{G_D(x,y)}{G_D(x,x_0)}g(y) - M_{D}^{x_{0}}(y, Q) g(y)\bigg |\, dy = 0. \end{aligned}$$

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

We can also establish the following identity, an analogue of [24, (3.16)].

Lemma 3.2

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

$$\begin{aligned} \mathbb {P}^{x} ( \tau _{D} > 1 ) = ( G_{D} P_{1}^{D} \kappa _{D} )(x). \end{aligned}$$

(3.1)

Proof

Let $x \in D$. Since our set D is Lipschitz, from Lemma 6 and the proof of Lemma 17 in [12],

$$\begin{aligned} \omega _{D}^{x} ( \partial D) = \mathbb {P}^{x} ( X_{\tau _{D}} \in \partial D )&= 0,\\ \mathbb {P}^{x} ( X_{\tau _{D}-} = X_{\tau _{D}})&= 0,\\ \mathbb {P}^x (X_{\tau _D-} \in D)&= 1. \end{aligned}$$

By the Ikeda–Watanabe formula (2.13) and the Chapman–Kolmogorov equations we have

$$\begin{aligned} \mathbb {P}^{x} ( \tau _{D}> 1 )&= \mathbb {P}^{x} \big [ \tau _{D} > 1, \ X_{\tau _{D}-} \in D, \ X_{\tau _{D}} \in D^{c} \big ] \\&= \int _{1}^{\infty } \int _{D^{c}} \int _{D} p_{s}^{D} (x,z) \nu (z - w) \, dz \, dw \, ds \\&= \int _{\mathbb {R}^{d}} \int _{D^{c}} \int _{0}^{\infty } p_{t + 1}^{D} (x,z) \nu (z - w) \, dt \, dw \, dz \\&= \int _{\mathbb {R}^{d}} \int _{D^{c}} \int _{0}^{\infty } \int _{D} p_{t}^{D} (x,y) p_{1}^{D} (y,z) \, dy \, \nu (z - w) \, dt \, dw \, dz \\&= \int _{D} \int _{0}^{\infty } p_{t}^{D} (x,y) \, dt \int _{\mathbb {R}^{d}} p_{1}^{D} (y,z) \int _{D^{c}} \nu (z - w) \, dw \, dz \, dy \\&= \int _{D} G_{D} (x,y) \int _{\mathbb {R}^{d}} p_{1}^{D} (y,z) \kappa _{D}(z) \, dz \, dy \\&= \int _{D} G_{D} (x,y) ( P_{1}^{D} \kappa _{D} ) (y) \, dy \\&= ( G_{D} P_{1}^{D} \kappa _{D} ) (x). \end{aligned}$$

For $x \in D^{c}$, both sides of (3.1) are equal to 0. This ends the proof. $\square$

We define

$$\begin{aligned} C_{1} {:}{=} \int _{D} \int _{D} M_{D}^{x_{0}} (y, Q) p_{1}^{D}(y,z) \kappa _{D}(z)\, dz\, dy. \end{aligned}$$

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

Lemma 3.3

We have $0< C_{1} < \infty$ and $\lim _{x \rightarrow Q} \frac{\mathbb {P}^{x} ( \tau _{D} > 1 )}{G_{D}(x,x_{0})} = C_{1}.$

Proof

By Lemma 3.2, $\mathbb {P}^x(\tau _D>1) = (G_DP_1^D\kappa _D)(x)$. Note that $(P_{1}^{D} \kappa _{D})(y)$ is bounded. Indeed, by (2.6),

$$\begin{aligned} ( P_{1}^{D} \kappa _{D} ) (y)&= \int _{D} p_{1}^{D} (y, z) \kappa _{D} (z) \, dz \nonumber \\&\approx \mathbb {P}^{y} (\tau _{D}> 1 ) \int _{D} \mathbb {P}^{z} (\tau _{D} > 1 ) p_{1}(y,z) \kappa _{D}(z) \, dz,\quad y\in D. \end{aligned}$$

(3.2)

Since D is bounded, by (2.4),

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

(3.3)

Hence (3.2) becomes

$$\begin{aligned} ( P_{1}^{D} \kappa _{D} ) (y) \approx \mathbb {P}^{y} (\tau _{D}> 1) \int _{D} \mathbb {P}^{z} (\tau _{D} > 1) \kappa _{D}(z) \, dz,\quad y\in D. \end{aligned}$$

(3.4)

Using (3.1), we see that for $x \in \mathbb {R}^{d}$,

$$\begin{aligned} \int _{D} G_{D}(x,y) ( P_{1}^{D} \kappa _{D} )(y)\, dy = ( G_{D} P_{1}^{D} \kappa _{D} )(x) = \mathbb {P}^{x} ( \tau _{D} > 1 ) \le 1. \end{aligned}$$

By (2.10), $G_{D}(x,y)$ is strictly positive for all $x, y \in D$. Thus $P_{1}^{D} \kappa _{D}$ has to be finite almost everywhere. Hence the integral in (3.4) is finite and

$$\begin{aligned} ( P_{1}^{D} \kappa _{D} ) (y) \approx \mathbb {P}^{y} (\tau _{D} > 1), \end{aligned}$$

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

$$\begin{aligned} \lim _{x \rightarrow Q} \frac{\mathbb {P}^{x} ( \tau _{D} > 1 )}{G_{D}(x,x_{0})}&= \lim _{x \rightarrow Q} \frac{( G_{D} P_{1}^{D} \kappa _{D} ) (x)}{G_{D}(x,x_{0})} \\&= \lim _{x \rightarrow Q} \int _{D} \frac{G_{D}(x,y)}{G_{D}(x,x_{0})} ( P_{1}^{D} \kappa _{D} )(y) \, dy\\&= \int _{D} M_{D}^{x_{0}} (y,Q) ( P_{1}^{D} \kappa _{D} )(y) \, dy = C_{1} < \infty . \end{aligned}$$

$\square$

We are now in a position to prove Theorem 1.1.

Proof of Theorem 1.1

Let us define

$$\begin{aligned} m_{x} (A) {:}{=} \frac{\int _{A} p_{1}^{D} (x,y) dy}{\mathbb {P}^{x} ( \tau _{D} > 1 )}, \quad x \in D, \ A \subseteq \mathbb {R}^{d}. \end{aligned}$$

(3.5)

First we note that the family $\{ m_{x}: x \in D \}$ is tight. Indeed, combining the factorization of $p_{1}^{D}(x,y)$ in (2.6) with the Eq. (3.3), we get

$$\begin{aligned} \frac{p_{1}^{D}(x,y)}{\mathbb {P}^{x} ( \tau _{D}> 1 )} \approx \mathbb {P}^{y} ( \tau _{D} > 1 ), \qquad x, y \in D. \end{aligned}$$

(3.6)

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

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

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

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

(3.7)

To show this, we first remark that $u_{\phi }\in C_0(\mathbb {R}^d)$ and that $( G_{D} u_{\phi } )(x) = \phi (x)$, see [37, Lemma 5.7] and [23, (11)]. By (2.4) it follows that

$$\begin{aligned} ( P_{1}^{D} | u_{\phi } | ) (x) = \int _{D} p_{1}^{D}(x,y) | u_{\phi }(y) |\, dy \le c<\infty . \end{aligned}$$

Therefore, since for a fixed $z\in D^c$ we have $\nu (y,z) \gtrsim 1$ for $y\in D$, by (2.12) we get

$$\begin{aligned} ( G_{D} P_{1}^{D} | u_{\phi } | ) (x)&= \int _{D} G_{D}(x,y) ( P_{1}^{D} | u_{\phi } | )(y)\, dy \\&\le c \int _{D} G_{D}(x,y) \, dy < \infty . \end{aligned}$$

As a result, we can apply Fubini–Tonelli theorem and establish (3.7) as follows:

$$\begin{aligned} ( G_{D} P_{1}^{D} u_{\phi } )(x)&= \int _{D} \int _{D} \int _{0}^{\infty } p_{t}^{D}(x,y) p_{1}^{D} (y,z) u_{\phi } (z)\, dt\, dz\, dy \nonumber \\&= \int _{D} \int _{0}^{\infty } p_{t + 1}^{D}(x,z) u_{\phi } (z)\, dt\, dz \nonumber \\&= \int _{D} \int _0^\infty \int _D p_1^D(x,y)p_t^D(y,z) u_\phi (z) \, dy \, dt \, dz\nonumber \\&= \int _D \int _D \int _0^\infty p_1^D(x,y) p_t^D(y,z) u_\phi (z) \, dt \, dz \, dy\nonumber \\&= ( P_{1}^{D} G_{D} u_{\phi } )(x) = ( P_{1}^{D} \phi )(x). \end{aligned}$$

Let us denote $m_{x} (\phi ) {:}{=} \int _{D} \phi (y)\, m_{x} (dy)$. Using (3.7), Lemmas 3.3, and 3.1, we get

$$\begin{aligned} \lim _{x \rightarrow Q} m_{x}(\phi )&= \lim _{x \rightarrow Q} \frac{( P_{1}^{D} \phi )(x)}{\mathbb {P}^{x} ( \tau _{D}> 1 )} \nonumber \\&= \lim _{x \rightarrow Q} \frac{( P_{1}^{D} G_{D} u_{\phi } )(x)}{\mathbb {P}^{x} ( \tau _{D}> 1 )} \nonumber \\&= \lim _{x \rightarrow Q} \frac{( G_{D} P_{1}^{D} u_{\phi } )(x)}{G_{D} (x,x_{0})} \frac{G_{D}(x,x_{0})}{\mathbb {P}^{x} ( \tau _{D} > 1 )} \nonumber \\&= \frac{1}{C_{1}} \int _{D} M_{D}^{x_{0}}(y,Q) ( P_{1}^{D} u_{\phi } ) (y)\, dy. \end{aligned}$$

(3.8)

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

For $t > 1$, we consider $\phi _{t,y}(\cdot ) {:}{=} p_{t-1}^{D}(\cdot , y) \in C_{0} ( \mathbb {R}^{d} )$, see [19] or [32, Proposition 1.19]. Using Chapman–Kolmogorov, we get

$$\begin{aligned} \eta _{t, Q}(y)&= \lim _{x \rightarrow Q} \frac{p_{t}^{D}(x,y)}{\mathbb {P}^{x} ( \tau _{D}> 1 )} \\&= \lim _{x \rightarrow Q} \frac{\int _{D} p_{t - 1}^{D} (z,y) p_{1}^{D} (x,z)\, dz}{\mathbb {P}^{x} ( \tau _{D}> 1 )} \\&= \lim _{x \rightarrow Q} \frac{( P_{1}^{D} p_{t - 1}^{D} ( \cdot ,y) )(x)}{\mathbb {P}^{x} ( \tau _{D} > 1 )} \\&= \lim _{x \rightarrow Q} m_{x} ( p_{t - 1}^{D} (\cdot , y) ). \end{aligned}$$

By (3.8), the existence of $\eta _{t, Q}(y)$ for $t > 1$ follows:

$$\begin{aligned} \eta _{t, Q}(y) = m_{Q} ( p_{t - 1}^{D} (\cdot , y) ). \end{aligned}$$

Note that the threshold $t>1$ is arbitrary, that is, 1 can be replaced with any $t_0>0$. Indeed, the results of this section can be readily reformulated with $t_0$ in place of 1, for instance, Lemma 3.3 may be strengthened to assert that for every $t_0>0$,

$$\begin{aligned} \lim \limits _{x\rightarrow Q} \frac{\mathbb {P}^x(\tau _D>t_0)}{G_D(x,x_0)} = \int _D\int _D M_D^{x_0}(y,Q)p_{t_0}^D(y,z)\kappa _D(z)\, dz\, dy. \end{aligned}$$

Accordingly, we get the existence of the limit

$$\begin{aligned} \lim \limits _{x\rightarrow Q} \frac{\mathbb {P}^x(\tau _D> 1)}{\mathbb {P}^x(\tau _D > t_0)}. \end{aligned}$$

(3.9)

We can also reuse the above arguments to get for all $t>t_0$, the existence of

$$\begin{aligned} \lim \limits _{x\rightarrow Q} \frac{p^D_t(x,y)}{\mathbb {P}^x(\tau _D > t_0)}. \end{aligned}$$

(3.10)

Of course, (3.9) and (3.10) give the existence of $\eta _{t,Q}(y)$ for all $t>0$.

The Eq. (1.8) follows from Eq. (3.6), and the Eq. (1.9) follows from the Chapman–Kolmogorov equations and the dominated convergence theorem (see [19, (27)]):

$$\begin{aligned} \eta _{t + s, Q}(y) = \lim _{x \rightarrow Q} \int _{D} \frac{p_{t}^{D} (x,z)}{\mathbb {P}^{x} ( \tau _{D} > 1 )} p_{s}^{D}(z,y)\, dz = \int _{D} \eta _{t,Q}(z) p_{s}^{D}(z,y)\, dz. \end{aligned}$$

The fact that $\widetilde{\eta }$ and $\eta ^{x_0}$ exist follows from the existence of $\eta$ and from Lemma 3.3. The continuity of $\eta$ follows from (1.8), (1.9), continuity of $p_t^D(x,y)$, and the dominated convergence theorem. $\square$

Corollary 3.4

The functions $(t,y)\mapsto \eta _{t,y}(x), \widetilde{\eta }_{t,y}(x), \eta ^{x_0}_{t,y}(x)$ are continuous on $(0,\infty )\times \overline{D}$ for all $x\in D$.

Proof

Fix $x\in D$. By Theorem 1.1 and the fact that $p_t^D(x,y)$ and $\mathbb {P}^y(\tau _D>1)$ are continuous for $(t,y)\in (0,\infty )\times D$, and separated from 0 in sufficiently small neighborhood of any point (t, y), it suffices to verify that for any sequence $((t_n,y_n))\subset (0,\infty )\times D$ such that $(t_n,y_n) \rightarrow (t,Q)\in (0,\infty )\times \partial D$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{p_{t_n}^D(x,y_n)}{\mathbb {P}^{y_n}(\tau _D>1)} = \eta _{t,Q}(x). \end{aligned}$$

(3.11)

Furthermore, by Theorem 1.1, in order to obtain (3.11) it suffices to prove that for any $t>0$ there exists a modulus of continuity $\omega$ independent of y such that

$$\begin{aligned} \bigg |\frac{p_{t+\varepsilon }^D(x,y)-p_{t}^D(x,y)}{\mathbb {P}^y(\tau _D>1)}\bigg | \le \omega (\varepsilon ),\quad \varepsilon >0. \end{aligned}$$

(3.12)

By Chapman–Kolmogorov, we have

$$\begin{aligned}&\bigg |\frac{p_{t+\varepsilon }^D(x,y)-p_{t}^D(x,y)}{\mathbb {P}^y(\tau _D>1)}\bigg | \int _{D\setminus B(x,\delta _D(x)/2)} \frac{|p_t^D(z,y) - p_t^D(x,y)| p_{\varepsilon }^D(x,z)}{\mathbb {P}^y(\tau _D>1)} \, dz\\&\quad + \int _{B(x,\delta _D(x)/2)} \frac{|p_t^D(z,y) - p_t^D(x,y)|p_{\varepsilon }^D(x,z)}{\mathbb {P}^y(\tau _D>1)} \, dz =: I_1+I_2. \end{aligned}$$

Then by (2.6),

$$\begin{aligned} I_1 \le \int _{D\setminus B(x,\delta _D(x)/2)} p_\varepsilon ^D(x,z)\, dz \le \int _{D\setminus B(x,\delta _D(x)/2)} p_\varepsilon (x,z)\, dz \le \omega (\epsilon ). \end{aligned}$$

For $I_2$, we use the gradient bounds of Kulczycki and Ryznar [51, Theorem 1.1] and (2.6):

$$\begin{aligned} I_2&\le \int _{B(x,\delta _D(x)/2)} \frac{|p_t^D(z,y) - p_t^D(x,y)|p_{\varepsilon }^D(x,z)}{\mathbb {P}^y(\tau _D>1)} \, dz\\&\le \int _{B(x,\delta _D(x)/2)} |x-z| \frac{\Vert \nabla _x p_t^D(\cdot ,y)\Vert _{L^\infty (B(x,\delta _D(x)/2))}}{\mathbb {P}^y(\tau _D>1)} p_\varepsilon ^D(x,z)\, dz\\&\lesssim \int _{B(x,\delta _D(x)/2)} |x-z| \frac{\Vert p_t^D(\cdot ,y)\Vert _{L^\infty (B(x,\delta _D(x)/2))}}{\mathbb {P}^y(\tau _D>1)} p_\varepsilon ^D(x,z)\, dz\\&\lesssim \int _{B(x,\delta _D(x)/2)} |x-z|p_\varepsilon ^D(x,z)\, dz \le \int _{B(x,\delta _D(x)/2)} |x-z|p_\varepsilon (x,z)\, dz \le \omega (\varepsilon ). \end{aligned}$$

Thus, $I_1+I_2 \le \omega (\varepsilon )$, which ends the proof for $\eta$. For $\widetilde{\eta }$ and $\eta ^{x_0}$, we use Lemma 3.2 and (1.8). $\square$

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

Lemma 3.5

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

Proof

By (2.6) and (2.4), we have

$$\begin{aligned} \eta _{s,Q}(x)&= \lim \limits _{D\ni \xi \rightarrow Q} \frac{p_s^D(x,\xi )}{\mathbb {P}^\xi (\tau _D>1)} \lesssim \limsup \limits _{D\ni \xi \rightarrow Q} \frac{\mathbb {P}^\xi (\tau _D>s)}{\mathbb {P}^\xi (\tau _D>1)}p_s(x,\xi )\mathbb {P}^x(\tau _D> s)\\&\lesssim |x-Q|^{-d-\alpha }\mathbb {P}^x(\tau _D> s)\limsup \limits _{D\ni \xi \rightarrow Q} \frac{s\mathbb {P}^\xi (\tau _D>s)}{\mathbb {P}^\xi (\tau _D>1)}. \end{aligned}$$

If $|x-Q|\ge \varepsilon$, then $\eta _{s,Q}(x)$ is bounded—it even converges to 0 as $s\rightarrow 0$—see Lemma B.2. If $s>\varepsilon$, then we use the approximate factorization of $p_t^D$ and the fact that $\mathbb {P}^x(\tau _D > s)\rightarrow 0$ as $x\rightarrow y\in \partial D$. $\square$

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

Corollary 3.6

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

$$\begin{aligned} \eta _{t,Q}(x) \approx {\left\{ \begin{array}{ll} \frac{1}{\sqrt{t}}\bigg (1\wedge \frac{\delta _D^{\alpha /2}(x)}{\sqrt{t}}\bigg )p_t(x,Q),\quad &{}t\in (0,1),\ x\in D,\ Q\in \partial D,\\ e^{-\lambda _1 t} \delta _D(x)^{\alpha /2},\quad &{}t\in [1,\infty ),\ x\in D,\ Q\in \partial D. \end{array}\right. } \end{aligned}$$

(3.13)

If D is Lipschitz, then

$$\begin{aligned} \eta _{t,Q}(x) \approx e^{-\lambda _1 t} \mathbb {P}^x(\tau _D > t),\quad t\in [1,\infty ),\ x\in D,\ Q\in \partial D, \end{aligned}$$

(3.14)

and

$$\begin{aligned} \eta _{t,Q}(x) \approx \frac{\mathbb {P}^x(\tau _D>t)p_t(x,Q)}{\Phi (A_{t^{1/\alpha }}(Q))}, \quad t\in (0,1),\ x\in D,\ Q\in \partial D. \end{aligned}$$

(3.15)

Furthermore, there exist $0<\sigma _1\le \sigma _2<1$ such that

$$\begin{aligned} t^{-\sigma _1}\lesssim \frac{\eta _{t,Q}(x)}{\mathbb {P}^x(\tau _D>t)p_t(x,Q)} \lesssim t^{-\sigma _2},\quad t\in (0,1),\ x\in D,\ Q\in \partial D. \end{aligned}$$

(3.16)

Proof

The estimate (3.13) follows from (2.7) and (2.9). By [49, Theorem 1.1] and (2.6), $\mathbb {P}^y(\tau _D>1) \approx \varphi _1(y)$, so (3.14) is a consequence of (2.8). It remains to prove (3.15) and (3.16). By (2.6),

$$\begin{aligned} \eta _{t,Q}(x) \approx \mathbb {P}^x(\tau _D>t)p_t(x,Q) \lim \limits _{y\rightarrow Q} \frac{\mathbb {P}^y(\tau _D>t)}{\mathbb {P}^y(\tau _D > 1)}. \end{aligned}$$

(3.17)

By [19, Theorem 2] and (2.11),

$$\begin{aligned} \frac{\mathbb {P}^y(\tau _D>t)}{\mathbb {P}^y(\tau _D>1)} \approx \frac{1}{\Phi (A_{t^{1/\alpha }}(y))}. \end{aligned}$$

By geometrical considerations, we can choose points $A_{t^{1/\alpha }}(y)$ converging to a point in $\mathcal A_{t^{1/\alpha }}(Q)$. This proves (3.15). By (3.17) and Lemma B.2, we get the upper bound in (3.16). The lower bound follows from (3.15) and [12, Lemma 3] with some $\sigma _1>0$. Of course, we must have $\sigma _1\le \sigma _2$ in (3.16). $\square$

A consequence of Theorem 1.1 is the Yaglom-type limit, obtained in the thesis of the first author [4].

Theorem 3.7

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

$$\begin{aligned} \lim _{t \rightarrow \infty } \mathbb {P}^{x} \bigg ( \frac{X_{t}}{t^{1/\alpha }} \in A \ \bigg | \ \bigg ( \frac{X_{s}}{t^{1/\alpha }} \bigg )_{0 \le s \le t} \subset D \bigg ) = m_{0}(A), \end{aligned}$$

where $\mathbb {P}^x(A_1|A_2) {:}{=} \mathbb {P}^x(A_1\cap A_2)/\mathbb {P}^x(A_2)$ is the conditional probability and $m_{0}(A) {:}{=} \int _A \eta _{1,{0}}(y)\, dy$.

Proof

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

$$\begin{aligned} \mathbb {P}^{x} \bigg ( \frac{X_{t}}{t^{1/\alpha }} \in A \ \bigg | \ \bigg ( \frac{X_{s}}{t^{1/\alpha }} \bigg )_{0 \le s \le t} \subset D \bigg )&= \frac{\mathbb {P}^{x} ( X_{t} \in t^{1/\alpha }A, \ ( X_{s})_{0 \le s \le t} \subset t^{1/\alpha }D )}{\mathbb {P}^{x} (( X_{s})_{0 \le s \le t} \subset t^{1/\alpha }D )}\\&= \frac{\int _{t^{1/\alpha } A} p_{t}^{t^{1/\alpha } D} (x,y)\, dy}{\int _{t^{1/\alpha }D} p_{t}^{t^{1/\alpha }D} (x,y)\, dy}\\&= \frac{\int _{t^{1/\alpha } A} t^{-d/\alpha } p_{1}^{D} ( t^{-1/\alpha } x, t^{-1/\alpha } y )\, dy}{\int _{t^{1/\alpha } D}t^{-d/\alpha } p_{1}^{D} ( t^{-1/\alpha } x, t^{-1/\alpha } y )\, dy} \\&= \frac{\int _{A} p_{1}^{D} ( t^{-1/\alpha } x, y )\, dy}{\int _{D} p_{1}^{D} ( t^{-1/\alpha } x, y )\, dy} = m_{t^{-1/\alpha } x}(A), \end{aligned}$$

where $m_{t^{-1/\alpha }x}$ is the measure defined in (3.5) above (note that $t^{-1/\alpha }x\in D$). Therefore, by Theorem 1.1, this probability approaches $m_{0}(A)$ as $t \rightarrow \infty$. $\square$

4 Hölder regularity

This section is devoted to proving Theorems 1.2 and 1.4. 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 Sect. 3, which enable us to relate the heat kernel regularity to the elliptic regularity.

Fix $n_0\ge 2$ such that the reference points $x_0$ belongs to $D_{n_0/2}$.

Lemma 4.1

There exists $p_0=p_0(d,\alpha ,\underline{D})>1$ such that the family $\{(G_D(y,\cdot )/G_D(x_0,y))^p: y\in D\}$ is uniformly integrable in D for all $p\in [1,p_0)$.

Proof

For $y\in D_{n_0}$ we have a crude bound:

$$\begin{aligned} G_D(y,z)/G_D(x_0,y) \le C(d,\alpha ,\underline{D}) |y-z|^{\alpha -d},\quad z\in D. \end{aligned}$$

Considering the functions on the right-hand side, we see that $p_0 = d/(d-\alpha )$ will do.

From now on assume that $y\in D{\setminus } D_{n_0}$. By (2.10), there exists $C=C(d,\alpha ,\underline{D})$ such that

$$\begin{aligned} \frac{G_D(y,z)}{G_D(x_0,y)} \le C \frac{|y-z|^{\alpha -d}}{|x_0-y|^{\alpha -d}} \frac{\Phi (z)\Phi (A_{x_0,y})^2}{\Phi (x_0)\Phi (A_{y,z})^2}. \end{aligned}$$

We immediately get that

$$\begin{aligned} \frac{G_D(y,z)}{G_D(x_0,y)} \le C' |y-z|^{\alpha -d} \frac{\Phi (z)}{\Phi (A_{y,z})^2}. \end{aligned}$$

By the Carleson estimate [48, Lemma 13], we further find that $\Phi (z)/\Phi (A_{y,z})\le C(d,\alpha ,\underline{D})$. If we let $U = D_{32/r_0}\cup (D{\setminus } B(y,r_0/32))$, then it follows that

$$\begin{aligned} \frac{G_D(y,z)}{G_D(x_0,y)} \le {\left\{ \begin{array}{ll} C(d,\alpha ,\underline{D}) |y-z|^{\alpha -d},\quad &{}z\in U,\\ C(d,\alpha ,\underline{D})|y-z|^{\alpha -d} \Phi (A_{y,z})^{-1},\quad &{}z\in D{\setminus } U.\end{array}\right. } \end{aligned}$$

The definition of $A_{y,z}$ implies that for $z\in D{\setminus } U$ there exists $Q = Q(z)\in \partial D$ such that $y,z\in B(Q,3r)$ and $B(A_{y,z},\kappa r) \subset D\cap B(Q,6r)$. Using [12, Lemma 5], we find that there exist $C=C(d,\alpha ,\underline{D})$ and $\gamma = \gamma (d,\alpha ,\underline{D})\in (0,\alpha )$ such that

$$\begin{aligned} \Phi (A_{y,z}) \ge C |A_{y,z} - Q(z)|^\gamma \ge C\kappa ^\gamma r^\gamma \ge C\kappa ^\gamma |y-z|^{\gamma }. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{G_D(y,z)}{G_D(x_0,y)} \le C (|y-z|^{\alpha -d}\vee |y-z|^{\alpha -\gamma -d}),\quad y\in D{\setminus } D_{n_0},\ z\in D, \end{aligned}$$

(4.1)

so the statement of the lemma holds for all $p\in [1,d/(d-\alpha +\gamma ))$. We can take $p_0 = d/(d-\alpha +\gamma )$. $\square$

The following lemma is a specific Carleson-type estimate.

Lemma 4.2

Let $0<r<\delta _D(y)$, $|z-y|\ge 2r$, and $|v-y|\le r$. There exists $C = C(d,\alpha ,\underline{D})$ such that

$$\begin{aligned} G_D(z,v) \le C G_D(z,y). \end{aligned}$$

Proof

Note that $2|z-v|\ge |z-y|$. By (2.10), there is $c = c(d,\alpha ,\underline{D})$ such that

$$\begin{aligned} G_D(z,v) \le c \frac{\Phi (z)\Phi (v)}{\Phi (A_{z,v})^2} |z-v|^{\alpha -d} \le 2^{d-\alpha } c \frac{\Phi (z)\Phi (v)}{\Phi (A_{z,v})^2} |z-y|^{\alpha -d}. \end{aligned}$$

By elementary calculations, we find that $2r_{z,v} \ge r_{z,y}$. By [48, Lemma 13] we therefore get $\Phi (A_{z,v})\ge c(d,\alpha ,\underline{D})\Phi (A_{z,y})$. Furthermore, by [12, Lemma 4 and 5] we get $\Phi (v) \le c(d,\alpha ,\underline{D})\Phi (y)$. This ends the proof. $\square$

The next lemma can be viewed as a more concrete, quantified version of [23, 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 [12, Lemma 16], but here the singularity can occur between the boundary and the arguments of the function, see Fig. 1.

Fig. 1

Illustration for Lemma 4.3. The boundary Harnack principle cannot be used to estimate increments between y and $y'$ because of the singularity at z. Instead we show regularity in the smaller ball using harmonicity in the larger ball

Lemma 4.3

Let $y\in D$ and $Q\in \partial D$ satisfy $|Q-y| = \delta _D(y)$. Assume that $z\in D\cap B(Q,3\delta _D(y))$ and let $r = |z-y|/4$, so that $\overline{B(y,r)}\subset D$. Then there exist constants $C \ge 1$, $k_0\ge 4$, $\sigma \in (0,1]$, and $\gamma \in (0,\alpha )$, depending only on $d,\alpha ,\underline{D}$, such that for every $y'\in B(y,2^{-k_0}r)$ we have

$$\begin{aligned} \bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg | \le C\bigg (\frac{|y-y'|}{r}\bigg )^{\sigma } r^{\alpha -d-\gamma }. \end{aligned}$$

Proof

Let $k_0\ge 4$ (further constraints on $k_0$ stem from the proof) and let

$$\begin{aligned}&B_k = B(y,(2^{k_0})^{-k}r),\quad k=0,1,\ldots ,\\&\Pi _k = B_k{\setminus } B_{k+1},\quad k=0,1,\ldots ,\quad \Pi _{-1} = D\setminus B_0,\\&u(y) = G_D(y,z),\quad v(y) = G_D(x_0,y). \end{aligned}$$

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

$$\begin{aligned} \sup \limits _{B_k} \frac{u}{v} \le (1 + c\zeta ^k)\inf \limits _{B_k} \frac{u}{v}. \end{aligned}$$

(4.2)

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

In order to obtain (4.2), for $-1\le l < k$ we define

$$\begin{aligned}&u_{k}^l(x) = \mathbb {E}^x[u(X_{\tau _{B_k}})\, ;\, X_{\tau _{B_k}}\in \Pi _l],\quad v_{k}^l(x) = \mathbb {E}^x[v(X_{\tau _{B_k}})\, ;\, X_{\tau _{B_k}}\in \Pi _l],\quad x\in \mathbb {R}^d, \end{aligned}$$

and we will prove the following two claims.

Claim 1

There exist $C = C(d,\alpha ,\underline{D})$ and $q = q(d,\alpha ,\underline{D})\in (0,1)$ such that for $-1\le l \le k-2$ and $x\in B_k$,

$$\begin{aligned} u_k^l(x) \le C(q^{k_0})^{k-l-1}u(x),\\ v_k^l(x) \le C (q^{k_0})^{k-l-1}v(x). \end{aligned}$$

We define the oscillation of function f as ${{\,\textrm{Osc}\,}}_{A} f = \sup \nolimits _A f - \inf \nolimits _A f.$

Claim 2

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

Using Claim 1 with $k_0$ large enough (see [12, (5.23)–(5.25)] for details) and Claim 2 we may repeat the final part of the proof in [12, Lemma 16] to get (4.2)—we skip those details.

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

$$\begin{aligned} u_k^l(x) = \int _{\Pi _l} G_D(z,{w}) P_{B_k}(x,{w})\, d{w} \le cG_D(z,y) \mathbb {P}^x(X_{\tau _{B_k}}\in \Pi _l). \end{aligned}$$

Furthermore, since $k\ge 1$, Lemma 4.2 yields $G_D(z,y)\le cG_D(z,x)$. Therefore,

$$\begin{aligned} u_k^l(x) \le cG_D(z,x) \mathbb {P}^x(X_{\tau _{B_k}}\in \Pi _l). \end{aligned}$$

(4.3)

Recall the explicit formula for the Poisson kernel of the ball—see, e.g., Landkof [54]:

$$\begin{aligned} P_{B(0,r)}(x,{w}) = c_{d,\alpha } \frac{(r^2 - |x|^2)^{\alpha /2}}{(|{w}|^2 - r^2)^{\alpha /2}}|x-{w}|^{-d},\quad x\in B(0,r),\ {w}\in B(0,r)^c. \end{aligned}$$

(4.4)

Using the formula, we find that

$$\begin{aligned} \mathbb {P}^x(X_{\tau _{B_k}}\in \Pi _l)&= \int _{\Pi _l} P_{B_k}(x,{w})\, d{w} {\le } c_{d,\alpha } (r(2^{k_0})^{-k})^\alpha \times \int _{\Pi _l} (|{w}{-}y|^2 {-} (r(2^{k_0})^{-k})^2)^{-\alpha /2}|x{-}{w}|^{-d}\, d{w}\\&\le \tilde{c}_{d,\alpha } \frac{(r(2^{k_0})^{-k})^\alpha }{(r(2^{k_0})^{-l-1})^{\alpha }} = \tilde{c}_{d,\alpha }(2^{-k_0\alpha })^{k-l-1}. \end{aligned}$$

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

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

$$\begin{aligned} u_k^{-1}(x)&\le C(d,\alpha ,\underline{D})\int _{D\setminus B(y,r)} G_D(z,{w}) \frac{((r(2^{k_0})^{-k})^2 - |x-y|^2)^{\alpha /2}}{(|{w}-y|^2 - (r(2^{k_0})^{-k})^2)^{\alpha /2}} |x-{w}|^{-d}\, d{w}\\&\le C(d,\alpha ,\underline{D})((2^{k_0})^{-k})^{\alpha }\int _{D\setminus B(y,r)} G_D(z,{w}) \frac{r^\alpha }{(|{w}-y|^2 - (r(2^{k_0})^{-k})^2)^{\alpha /2}}|x-w|^{-d}\, d{w}\\&\le c(d,\alpha )C(d,\alpha ,\underline{D})(2^{-\alpha k_0})^{k}\int _{D\setminus B(y,r)} G_D(z,{w}) \frac{(r^2 - |x-y|^2)^{\alpha /2}}{(|{w}-y|^2 - r^2)^{\alpha /2}}|x-w|^{-d}\, d{w}\\&\le \tilde{c}(d,\alpha )C(d,\alpha ,\underline{D})(2^{-\alpha k_0})^{k}\int _{D\setminus B(y,r)} G_D(z,{w}) P_{B(y,r)}(x,{w})\, d{w}. \end{aligned}$$

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

It remains to prove Claim 2, which we do now. Let $a_1 = \inf \nolimits _{B_k} g$ and $a_2 = \sup \nolimits _{B_k} g$. Without any loss of generality we may assume $a_1\ne a_2$. Then, we let

$$\begin{aligned} g'(x) = \frac{g(x) - a_1}{a_2 - a_1},\quad x\in B_k. \end{aligned}$$

We have $0\le g'\le 1$, ${{\,\textrm{Osc}\,}}_{B_k} g' = 1$, and ${{\,\textrm{Osc}\,}}_{B_{k+2}} g = {{\,\textrm{Osc}\,}}_{B_{k+2}} g' {{\,\textrm{Osc}\,}}_{B_k} g$. If $\sup \nolimits _{B_{k+2}} g' \le \tfrac{1}{2}$, then we are done, so assume otherwise. Note that

$$\begin{aligned} g'(x) = \frac{\frac{u_{k+1}^k(x) - a_1v_{k+1}^k(x)}{a_2-a_1}}{v_{k+1}^k(x)} =: \frac{g_1(x)}{g_2(x)},\quad x\in B_{k+2}. \end{aligned}$$

By (4.4), we have

$$\begin{aligned} 1\le \frac{\sup _{B_{k+2}} g_2}{\inf _{B_{k+2}} g_2}= \frac{\sup _{B_{k+2}} v_{k+1}^k}{\inf _{B_{k+2}} v_{k+1}^k} \le C(d,\alpha ). \end{aligned}$$

(4.5)

Furthermore, since $v_{k+1}^k(x) \le \sup \nolimits _{B_0}v\le C(d,\alpha ,\underline{D})$ for all $x\in \mathbb {R}^d$, we get

$$\begin{aligned} g_1(x) = v_{k+1}^k(x) g'(x) \le C(d,\alpha ,\underline{D}),\quad x\in B_k. \end{aligned}$$

If we extend $g_1$ to be equal to 0 on $\mathbb {R}^d{\setminus } B_k$, then $g_1$ is regular harmonic on $B_{k+1}$, nonnegative and bounded. Therefore, by the Harnack inequality in an explicit scale invariant formulation [14, Lemma 1]; see also Bass and Levin [8, Theorem 3.6] or Grzywny [43],

$$\begin{aligned} 1\le \frac{\sup _{B_{k+2}} g_1}{\inf _{B_{k+2}} g_1} \le C(d,\alpha ,\underline{D}). \end{aligned}$$

(4.6)

By (4.5) and (4.6), we get

$$\begin{aligned} \inf \limits _{B_{k+2}} g' \ge C^{-2} \sup \limits _{B_{k+2}} g' \ge \tfrac{1}{2} C^{-2}. \end{aligned}$$

Therefore,

$$\begin{aligned} {{\,\textrm{Osc}\,}}_{B_{k+2}} g' \le \max (\tfrac{1}{2}, 1 - \tfrac{1}{2} C^{-2}) = 1 - \tfrac{1}{2} C^{-2}, \end{aligned}$$

which ends the proof of Claim 2, and thus the lemma is proved. $\square$

Proof of Theorem 1.4

By Lemma 4.1, we can assume without loss of generality that $|y-y'|\le 1/16$.

We first consider the case $2^{k_0}|y' - y|^{1/2} \ge \delta _D(y)$, with $k_0$ from Lemma 4.3), and let $Q\in \partial D$ be such that $|y-Q| = \delta _D(y)$. Note that $y,y'\in B(Q,2^{k_0+1}|y-y'|^{1/2})$, because $|y-y'|>1$ implies $|y-y'| < |y-y'|^{1/2}$. We split the integral as follows:

$$\begin{aligned} \int _D \bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\, dz = \int _{D\cap B(Q,2^{k_0+2}|y-y'|^{1/2})} + \int _{D\setminus B(Q,2^{k_0+2}|y-y'|^{1/2})}. \end{aligned}$$

(4.7)

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

$$\begin{aligned}&\int _{D\cap B(Q,2^{k_0+2}|y-y'|^{1/2})}\bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\, dz\\ {}&\le 2^p\int _{D\cap B(Q,2^{k_0+2}|y-y'|^{1/2})}\bigg (\bigg |\frac{G_D(y,z)}{G_D(x_0,y)}\bigg |^p + \bigg |\frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\bigg )\, dz\\&\le c \int _{B(0,2^{k_0+2}|y-y'|^{1/2})} |z|^{p(\alpha -\gamma -d)}\, dz\\&= c C(d,\alpha ,p) |y-y'|^{(d+p(\alpha -\gamma -d))/2}. \end{aligned}$$

In the second integral of (4.7) we use the boundary Harnack principle given in [13, Lemma 3]: we let $u(y) = G_D(y,z)$, $v(y) = G_D(x_0,y)$ and $r = 2^{k_0+1}|y-y'|^{1/2}$ there. By the Green function estimates (2.10) and arguments similar to the proof of Lemma 4.1 we find that for $z\in D\cap (B(Q,2^{k+k_0+3}|y-y'|^{1/2}){\setminus } B(Q,2^{k+k_0+2}|y-y'|^{1/2}))$ we have $u(A_r(Q))/v(A_r(Q))\le C(d,\alpha ,\underline{D}) (2^k|y-y'|^{1/2})^{\alpha -\gamma -d}$, for all $k\in \{0,1,\ldots ,N_0\}$, where $N_0 = \lceil \log _2 (\textrm{diam}(D)/2^{k_0+2}|y-y'|^{1/2})\rceil$ and we define u/v to be 0 outside D. Therefore, by [13, Lemma 3], there exist c and $\sigma >0$ depending only on $d,\alpha ,\underline{D}$ such that

$$\begin{aligned} \bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p \le c (2^k|y-y'|^{1/2})^{p(\alpha -\gamma -d)} |y-y'|^{\sigma p/2} \end{aligned}$$

holds for all $z\in D\cap (B(Q,2^{k+k_0+3}|y-y'|^{1/2}){\setminus } B(Q,2^{k+k_0+2}|y-y'|^{1/2}))$. Hence,

$$\begin{aligned}&\int _{D\setminus B(Q,2^{k_0+2}|y-y'|^{1/2})}\bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\, dz\\&=\sum \limits _{k=0}^{N_0}\int _{D\cap (B(Q,2^{k+k_0+3}|y-y'|^{1/2})\setminus B(Q,2^{k+k_0+2}|y-y'|^{1/2}))}\bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\, dz\\&\le c |y-y'|^{\sigma p/2} \sum \limits _{k=0}^{N_0} (2^k|y-y'|^{1/2})^{p(\alpha -\gamma -d)} (2^k|y-y'|^{1/2})^d \\&= c|y-y'|^{\sigma p/2} \sum \limits _{k=0}^{N_0} (2^k|y-y'|^{1/2})^{d + p(\alpha -\gamma -d)}. \end{aligned}$$

The last sum is comparable to $\textrm{diam}(D)^{d+p(\alpha -\gamma -d)}$, so the proof is complete when $2^{k_0}|y-y'|^{1/2} \ge \delta _D(y)$.

Now assume that $2^{k_0}|y-y'|^{1/2}<\delta _D(y)$. We split the integral as follows:

$$\begin{aligned} \begin{aligned}&\int _D \bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg |^p\, dz \\&= \int _{D\cap B(y,2^{k_0}|y-y'|^{1/2})} + \int _{D\cap B(y,2^{k_0}|y-y'|^{1/2})^c\cap B(Q,3\delta _D(y))^c} \\&+ \int _{D\cap B(y,2^{k_0}|y-y'|^{1/2})^c\cap B(Q,3\delta _D(y))}. \end{aligned} \end{aligned}$$

(4.8)

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:

$$\begin{aligned} \int _{D\cap B(y,2^{k_0}|y-y'|^{1/2})^c\cap B(Q,3\delta _D(y))}&\le \sum \limits _{k=0}^{M_0} \int _{D\cap B(Q,3\delta _D(y))\cap (B(y,2^{k+k_0+1}|y-y'|^{1/2})\setminus B(y,2^{k+k_0}|y-y'|^{1/2}))} \\&=: \sum \limits _{k=0}^{M_0} I_k, \end{aligned}$$

where $M_0 = \lceil \log _2 (3\delta _D(y)/(2^{k_0}|y-y'|^{1/2}))\rceil$. We then use Lemma 4.3 with $r = r_k = 2^{k_0+k}|y-y'|^{1/2}/4$:

$$\begin{aligned} I_k&\le C(d,\alpha ,\underline{D}) |y-y'|^{\sigma p/2} \!\!\!\!\!\!\!\!\!\!\int \limits _{ B(y,2^{k+k_0+1}|y-y'|^{1/2})\setminus B(y,2^{k+k_0}|y-y'|^{1/2})} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!r_k^{p(\alpha -d-\gamma )}\, dz\\&\le \widetilde{C}(d,\alpha ,\underline{D}) |y-y'|^{\sigma p/2} (2^{k+k_0}|y-y'|^{1/2})^{d-p(\alpha -\gamma -d)},\quad k=0,\ldots ,M_0, \end{aligned}$$

since for $|y-y'|\le 1/16$, we have $|y-y'|\le |y-y'|^{1/2}/4$, so $y'\in B(y,2^{-k_0}r)$. Therefore we get

$$\begin{aligned} \sum \limits _{k=0}^{M_0} I_k \le C(d,\alpha ,\underline{D}) |y-y'|^{\sigma p/2} \delta _D(y)^{d-p(\alpha -\gamma -d)}, \end{aligned}$$

which ends the proof. $\square$

Proof of Theorem 1.2

Fix $x\in D$ and $t\in [T_1,T_2]$. First, we investigate $\eta ^{x_0}$. By the results of Sect. 2.3,

$$\begin{aligned} p_t^D(x,y) = G_D\Delta ^{\alpha /2}_yp_t^D(x,y). \end{aligned}$$

Furthermore, by Corollary 2.6, the function $f(y) = \Delta ^{\alpha /2}_yp_t^D(x,y)$ is bounded and the bound does not depend on $x\in D$. Therefore, by Theorem 1.4, for $y,y'\in D\setminus B(x_0,r_1)$,

$$\begin{aligned} \bigg |\frac{p_t^D(x,y)}{G_D(x_0,y)} - \frac{p_t^D(x,y')}{G_D(x_0,y')}\bigg |&\le \int _D \bigg |\frac{G_D(y,z)}{G_D(x_0,y)} - \frac{G_D(y',z)}{G_D(x_0,y')}\bigg | |f(z)|\, dz\\ {}&\le \bigg \Vert \frac{G_D(y,\cdot )}{G_D(x_0,y)} - \frac{G_D(y',\cdot )}{G_D(x_0,y')}\bigg \Vert _{L^1(D)} \Vert f\Vert _{\infty } \le C|y-y'|^{\sigma }, \end{aligned}$$

where the constants $C,\sigma$ depend only on $d,\alpha ,\underline{D},T_1,T_2,x_0,$ and $r_1$.

We now proceed to $\widetilde{\eta }$. Note that there exist $x_1\in D$ and $r = r(\underline{D})$ such that $B(x_1,2r)\subset D$. Without loss of generality, we can assume that $|y-y'| < r/4$. Then, for any fixed $y,y'$, there exists $x_2$ such that $B(x_2,r/4)\subset D$ and $y,y'\notin B(x_2,r/4)$. This means that $G_D(x_2,y),G_D(x_2,y'){\le } C$, where $C{\ge } 1$ depends only on $d,\alpha ,$ and $\underline{D}$. We then split as follows:

$$\begin{aligned} \bigg |\frac{p_t^D(x,y)}{p_{t_0}^D(x_0,y)} - \frac{p_t^D(x,y')}{p_{t_0}^D(x_0,y')}\bigg |&= \bigg |\frac{p_t^D(x,y)}{G_D(x_2,y)} \frac{G_D(x_2,y)}{p_{t_0}^D(x_0,y)}- \frac{p_t^D(x,y')}{G_D(x_2,y')}\frac{G_D(x_2,y')}{p_{t_0}^D(x_0,y')}\bigg |\nonumber \\&\le \frac{p_t^D(x,y)}{G_D(x_2,y)} \bigg |\frac{G_D(x_2,y)}{p_{t_0}^D(x_0,y)}-\frac{G_D(x_2,y')}{p_{t_0}^D(x_0,y')}\bigg |\nonumber \\&\quad + \frac{G_D(x_2,y')}{p_{t_0}^D(x_0,y')}\bigg |\frac{p_t^D(x,y')}{G_D(x_2,y')} - \frac{p_t^D(x,y)}{G_D(x_2,y)}\bigg |. \end{aligned}$$

(4.9)

By using Lemma 3.2 and (2.6), we find that

$$\begin{aligned} \frac{p_t^D(x,y)}{G_D(x_2,y)} \lesssim \frac{\mathbb {P}^y(\tau _D > t)}{G_D(x_2,y)} = \frac{G_DP_t^D\kappa _D(y)}{G_D(x_2,y)} \le C(d,\alpha ,\underline{D},T_1,T_2) < \infty . \end{aligned}$$

(4.10)

By similar arguments,

$$\begin{aligned} \frac{p_t^D(x_0,y)}{G_D(x_2,y)} \ge c(d,\alpha ,\underline{D},T_1,T_2,x_0)> 0. \end{aligned}$$

(4.11)

From (4.9), (4.10), (4.11), and the Hölder regularity of $\eta ^{x_0}$ obtained above, we arrive at

$$\begin{aligned} \bigg |\frac{p_t^D(x,y)}{p_{t_0}^D(x_0,y)} - \frac{p_t^D(x,y')}{p_{t_0}^D(x_0,y')}\bigg | \le C|y-y'|^\sigma , \end{aligned}$$

with C and $\sigma$ depending on $d,\alpha ,\underline{D},T_1,T_2,x_0$.

The arguments for $\eta$ are similar to the ones for $\widetilde{\eta }$, with no dependence on $x_0$. The proof is complete. $\square$

5 Space-time stable processes and caloric functions

5.1 Preliminaries

Recall that $(X_s)_{s\ge 0}$ is the isotropic $\alpha$-stable Lévy process. Like for the space-time Brownian motion [36], we define the space-time $\alpha$-stable process as the following Lévy process on $\mathbb {R}^{d+1}$:

$$\begin{aligned} \dot{X}_s {:}{=} (-s,X_s),\quad s\ge 0. \end{aligned}$$

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

$$\begin{aligned} \mathbb {P}^{(t,x)}(\dot{X}_s \in A) {:}{=} \mathbb {P}((t-s,X_s+x)\in A), \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}^{(t,x)}[f(\dot{X}_s)] = \mathbb {E}[f(t-s,X_s+x)]. \end{aligned}$$

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

$$\begin{aligned} \widetilde{p}_s(t,x,du,dy) = p_s(x,y)\, dy\otimes \delta _{\{t-s\}}(du),\quad s\ge 0,\ (t,x),(u,y)\in \mathbb {R}\times \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned} C^{1,2}_b([0,\infty )\times \mathbb {R}^d) = \{u\in C_b([0,\infty )\times \mathbb {R}^d): \partial _t u, \nabla _x u, D^2_x u\in C_b([0,\infty )\times \mathbb {R}^d)\}, \end{aligned}$$

with the norm $\Vert u\Vert _{C^{1,2}} = \Vert u\Vert _{\infty } + \Vert \partial _t u\Vert _{\infty } + \Vert \nabla _x u\Vert _{\infty } + \Vert D^2_xu\Vert _{\infty }$.

Lemma 5.1

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

Proof

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

$$\begin{aligned} \frac{1}{s} (\widetilde{P}_s u(t,x) - u(t,x))&= \frac{1}{s} \int _{\mathbb {R}^d}\int _{[0,\infty )} (u(r,y) - u(t,x)) \widetilde{p}_{s}(t,x,dy,dr)\nonumber \\&= \frac{1}{s} \int _{\mathbb {R}^d} (u(t-s,y) -u(t,x)) p_{s}(x,y) \, dy\nonumber \\&= \frac{1}{s} \int _{\mathbb {R}^d} (u(t-s,y) - u(t-s,x))p_{s}(x,y) \, dy \end{aligned}$$

(5.1)

$$\begin{aligned}&\quad + \frac{1}{s} (u(t-s,x) - u(t,x)). \end{aligned}$$

(5.2)

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

$$\begin{aligned} \frac{1}{s} \int _{\mathbb {R}^d} ((u(t,y) - u(t,x)) - (u(t-s,y) - u(t-s,x)))p_{s}(x,y) \, dy \end{aligned}$$

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

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

By Taylor’s expansion, $u(t-s,x) = u(t,x) - s\partial _t u(t,x) + o(s)$ as $s\rightarrow 0$, and similarly for y, so

$$\begin{aligned}&\bigg |\frac{1}{s} \int _{B(0,\delta )}((u(t,y) - u(t,x)) - (u(t-s,y) - u(t-s,x)))p_{s}(x,y) \, dy\bigg |\\&\quad =\bigg |\int _{B(0,\delta )} (\partial _t u(t,x) - \partial _t u(t,y) + \frac{o(s)}{s}) p_{s}(x,y) \, dy\bigg |\\&\quad \le \delta \Vert u\Vert _{C^{1,2}} + o(1). \end{aligned}$$

This ends the proof. $\square$

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

$$\begin{aligned} \tau _{G} {:}{=} \inf \{t>0: \dot{X}_t \notin G\}. \end{aligned}$$

Lemma 5.2

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

$$\begin{aligned}&\mathbb {P}^{(t,x)} (\dot{X}_{\tau _{\dot{D}}} \in (ds, dy)) \\ {}&\quad = {\left\{ \begin{array}{ll}{{\textbf {1}}}_{[r,t)}(s)\, ds\otimes J^D(t,x,s,y)\, dy + \delta _{t-r}(ds)\otimes p_{t-s}^D(x,y) \, dy,\quad &{}r>-\infty ,\\ {\textbf {1}}_{(-\infty ,t)}(s)\, ds\otimes J^D(t,x,s,y)\, dy,\quad &{}r=-\infty , \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} J^D(t,x,s,y) {:}{=} \int _D p_{t-s}^D(x,\xi ) \nu (\xi ,y)\, d\xi ,\quad s<t,\ x\in D,\ y\in D^c. \end{aligned}$$

We call $J^D$ the lateral Poisson kernel.

Remark 5.3

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

Proof of Lemma 5.2

Let $r>-\infty$. We have

$$\begin{aligned} \mathbb {P}^{(t,x)} (\dot{X}_{\tau _{\dot{D}}} \in (ds,dy))&=\mathbb {P}^{(t,x)} (\dot{X}_{\tau _{\dot{D}}} \in (ds,dy) , \tau _{\dot{D}} > \tau _D) \end{aligned}$$

(5.3)

$$\begin{aligned}&\quad +\mathbb {P}^{(t,x)} (\dot{X}_{\tau _{\dot{D}}} \in (ds,dy) , \tau _{\dot{D}} = \tau _D) \end{aligned}$$

(5.4)

$$\begin{aligned}&\quad + \mathbb {P}^{(t,x)} (\dot{X}_{\tau _{\dot{D}}} \in (ds,dy) , \tau _{\dot{D}} < \tau _D). \end{aligned}$$

(5.5)

Note that (5.3) vanishes, because $\mathbb {P}^{(t,x)}(\tau _{\dot{D}} > \tau _D) = 0$.

By the Ikeda–Watanabe formula (2.13), the term (5.4) is equal to

$$\begin{aligned} \mathbb {P}^{(t,x)} (X_{\tau _D} \in A,\, \tau _{D} \le t-r,\, \tau _D\in ds) = {{\textbf {1}}}_{[r,t)}(s)\, ds\otimes J^D(t,x,s,y)\, dy. \end{aligned}$$

In (5.5) we have $\tau _D > \tau _{\dot{D}} = t-r$, so by the definition of the Dirichlet heat kernel, this term is equal to

$$\begin{aligned} \delta _{t-r}(ds)\otimes p_{t-r}^D(x,y), \end{aligned}$$

see [32, Chapter 2]. The case of $r=-\infty$ is left to the reader. $\square$

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

$$\begin{aligned} \mathbb {E}^{(t,x)}u(\dot{X}_{\tau _{\dot{D}}}) = \int _r^t\int _{D^c} J^D(t,x, s,z)u(s,z) \, dz \, ds + \int _{D} p_{t-r}^D(x,y)u(r,y) \, dy, \end{aligned}$$

(5.6)

whenever this integral makes sense, e.g., for nonnegative u. By analogy to the elliptic equations, we call the right-hand side of (5.6) the Poisson integral, and the first term on the right-hand side of (5.6)—the lateral Poisson integral.

Remark 5.4

Another motivation for calling $J^D(t,x,s,z)$ the lateral Poisson kernel comes from the fact that it is the nonlocal normal derivative of $p_{t-s}^D$, whereas $p_{t-s}^D$ serves as the Green function for the fractional heat equation. Indeed, using the definition of the nonlocal normal derivative from [34]:

$$\begin{aligned}{}[\partial _{\textbf{n}} u](x) {:}{=} \int _{D} (u(y) - u(x))\nu (x,y)\, dy,\quad x\in D^c, \end{aligned}$$

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

$$\begin{aligned} \partial _{\textbf{n}} p_{t-s}^D(x,\cdot )(z) = \int _D p_{t-s}^D(x,y) \nu (y,z) \, dy = J^D(t,x,s,z), \quad x\in D. \end{aligned}$$

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 $-\infty<T_1<T_2<\infty$. We say that $u:(T_1,T_2)\times \mathbb {R}^d\rightarrow [0,\infty )$ is caloric in $(T_1,T_2)\times D$, if the mean value property:

$$\begin{aligned} u(t,x) = \mathbb {E}^{(t,x)} u(\dot{X}_{\tau _{G}}),\qquad (t,x)\in (T_1,T_2)\times D, \end{aligned}$$

(5.7)

holds for every open set $G\subset \subset (T_1,T_2)\times D$.

We say that $u:[T_1,T_2)\times \mathbb {R}^d\rightarrow [0,\infty )$ is caloric in $[T_1,T_2)\times D$ if (5.7) holds for every open $G\subset \subset [T_1,T_2)\times D$. If u is caloric in $[T_1,T_2)\times D$ and satisfies (5.7) for $G=(T_1,T_2)\times D$, then we say that u is regular caloric. If u is caloric in $[T_1,T_2)\times D$ and $u\equiv 0$ on the parabolic boundary

$$\begin{aligned} D^p {:}{=} (\{T_1\}\times D)\cup ((T_1,T_2)\times D^c), \end{aligned}$$

then we say that u is singular caloric.

Remark 5.6

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

• (b) We may also consider $T_1=-\infty$ or $T_2 = \infty$, where appropriate, in particular when defining functions caloric on $(T_1,T_2)\times D$.

• (c) The condition $G\subset \subset [T_1,T_2)\times D$ allows G to touch $\{T_1\}\times D$. Caloricity in $[T_1,T_2)\times D$ may be considered as a (new) relaxation of regular caloricity, localized near the part $\{T_1\}\times D$ of the boundary of $(T_1,T_2)\times D$, 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).

• (d) The caloricity in $[T_1,T_2)\times D$ helps to handle initial conditions which are functions, but also rules out some interesting cases, e.g., $(t,y)\mapsto p_t^D(x,y)$. See also [23]. Remarkably, every (nonnegative) function caloric in $(T_1,T_2)\times D$ has a certain initial condition which is a measure, see Sect. 6.

• (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 [29, Section 2.4.1] for viscosity solutions. See also Remark 5.12 below.

Lemma 5.7

Regular caloricity implies caloricity in $[T_1,T_2)\times D$, which in turn implies caloricity in $(T_1,T_2)\times D$. Furthermore, (5.7) only needs to be verified for cylinders G.

Proof

Assume that (5.7) holds for G. By the strong Markov property of $\dot{X}$, (5.7) then holds for every open $G' \subset G$:

$$\begin{aligned} u(t,x) = \mathbb {E}^{(t,x)} u(\dot{X}_{\tau _{G}}) = \mathbb {E}^{(t,x)} \mathbb {E}^{\dot{X}_{\tau _{G'}}} u(\dot{X}_{\tau _G}) = \mathbb {E}^{(t,x)} u(\dot{X}_{\tau _{G'}}). \end{aligned}$$

This first two assertions follow immediately. To clarify the third one, note that every open $G'\subset \subset [T_1,T_2)\times D$ is contained in an open cylinder, relatively compact in $[T_1,T_2)\times D$. Similarly for $(T_1,T_2)\times D$. $\square$

We continue with several examples of caloric functions.

Example 5.8

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

Example 5.9

If we let

$$\begin{aligned} \eta _{t,Q}(x) {:}{=} 0,\quad (t,x)\in (-\infty ,0]\times \mathbb {R}^d \cup (0,\infty )\times D^c,\ Q\in \partial D, \end{aligned}$$

(5.8)

then for every fixed $Q\in \partial D$, the function $(t,x)\mapsto \eta _{t,Q}(x)$ is caloric in $(-\infty ,\infty )\times D$. Indeed, the mean value property in $(\varepsilon ,T)\times D$, with $0<\varepsilon<T<\infty$ is a consequence of (1.9). Then, by Lemma 3.5,

$$\begin{aligned} \eta _{t,Q}(x)&= \int _0^t \int _{U^c} J^D(t,x,s,z) \eta _{s,Q}(z)\, dz\, ds\\&= \int _{-R}^t \int _{U^c} J^D(t,x,s,z) \eta _{s,Q}(z)\, dz\, ds, \end{aligned}$$

for any $R\ge 0$.

Example 5.10

If $f:\mathbb {R}^d \rightarrow [0,\infty )$ is a nonnegative measurable function and $P_{1}^D f(x)$ is finite for all $x\in D$, then $(t,x)\mapsto P_t^D f(x)$ is caloric in $[0,\infty )\times D$, with the usual convention $P_0^Df {:}{=} f$.

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

$$\begin{aligned} h(t,x) {:}{=} {\left\{ \begin{array}{ll}\int _{[0,t)} \int _{\partial D} \eta _{t-\tau ,Q}(x)\mu (dQ\, d\tau ),\quad &{}t>0,\ x\in D,\\ 0,\quad &{} \text{ elsewhere },\end{array}\right. } \end{aligned}$$

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

Proof

By Lemma 3.5, h is finite for all $t>0$ and $x\in D$, and by (5.8), we have

$$\begin{aligned} \int _{[0,t)} \int _{\partial D} \eta _{t-\tau ,Q}(x)\mu (dQ\, d\tau ) = \int _{[0,\infty )} \int _{\partial D} \eta _{t-\tau ,Q}(x)\mu (dQ\, d\tau ),\quad t\ge 0,\ x\in D. \end{aligned}$$

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

Remark 5.12

We note that the viscosity solution considered in [29, Section 2.4.1], although non-differentiable, is Lipschitz in time. The function $n_{t,Q}$ is not even Lipschitz in t because for $t\in (0,1)$ and fixed $x\in D$,

$$\begin{aligned} \frac{\eta _{t,Q}(x)}{t}&= \frac{1}{t} \lim \limits _{y\rightarrow Q}\frac{p_t^D(x,y)}{\mathbb {P}^y(\tau _D>t)}\frac{\mathbb {P}^y(\tau _D>t)}{\mathbb {P}^y(\tau _D>1)} \gtrsim \frac{p_t(x,Q)}{t} \lim _{y\rightarrow Q} \frac{\mathbb {P}^y(\tau _D>t)}{\mathbb {P}^y(\tau _D>1)} \\ {}&\gtrsim |x-Q|^{-d-\alpha }\lim _{y\rightarrow Q} \frac{\mathbb {P}^y(\tau _D>t)}{\mathbb {P}^y(\tau _D>1)}. \end{aligned}$$

We see, indeed, that the last limit is comparable to $t^{-1/2}$ if D is $C^{1,1}$ by (2.7). Furthermore, for Lipschitz D it also explodes as $t\rightarrow 0^+$ because of the proof of Lemma B.2 and [12, Lemma 3].

Lemma 5.13

If u is caloric in $(T_1,T_2)\times D$ for $T_1<T_2$, then $u\in L^1_\textrm{loc}((T_1,T_2)\times \mathbb {R}^d)$.

Proof

The proof is similar to the one of [18, Lemma 4.5]. First note that for any fixed $x\in D$, $r>0$, and $B=B(x,r)$, by (2.6) we have

$$\begin{aligned} J^{B}(t,x, s,z)&= \int _{B} p_{t-s}^{B}(x,y)\nu (y,z)\, dy \approx \int _B p_{t-s}(x,y)\mathbb {P}^y(\tau _B> t-s)\nu (y,z)\, dy\\ {}&\ge c\int _{B(x,r/2)} p_{t-s}(x,y)\, dy \ge C>0, \end{aligned}$$

with C depending only on r and R, where $\delta _B(z),t-s\le R$. Thus, $J^B(t,x,\cdot ,\cdot )$ is locally bounded from below. Now, take two disjoint balls $B_1,B_2\subseteq D$, centered at some points $x_1,x_2\in D$ respectively, and let $T_1<t_0<t<T_2$ and $R>0$. Since u is nonnegative and caloric, for $i=1,2$ we get

$$\begin{aligned} \infty > u(t,x) \ge \int _{t_0}^t\int _{B_i^c} u(s,z) J^{B_i}(t,x,s,z)\, dz\, ds \ge C\int _{t_0}^t\int _{B(0,R)\setminus B_i} u(s,z) \, dz\, ds. \end{aligned}$$

Therefore $u\in L^1((t_0,t)\times (B(0,R){\setminus } B_i))$ for $i=1,2$. But $B_1\cap B_2 = \emptyset$, so $u\in L^1((t_0,t)\times B(0,R))$. Since R can be chosen arbitrarily large, the proof is complete. $\square$

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

Lemma 5.14

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

$$\begin{aligned} u(t,x) = \mathbb {E}^{(t,x)}[u(\tau _{(-\infty ,t)\times U},X_{\tau _{(-\infty ,t)\times U}})] = \int _{-\infty }^t\int _{U^c} J^U(t,x,s,z)u(s,z)\, dz \, ds.\nonumber \\ \end{aligned}$$

(5.9)

In particular, the integral on the right-hand side of (5.9) is finite.

Proof

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

$$\begin{aligned} u(t,x) = \int _v^t\int _{U^c} J^U(t,x, s,z)u(s,z)\, dz\, ds + \int _U p_{t-v}^U(x,y) u(v,y)\, dy. \end{aligned}$$

The first integral on the right-hand side increases to the right-hand side of (5.9) by the monotone convergence theorem and the second integral decreases. It suffices to prove that

$$\begin{aligned} a{:}{=}\lim \limits _{v\rightarrow -\infty } \int _U p_{t-v}^U(x,y) u(v,y)\, dy = 0. \end{aligned}$$

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

$$\begin{aligned} \int _U p_{t-v}^U(x,y)u(v,y) \, dy \ge a. \end{aligned}$$

Let $n >0$ be so large that $U\subset \subset D_n$ (see (2.1)). Recall that $\lambda _1(V)$ is the first eigenvalue of the Dirichlet fractional Laplacian for an open set V. We claim that

$$\begin{aligned} \lambda _1(D_n) < \lambda _1 (U). \end{aligned}$$

(5.10)

A weak inequality is well known as the domain monotonicity. In order to prove the strict inequality, assume without loss of generality that $0\in U$. Then there exists $q>1$ such that $qU\subset \subset D_n$, so, by domain monotonicity, $\lambda _1(D_n)\le \lambda _1(qU)=q^{-\alpha } \lambda _1(U)$, which yields (5.10).

By (2.9), (2.8), and the fact that each eigenfunction is bounded from above and bounded from below away from the boundary, for $s<t$, $s\rightarrow -\infty$, we get

$$\begin{aligned} \infty >u(t,x)&\ge \int _{D_n} u(s,y) p_{t-s}^{D_n}(x,y) \, dy \ge \int _U u(s,y) p_{t-s}^{D_n}(x,y)\, dy \\ {}&\approx \int _{U} u(s,y)e^{-\lambda _1(D_n)(t-s)}\, dy\\&= e^{(-\lambda _1(D_n) + \lambda _1(U))(t-s)}\int _U u(s,y)e^{-\lambda _1(U)(t-s)}\, dy\\ {}&\gtrsim e^{(-\lambda _1(D_n) + \lambda _1(U))(t-s)}\int _U u(s,y) p_{t-s}^U(x,y)\, dy. \end{aligned}$$

By (5.10), we must have $a=0$. $\square$

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 $J^D$, which may be of independent interest. Let us note in passing that bounded caloric functions are known to be locally Hölder continuous [31, Theorem 4.14].

Proposition 5.15

Assume that u is a nonnegative caloric function in $(T_0,T_1)\times D$ for some $T_0<T_1$. Then, u is continuous and locally bounded therein.

We fix arbitrary $(t_0,x_0)\in (T_0,T_1)\times D$, $r\in (0,\delta _D(x_0)/2)$, and let $B_\rho = B(x_0,\rho )$ for $\rho >0$. We first establish some basic facts about the lateral Poisson kernel. With a slight conflict of notation, we introduce the Euclidean distance between $A,B \subseteq {\mathbb {R}}^d$,

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

Lemma 5.16

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

$$\begin{aligned} J^D(t,x,s,z) \approx J^D(t,x_0,s,z),\quad x\in U,\ z\in D^c,\ 0<t-s<T, \end{aligned}$$

(5.11)

and

$$\begin{aligned} J^D(t,x,s,z) \lesssim J^D(t',x,s,z),\quad x\in U,\ z\in D^c,\ 0<t-s\le t'-s<T, \end{aligned}$$

(5.12)

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

Proof

Let $U'$ be such that $U\subset \subset U' \subset \subset D$. We pick $U'$ so that the constants below depend only on D and U, e.g., by assuming $d(U,D^c)/2 \ge d(U',D^c) \ge d(U,D^c)/3$. We first prove (5.11). By (2.6),

$$\begin{aligned} J^D(t,x,s,z)&= \int _D p_{t-s}^D(x,y)\nu (y,z)\, dy \nonumber \\&\approx \mathbb {P}^x(\tau _D>t-s)\int _D p_{t-s}(x,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy\nonumber \\&\approx \mathbb {P}^{x_0}(\tau _D>t-s)\bigg (\int _{D\setminus U'} + \int _{U'}\bigg ) p_{t-s}(x,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy, \end{aligned}$$

(5.13)

with constants depending on $d,\alpha ,\underline{D},d(U,D^c)$, and T. For $y\in D\setminus U'$, we have $|x-y| \approx |x_0-y|$, so by (2.4),

$$\begin{aligned}&\int _{D\setminus U'}p_{t-s}(x,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy\\&\quad \approx \int _{D\setminus U'}p_{t-s}(x_0,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy. \end{aligned}$$

For $y\in U'$, $\mathbb {P}^y(\tau _D>t-s)\approx 1$ and $\nu (y,z)\approx \nu (x_0,z)$. Using this and the fact that $U\subset \subset U'$, we find that

$$\begin{aligned} \int _{U'} p_{t-s}(x,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy&\approx \nu (x_0,z)\int _{U'}p_{t-s}(x,y)\, dy \\ {}&\approx \nu (x_0,z)\int _{U'}p_{t-s}(x_0,y)\, dy\\&\approx \int _{U'} p_{t-s}(x_0,y)\mathbb {P}^y (\tau _D>t-s)\nu (y,z)\, dy. \end{aligned}$$

Coming back to (5.13), we obtain (5.11). We now proceed to proving (5.12). We split in a similar way:

$$\begin{aligned} J^D(t,x,s,z) = \bigg (\int _{U'} + \int _{D\setminus U'}\bigg ) p_{t-s}^D(x,y)\nu (y,z)\, dy. \end{aligned}$$

By Lemma B.1,

$$\begin{aligned} \int _{D\setminus U'} p_{t-s}^D(x,y)\nu (y,z)\, dy \lesssim \int _{D\setminus U'} p_{t'-s}^D(x,y)\nu (y,z)\, dy. \end{aligned}$$

For the integral over $U'$ we use (2.6):

$$\begin{aligned}&\int _{U'} p_{t-s}^D(x,y)\nu (y,z)\, dy \\&\quad \approx \nu (x_0,z) \int _{U'}p_{t-s}^D(x,y)\, dy \\&\quad \approx \nu (x_0,z) \int _{U'}p_{t-s}(x,y) \mathbb {P}^x(\tau _D>t-s)\mathbb {P}^y(\tau _D>t-s)\, dy. \end{aligned}$$

For $w\in U'$ and $0<t-s<T$, we have $\mathbb {P}^w(\tau _D > t-s)\approx 1$ and by (2.4), $\int _{U'}p_{t-s}(x,y)\, dy \approx 1$, with comparability constants depending only on $T,U'$, and $\underline{D}$. It follows that

$$\begin{aligned}&\nu (x_0,z) \int _{U'}p_{t-s}(x,y) \mathbb {P}^x(\tau _D>t-s)\mathbb {P}^y(\tau _D>t-s)\, dy\\ {}&\quad \approx \,\nu (x_0,z) \int _{U'}p_{t'-s}(x,y) \mathbb {P}^x(\tau _D>t'-s)\mathbb {P}^y(\tau _D>t'-s)\, dy\\&\quad \approx \int _{U'} p_{t'-s}^D(x,y)\nu (y,z)\, dy, \end{aligned}$$

which ends the proof. $\square$

Proof of Proposition 5.15

We will show continuity at the fixed point $(t_0,x_0)$. Let $x\in B_{r/2}$, $t_1\in (T_0,t_0)$ and $t\in (t_1,T_1)$, so that $T_1<t_1<t<T_0$. We have

$$\begin{aligned} u(t,x)&= \int _{B_r} u(t_1,y)p_{t-t_1}^{B_r}(x,y)\, dy + \int _{t_1}^t\int _{B_r} u(\tau ,z) J^{B_r}(t,x,\tau ,z)\, dz\, d\tau ,\\ u(t_0,x_0)&= \int _{B_r} u(t_1,y)p_{t_0-t_1}^{B_r}(x_{0},y)\, dy + \int _{t_1}^{t_0}\int _{B_r} u(\tau ,z) J^{B_r}(t_0,x_{0},\tau ,z)\, dz\, d\tau . \end{aligned}$$

Since u is nonnegative and caloric, all integrals above are finite. For (t, x) sufficiently close to $(t_0,x_0)$, we have $p_{t-t_1}^{B_r}(x,y) \approx p_{t_0-t_1}^{B_r}(x_0,y)$ uniformly in y. Therefore, by the dominated convergence theorem,

$$\begin{aligned} \int _{B_r} u(t_1,y)p_{t-t_1}^{B_r}(x,y)\, dy \mathop {\longrightarrow }\limits _{(t,x)\rightarrow (t_0,x_0)} \int _{B_r} u(t_1,y)p_{t_0-t_1}^{B_r}(x_0,y)\, dy. \end{aligned}$$

Therefore it remains to show that

$$\begin{aligned} \int _{t_1}^t\int _{B_r} u(\tau ,z) J^{B_r}(t,x,\tau ,z)\, dz\, d\tau \mathop {\longrightarrow }\limits _{(t,x)\rightarrow (t_0,x_0)} \int _{t_1}^{t_0}\int _{B_r} u(\tau ,z) J^{B_r}(t_0,x_0,\tau ,z)\, dz\, d\tau . \end{aligned}$$

Assume that $t>t_0$ (we skip the other case, as it is similar). Then,

$$\begin{aligned}&\bigg |\int _{t_1}^t\int _{B_r} u(\tau ,z) J^{B_r}(t,x,\tau ,z)\, dz\, d\tau - \int _{t_1}^{t_0}\int _{B_r} u(\tau ,z) J^{B_r}(t_0,x_0,\tau ,z)\, dz\, d\tau \bigg |\\&\quad \le \int _{t_1}^{t_0}\int _{B_r} u(\tau ,z) |J^{B_r}(t,x,\tau ,z) - J^{B_r}(t_0,x_0,\tau ,z)|\, dz\, d\tau \\&\qquad + \int _{t_0}^t\int _{B_r} u(\tau ,z) J^{B_r}(t,x,\tau ,z)\, dz\, d\tau =: I_1 + I_2. \end{aligned}$$

By Lemma 5.16, we have $J^{B_r}(t,x,\tau ,z) \lesssim J^{B_r}(t_0+\varepsilon ,x_0,\tau ,z)$ for $t_1\le \tau \le t\le t_0+\varepsilon$, $x\in B_{r/2}$, and $z\in B_r^c$. Therefore by the dominated convergence theorem, $I_2 \rightarrow 0$. Furthermore, by the properties of $p_t^{B_r}$ and the dominated convergence theorem, it is easy to see that $J^{B_r}(\cdot ,\cdot ,\tau ,z)$ is continuous on $(\tau ,\infty )\times B_r$ for all $\tau \in \mathbb {R}$ and $z\in D^c$. Therefore, using the bounds of Lemma 5.16 and the dominated convergence theorem once again, we find that $I_1\rightarrow 0$ as well. This ends the proof. $\square$

6 Representation of caloric functions in Lipschitz open sets

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

6.1 Functions caloric up to time 0

Lemma 6.1

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

Proof

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

$$\begin{aligned} u(t,x) = \mathbb {E}^{(t,x)} u(\dot{X}_{\tau _{\dot{D}_n}}) =: i_n(t,x) + l_n(t,x) + s_n(t,x), \end{aligned}$$

where

$$\begin{aligned}&i_n(t,x) = \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}_n}})\, ; \, \tau _{D_n} > t],\\&l_n(t,x) = \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}_n}})\, ; \, \tau _{D_n}< t,\, \tau _{D_n} = \tau _D],\\&s_n(t,x) = \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}_n}})\, ; \, \tau _{D_n}< t,\, \tau _{D_n} < \tau _D]. \end{aligned}$$

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

$$\begin{aligned} i_n(t,x) = \mathbb {E}^{(t,x)}(u(\dot{X}_t)\,; \, \tau _{D_n}> t) \nearrow \mathbb {E}^{(t,x)}[u(\dot{X}_t)\,; \, \tau _{D}>t] =: i(t,x), \end{aligned}$$

and by [12, (5.40)],

$$\begin{aligned} l_n(t,x) = \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}}})\,; \, \tau _D< t,\, \tau _{D_n} = \tau _D] \nearrow \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}}})\,; \, \tau _D < t] =: l(t,x), \end{aligned}$$

the limits being finite because all $i_n$, $l_n$, and $s_n$ are nonnegative. So, $s_n(t,x)$ converges to some s(t, x). Since $r(t,x){:}{=}i(t,x) + l(t,x) = \mathbb {E}^{(t,x)}u(\dot{X}_{\tau _{\dot{D}}})$, r is regular caloric. By inspecting the definition of $s_n$, we find that s is singular caloric: indeed, if $X_t$ starts from $x\in D^c$, then the event $\tau _{D_n} < \tau _D$ has probability 0, so $s_n(t,x)=0$ for $x\in D^c$, and if $\dot{X}$ starts from (0, x), $x\in D$, then $s_n(0,x) = 0$ because $\tau _{D_n} \ge 0$.

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

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

$$\begin{aligned} \bigg |\frac{p_t^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} - \frac{p_t^{D}(x,y)}{\mathbb {P}^y(\tau _D>1)}\bigg | \le \omega \bigg (\frac{1}{n}\bigg ),\quad y\in D_n,\ t\in [\varepsilon ,T]. \end{aligned}$$

(6.1)

Proof

First note that the expression on the left-hand side of (6.1) converges to 0 as $n\rightarrow \infty$ for every fixed $y\in D$ (the expression is considered only when $1/n<\delta _D(y)$). In order to get (6.1) we will show that the convergence is uniform by using the Arzelà–Ascoli theorem. Indeed, by Theorem 1.2, we find that $\overline{D_n}\ni y\mapsto p_t^{D_n}(x,y)/\mathbb {P}^y(\tau _{D_n}>1)$ are uniformly Hölder continuous for n large and $t\in [\varepsilon ,T]$. Furthermore, it is well-known that a Hölder continuous function in $\overline{D_n}$ can be extended to a function on $\overline{D}$ with the same Hölder regularity, see, e.g., Banach [5, IV (7.5)]. If we denote the corresponding extensions by $f_n$, then by the Arzelà–Ascoli theorem, we find that

$$\begin{aligned} \bigg |f_n(t,y) - \frac{p_t^{D}(x,y)}{\mathbb {P}^y(\tau _D>1)}\bigg | \le \omega \bigg (\frac{1}{n}\bigg ),\quad y\in D,\ t\in [\varepsilon ,T]. \end{aligned}$$

In particular, (6.1) follows. $\square$

Theorem 6.3

Assume that u is singular caloric in $[0,T)\times D$. Then there exists a nonnegative Borel measure $\mu$ on $\partial D\times [0,T)$ such that representation (1.2) holds.

Proof

Let $D_n$ be as in Lemma 6.1 and let N be large enough, so that $x,x_0\in D_N$. Since u is singular caloric, for natural $n>N$ we have

$$\begin{aligned} u(t,x)&= \mathbb {E}^{(t,x)}[u(\dot{X}_{\tau _{\dot{D}_n}})\, ; \, \tau _{D_n} < t, X_{\tau _{D_n}} \in D\setminus D_n]\\&= \int _0^t\int _{D\setminus D_n} u(s,z)\int _D p_{t-s}^{D_n}(x,y)\nu (y,z)\, dy \, dz\, ds\\&= \int _0^t\int _D \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\int _{D\setminus D_n} \mathbb {P}^y(\tau _{D_n}>1)u(s,z) \nu (y,z)\, dz \, dy\, ds. \end{aligned}$$

We define

$$\begin{aligned} \mu _n(dy\, ds) = \int _{D\setminus D_n} \mathbb {P}^y(\tau _{D_n}>1) u(s,z) \nu (y,z) \, dz \, dy\, ds. \end{aligned}$$

Note that by (2.6), if we fix $\theta >0$, then we have $\mathbb {P}^y(\tau _{D_n}>1)\lesssim p_{s+\theta }^{D_n}(x_0,y)$ uniformly in $s\in (0,t)$. Therefore, since u is caloric, for $\theta$ sufficiently small we have

$$\begin{aligned} \int _0^t \int _{\mathbb {R}^d}\mu _n(dy\, ds) \lesssim \int _0^t \int _{\mathbb {R}^d}\ \int _{D\setminus D_n} p_{t+\theta -s}^{D_n}(x_0,y) u(s,z) \nu (y,z) \, dz \, dy\, ds \le u(x_0,t+\theta ), \end{aligned}$$

which means that the masses of $\mu _n$ are uniformly bounded. With this notation we have

$$\begin{aligned} u(t,x) = \int _0^t \int _D \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\, \mu _n(dy\, ds). \end{aligned}$$

The goal is then to show that the right-hand side converges to the right-hand side of (1.2). To this end we will isolate small times and look separately at $D_N$ and $D{\setminus } D_N$.

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

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

$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\int _0^t\int _{D_{N}} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} \mu _n(dy\, ds)\nonumber \\{} & {} \quad \lesssim \lim \limits _{n\rightarrow \infty }\int _0^t\int _{D\setminus D_n} u(s,z)\int _{D_N} p_{t-s}^{D}(x,y)\nu (y,z)\, dy \, dz\, ds = 0. \end{aligned}$$

(6.2)

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

$$\begin{aligned} \int _{t-\epsilon }^t\int _{D} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} \mu _n(dy\, ds) < \omega (\epsilon ). \end{aligned}$$

(6.3)

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

$$\begin{aligned} \int _{t-\epsilon }^t\int _{D} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} \mu _n(dy\, ds)&= \int _{t-\epsilon }^t\int _{D\setminus D_n} J^{D_n}(t,x,s,z)u(s,z)\, dz\, ds \\&= u(t,x) - \int _{D_n} p_{\epsilon }^{D_n}(x,y)u(t-\epsilon ,y)\, dy. \end{aligned}$$

The last expression converges to 0 for $\epsilon \rightarrow 0^+$ for all fixed n, because u is continuous in both variables, and it is nonincreasing with respect to n because of the domain monotonicity. This proves (6.3).

Note also that the right-hand side of (1.2) is finite because $\mu$ is a finite measure and $\eta _{s,Q}(x)$ is bounded in s and Q for fixed x. Therefore,

$$\begin{aligned} \lim \limits _{\epsilon \rightarrow 0^+}\int _{[t-\epsilon ,t)}\int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds) = 0. \end{aligned}$$

(6.4)

By (6.2), (6.3), and (6.4), for any $\delta >0$ there exist $\epsilon$ (small) and $N_0$ (large) such that for $n>N_0$,

$$\begin{aligned}&\bigg |\int _0^t \int _D \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\, \mu _n(dy\, ds) - \int _{[0,t)} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg |\\&\quad \le \,\bigg |\int _{[t-\epsilon ,t)}\int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg | + \bigg |\int _{t-\epsilon }^t\int _{D\setminus D_{N}} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} \mu _n(dy\, ds)\bigg | \\ {}&\qquad + \bigg |\int _0^t\int _{D_{N}} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)} \mu _n(dy\, ds)\bigg |\\&\qquad +\,\bigg |\int _0^{t-\epsilon } \int _{D\setminus D_N} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\, \mu _n(dy\, ds) - \int _{[0,t-\epsilon )} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg |\\&\quad \le 3\delta + \bigg |\int _0^{t-\epsilon }\int _{D\setminus D_N} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\, \mu _n(dy\, ds) - \int _{[0,t-\epsilon )} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg |. \end{aligned}$$

Furthermore, if $N_0$ is large enough, then by Lemma 6.2,

$$\begin{aligned}&\bigg |\int _0^{t-\epsilon } \int _{D\setminus D_N} \frac{p_{t-s}^{D_n}(x,y)}{\mathbb {P}^y(\tau _{D_n}>1)}\, \mu _n(dy\, ds) - \int _{[0,t-\epsilon ]} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg |\\ {}&\quad \le \delta + \bigg |\int _0^{t-\epsilon } \int _{D\setminus D_N} \frac{p_{t-s}^{D}(x,y)}{\mathbb {P}^y(\tau _{D}>1)}\, \mu _n(dy\, ds) - \int _{[0,t-\epsilon ]} \int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds)\bigg |. \end{aligned}$$

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

Theorem 6.4

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

Proof

Following [12, 23], we start by showing that the measures $\mu _n^Q$ corresponding to $\eta _{t,Q}$ converge to $\delta _Q\otimes \delta _0$ for $t>0$, $Q\in \partial D$. To this end, fix $Q\in \partial D$ and let

$$\begin{aligned} \mu _n^Q(y,s) = \mathbb {P}^y(\tau _{D_n}>1) \int _{D\setminus D_n} \eta _{s,Q}(z) \nu (y,z)\, dz,\quad s>0,\ y\in \mathbb {R}^d. \end{aligned}$$

By Lemma 3.5, $\mu _n^Q((B(Q,\varepsilon )\times [0,\varepsilon ))^c) \rightarrow 0$ as $n\rightarrow \infty$, for any $\varepsilon >0$. So, $\mu _n$ converges weakly to $\delta _Q\otimes \delta _0$.

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

$$\begin{aligned} u(t,x) = \int _{[0,t)}\int _{\partial D} \eta _{t-s,Q}(x) \mu (dQ\, ds). \end{aligned}$$

Let $\mu _n(y,s) = \int _{D{\setminus } D_n} \mathbb {P}^y(\tau _D>1)u(s,z)\nu (y,z)\, dz$. By Fubini–Tonelli,

$$\begin{aligned} \mu _n(y,s)&= \int _{D\setminus D_n} \mathbb {P}^y(\tau _D>1) \nu (y,z)\int _{[0,s)}\int _{\partial D} \eta _{s-\tau ,Q}(z)\mu (dQ\, d\tau )\, dz\\&=\int _{[0,s)}\int _{\partial D} \mu _n^Q(y,s-\tau ) \mu (dQ\, d\tau ). \end{aligned}$$

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

$$\begin{aligned}&\int _0^t \int _{D} f(y,s) \mu _n(y,s)\, dy\, ds\\&\quad = \int _0^t \int _{D} f(y,s)\int _{[0,s)} \int _{\partial D} \mu _n^Q(y,s-\tau )\mu (dQ\, d\tau )\, dy\, ds\\&\quad = \int _{[0,t)} \int _{\partial D} \int _0^{t-\tau }\int _{D} f(y,s+\tau ) \mu _n^Q(y,s) \, dy\, ds\, \mu (dQ\, d\tau ). \end{aligned}$$

Since $\mu _n^Q \implies \delta _Q\otimes \delta _0$, the above integral with respect to $dy\, ds$ converges to $f(Q,\tau )$. Therefore, by the dominated convergence theorem,

$$\begin{aligned} \int _0^t \int _{D} f(y,s) \mu _n(y,s)\, dy\, ds \mathop {\longrightarrow }_{n\rightarrow \infty } \int _{[0,t)} \int _{\partial D} f(Q,s) \mu (dQ\,ds), \end{aligned}$$

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

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

Theorem 6.5

Assume that u is caloric on $(0,T)\times D$ and let $g=u|_{D^c}$. Then there exist unique bounded nonnegative measures $\mu$ on $[0,T)\times \partial D$ and $\mu _0$ on D such that for all $0<t<T$ and $x\in D$, (1.3) holds.

Proof

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

$$\begin{aligned} u(t,x)&= P_{t-\varepsilon }^D u(\varepsilon ,\cdot )(x) + \int _{[\varepsilon ,t)}\int _{\partial D} \eta _{t-s,Q}(x)\, \mu (dQ\, ds)\\&\quad + \int _\varepsilon ^t\int _{D^c} g(s,z)J^D(t,x,s,z)\, dz\, ds. \end{aligned}$$

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

$$\begin{aligned} u(t,x)&= \lim \limits _{\varepsilon \rightarrow 0^+}P_{t-\varepsilon }^D u(\varepsilon ,\cdot )(x) + \int _{(0,t)}\int _{\partial D} \eta _{t-s,Q}(x)\, \mu (dQ\, ds)\\&\quad + \int _0^t\int _{D^c} g(s,z)J^D(t,x,s,z)\, dz\, ds, \end{aligned}$$

where the remaining limit exists and the expression under it decreases. Since $p_{t-\varepsilon }^D(x,y) \approx p_{t}^D(x,y)$ and $p_t^D(x,\cdot )\approx 1$ for any $U\subset \subset D$ we find that $u(\varepsilon ,\cdot )$ have bounded integral on U. Therefore, by the Prokhorov theorem, there is a sequence $(\varepsilon _n)$ such that $u(\varepsilon _n,\cdot )$ converge weakly on compact subsets of D to a measure $\mu _0$, locally finite on D. Furthermore, we have

$$\begin{aligned} P_{t-\varepsilon }^D u(\varepsilon ,\cdot )(x) = \int _D p_{t-\varepsilon }^D(x,y)u(\varepsilon ,y)\, dy = \int _D \frac{p_{t-\varepsilon }^D(x,y)}{\mathbb {P}^y(\tau _D>1)} \mathbb {P}^y(\tau _D>1)u(\varepsilon ,y)\, dy. \end{aligned}$$

Since $\frac{p_{t-\varepsilon }^D(x,y)}{\mathbb {P}^y(\tau _D>1)} \approx \frac{p_{t}^D(x,y)}{\mathbb {P}^y(\tau _D>1)} \approx 1$ we find that the functions $y\mapsto \mathbb {P}^y(\tau _D>1)u(\varepsilon ,y)$ have bounded mass. By Prokhorov theorem, we can infer without loss of generality that $\mathbb {P}^y(\tau _D>1)u(\varepsilon _n,y)$ converge weakly to a finite measure $\widetilde{\mu }$ on $\overline{D}$. We have $\widetilde{\mu }(dy) = {\mathbb {P}}^y(\tau _D > 1)\mu _0(dy)$ on D. By (3.12),

$$\begin{aligned}&\lim \limits _{\varepsilon \rightarrow 0^+} \int _D \frac{p_{t-\varepsilon }^D(x,y)}{\mathbb {P}^y(\tau _D>1)} \mathbb {P}^y(\tau _D>1)u(\varepsilon ,y)\, dy = \int _{\overline{D}} \frac{p_{t}^D(x,y)}{\mathbb {P}^y(\tau _D>1)}\, \widetilde{\mu }(dy)\\&\quad =\int _D p_{t}^D(x,y)\, \mu _0(dy) + \int _{\partial {D}} \eta _{t,Q}(x)\, \widetilde{\mu }(dQ). \end{aligned}$$

We end the proof by defining $\mu$ on $\partial D \times [0,T)$ as $\mu {\textbf {1}}_{\partial D\times (0,T)} +\widetilde{\mu }\otimes \delta _0(dt)$. $\square$

Appendices

Appendix A: Weak and classical formulations for caloric functions

The following result seems to be well-known, but we were unable to locate a proof. The arguments are very similar to the case of the Laplacian discussed by Hunt [46].

Lemma A.1

For any $x\in D$ the function $(t,y) \mapsto p_t^D(x,y)$ is a classical solution to the fractional heat equation with the Dirichlet condition:

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t - \Delta ^{\alpha /2}_y)p_t^D(x,y) = 0\quad &{}t>0,\ y\in D,\\ p_t^D(x,y) = 0\quad &{}t>0,\ y\in D^c. \end{array}\right. } \end{aligned}$$

(A.1)

It is also a weak solution in the sense that for $\phi \in C_c^\infty ([0,\infty )\times \mathbb {R}^d)$ and $0<t_1<t_2<\infty$ we have

$$\begin{aligned} \int _{t_1}^{t_2} \int _D (\partial _t + \Delta ^{\alpha /2})\phi (t,y) p_t^D(x,y)\, dy\, dt&= \int _D \phi (t_2,y) p_{t_2}^D(x,y)\, dy\\&\quad - \int _D \phi (t_1,y) p_{t_1}^D(x,y)\, dy. \end{aligned}$$

Proof

By definition, the exterior condition is satisfied, so it suffices to verify that $(\partial _t - \Delta ^{\alpha /2})p_t^D(x,y) = 0$. To this end we will differentiate the Hunt formula. Using the subordination and Fourier inversion formulas (see, e.g., Bogdan and Jakubowski [22, Lemma 5]) it is easy to see that $p_t$ is smooth in x for $t>0$ and $\partial ^{\beta }_yp_t(x,y)$ is bounded whenever $|x-y|$ is separated from 0 for any $\beta \in \mathbb {N}_0$. Note that this is the case for $|X_{\tau _D}-y|$. Therefore, for fixed (t, y), by the dominated convergence theorem we find

$$\begin{aligned} \partial ^\beta _y p_t^D(x,y)&= \partial ^\beta _y p_t(x,y) - \partial ^\beta _y \mathbb {E}^x[ p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]\\&=\partial ^\beta _y p_t(x,y) - \mathbb {E}^x[ \partial ^\beta _yp_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]. \end{aligned}$$

Furthermore,

$$\begin{aligned} \Vert p_t^D(x,\cdot )\Vert _{C^2(B(y,\delta _D(y)/2))} < \infty , \end{aligned}$$

(6.2)

hence $\Delta ^{\alpha /2}_y p_t^D(x,y)$ is well defined for $(t,y)\in D\times (0,\infty )$ and we have

$$\begin{aligned} \Delta ^{\alpha /2}_y p_t^D(x,y) = \Delta ^{\alpha /2}_y p_t(x,y) - \Delta ^{\alpha /2}_y \mathbb {E}^x[ p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]. \end{aligned}$$

We can also interchange $\Delta ^{\alpha /2}_y$ with the expectation. The easiest way to see that is by using Fubini–Tonelli, (6.2) and the Taylor expansion in the following (symmetrized) representation of the fractional Laplacian:

$$\begin{aligned} \Delta ^{\alpha /2}_y u(x) = \int _{\mathbb {R}^d} (u(x+y) - 2u(x) + u(x-y))\nu (y)\, dy. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \Delta ^{\alpha /2}_y p_t^D(x,y) = \Delta ^{\alpha /2}_y p_t(x,y) - \mathbb {E}^x[\Delta ^{\alpha /2}_y p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]. \end{aligned}$$

We now compute the time derivative. Note that $\partial _t p_t(x,y)$ exists and is equal to $\Delta ^{\alpha /2}_y p_t(x,y)$, so it is bounded for $|x-y|$ separated from 0. We have

$$\begin{aligned} \partial _t p_t^D(x,y) = \partial _t p_t(x,y) - \partial _t \mathbb {E}^x[ p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t], \end{aligned}$$

provided the last expression exists, which we now prove. Without loss of generality, let $h>0$. We have

$$\begin{aligned}&\frac{1}{h} \big [\mathbb {E}^x[p_{t+h-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t+h] - \mathbb {E}^x[p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]\big ]\\&\quad =\frac{1}{h} \mathbb {E}^x[p_{t+h-\tau _D}(X_{\tau _D},y) - p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t] \\ {}&\qquad + \frac{1}{h} \mathbb {E}^x[p_{t+h-\tau _D}(X_{\tau _D},y)\, ; \, t\le \tau _D<t+h]. \end{aligned}$$

By the dominated convergence theorem, we get that

$$\begin{aligned}&\lim \limits _{h\rightarrow 0^+}\frac{1}{h} \mathbb {E}^x[p_{t+h-\tau _D}(X_{\tau _D},y) - p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]\\&\quad = \mathbb {E}^x[ \partial _t p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]. \end{aligned}$$

Furthermore, by (2.4),

$$\begin{aligned} \frac{1}{h} \mathbb {E}^x[p_{t+h-\tau _D}(X_{\tau _D},y)\, ; \, t\le \tau _D<t+h] \le C \mathbb {P}^x(\tau _D \in [t,t+h)), \end{aligned}$$

and the last expression converges to 0 by the dominated convergence theorem. Therefore we get

$$\begin{aligned} \partial _t p_t^D(x,y)&= \partial _t p_t(x,y) - \mathbb {E}^x[ \partial _tp_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t]\\ {}&= \Delta ^{\alpha /2}_y p_t(x,y) - \mathbb {E}^x[\Delta ^{\alpha /2}_y p_{t-\tau _D}(X_{\tau _D},y)\, ; \, \tau _D<t] = \Delta ^{\alpha /2}_y p_t^D(x,y), \end{aligned}$$

so $p_t^D$ is a classical solution to the fractional heat equation (A.1). It is also a weak solution, as follows from integration by parts and the fact that the support of the test function $\phi$ is separated from $\partial D$. $\square$

Appendix B: Almost-increasingness

The following result is used in the proof of Lemma 5.16.

Lemma B.1

For open $U\subset \subset U'\subset \subset D$ and $T>0$, there exists $C = C(d,\alpha ,\underline{D},U,d(U,(U')^c),T)$ such that

$$\begin{aligned} p_s^D(x,y) \le Cp_t^D(x,y),\quad x\in U,\ y\in D\setminus U',\ 0<s<t<T. \end{aligned}$$

The proof (given below) relies on approximate factorization of $p_t^D$ and the next lemma on the survival probability $\mathbb {P}^y(\tau _D>s)$. Of course, the latter is nonincreasing in $s\in (0,\infty )$. The following relative upper bound is a partial converse and may be independent interest.

Lemma B.2

Let $T>0$. There exists $C = C(d,\alpha ,\underline{D},T)$ and $\sigma \in (0,1)$ such that

$$\begin{aligned} \frac{\mathbb {P}^y(\tau _D>s)}{\mathbb {P}^y(\tau _D > t)}&\le C \bigg (\frac{s}{t}\bigg )^{-\sigma },\quad y\in D,\ 0<s<t<T. \end{aligned}$$

(B.1)

Proof

By [19, Remark 3] and scaling, $\mathbb {P}^y(\tau _D>t_1)\approx \mathbb {P}^y(\tau _D>t_2)$, uniformly in $y\in D$ and $t_1,t_2$ in each compact subset of $(0,\infty )$. Therefore we may assume that s and t in (B.1) are small. Then we can also assume that y is close to the boundary, otherwise the terms on the left-hand side of (B.1) are bounded from below by the survival probability of a sufficiently small ball (and above by 1). In this setting, recalling the notation of Sect. 2.1, by [19, Theorem 2] and [48, Lemma 17] we get

$$\begin{aligned} \frac{\mathbb {P}^y(\tau _D>s)}{\mathbb {P}^y(\tau _D>t)} \approx \frac{\mathbb {E}^{A_{t^{1/\alpha }}(y)}\tau _D}{\mathbb {E}^{A_{s^{1/\alpha }}(y)}\tau _D} \approx \frac{\Phi (A_{t^{1/\alpha }}(y))}{\Phi (A_{s^{1/\alpha }}(y))}, \end{aligned}$$

(6.2)

uniformly for the considered point y and times s, t.

Let $Q\in \partial D$ be closest to y. To estimate the rightmost ratio in (6.2), we consider three geometric situations:

Case 1. If $y\in \mathcal {A}_{t^{1/\alpha }}(y)\cap \mathcal {A}_{s^{1/\alpha }}(y)$, then we can take $A_{t^{1/\alpha }}(y) = A_{s^{1/\alpha }}(y)=y$, proving (B.1).

Case 2. If $y\in \mathcal {A}_{s^{1/\alpha }}(y)$, but $y\notin \mathcal {A}_{t^{1/\alpha }}(y)$, then $\kappa s^{1/\alpha } \le \delta _D(y) =|y-Q|< \kappa t^{1/\alpha }$, so

$$\begin{aligned} |A_{t^{1/\alpha }}(y) - A_{t^{1/\alpha }}(Q)|&\le |A_{t^{1/\alpha }}(y) - y| + |y-Q| + |Q- A_{t^{1/\alpha }}(Q)| \le (2+\kappa )t^{1/\alpha }. \end{aligned}$$

By definition, $\delta _D(A_{t^{1/\alpha }}(y))\wedge \delta _D(A_{t^{1/\alpha }}(Q)) \ge \kappa t^{1/\alpha }$. Therefore, by the Harnack inequality [14, Lemma 1], we find that $\Phi (A_{t^{1/\alpha }}(y))\approx \Phi (A_{t^{1/\alpha }}(Q))$.

On the other hand, since $\kappa \le 1/2$, we have $y\in \mathcal {A}_{\delta _D(y)/\kappa }(Q)$. In particular, we can take ${A}_{s^{1/\alpha }}(y)={A}_{\delta _D(y)/\kappa }(Q)=y$. Then, by [12, Lemma 5], we get

$$\begin{aligned} \frac{\Phi (A_{t^{1/\alpha }}(y))}{\Phi (A_{s^{1/\alpha }}(y))} \approx \frac{\Phi (A_{t^{1/\alpha }}(Q))}{\Phi (A_{\delta _D(y)/\kappa }(Q))} \lesssim \bigg (\frac{t^{1/\alpha }}{\delta _D(y)/\kappa }\bigg )^{\gamma } \le \bigg (\frac{s}{t}\bigg )^{-\gamma /\alpha }, \end{aligned}$$

where $\gamma = \gamma (d,\alpha ,\underline{D})\in (0,\alpha )$. This ends the proof in this case.

Case 3. If $y\notin \mathcal {A}_{s^{1/\alpha }}(y) \cup \mathcal {A}_{t^{1/\alpha }}(y)$, then $\delta _D(y)<\kappa s^{1/\alpha }<\kappa t^{1/\alpha }$. By the same argument as in the previous case, $\Phi (A_{t^{1/\alpha }}(y))\approx \Phi (A_{t^{1/\alpha }}(Q))$, $\Phi (A_{s^{1/\alpha }}(y))\approx \Phi (A_{s^{1/\alpha }}(Q))$, and

$$\begin{aligned} \frac{\Phi (A_{t^{1/\alpha }}(y))}{\Phi (A_{s^{1/\alpha }}(y))} \approx \frac{\Phi (A_{t^{1/\alpha }}(Q))}{\Phi (A_{s^{1/\alpha }}(Q))} \lesssim \bigg (\frac{s}{t}\bigg )^{-\gamma /\alpha }. \end{aligned}$$

The proof is complete. $\square$

Let us explain why (B.1) is a partial converse to nonincreasingness of the survival probability. We may interpret (B.1) as weak lower scaling (with exponent $-\sigma$) near $s=0$ of the survival probability $f(s){:}{=}\mathbb {P}^x(\tau _D>s)$, uniform in $x\in D$. Such scaling is defined as almost-increasingness $f(s)/s^{-\sigma }\le C f(t)/t^{-\sigma }$ for (bounded arguments) $0<s\le t$; see, e.g., [20].

Proof of Lemma B.1

We use (2.6):

$$\begin{aligned} p_s^D(x,y) \approx \mathbb {P}^x(\tau _D>s) p_s(x,y)\mathbb {P}^y(\tau _D>s). \end{aligned}$$

Since $x\in U\subset \subset D$, we have $\mathbb {P}^x(\tau _D>s) \lesssim \mathbb {P}^x(\tau _D>t)$ because the latter quantity is bounded from below by a constant depending only on $U,d,\alpha ,\underline{D}$, and T. Furthermore, since $y\in D\setminus U'$, by (2.4) we have $p_s(x,y) \approx s$. Therefore the statement of the lemma follows from Lemma B.2. $\square$

Appendix C: No mass concentration forward in time

Let $\mu _n$ be the sequence of measures converging to $\mu$ constructed in the proof of Theorem 6.3. We will show that $\mu _n\cdot {\textbf {1}}_{[0,t]\times D}$ do not accumulate mass at time t. Here is the precise formulation.

Lemma C.1

Let $0<t<T$ and let $f\in C([0,t]\times \overline{D})$. Then

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \int _0^t\int _{D} f(s,y)\mu _n(dy\, ds) = \int _{[0,t)}\int _{\partial D} f(s,Q)\mu (dQ\, ds). \end{aligned}$$

Proof

It suffices to show that

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^ +}\limsup \limits _{n\rightarrow \infty } \int _{t-\varepsilon }^t\int _{D} \mu _n(dy \, ds) = 0. \end{aligned}$$

(6.3)

Fix $\theta \in (0,T-t)$ and $x\in D$. By (2.6) we have

$$\begin{aligned} \int _{t-\varepsilon }^t \int _{D} \mu _n(dy\, ds)&= \int _{t-\varepsilon }^t\int _{D\setminus D_n}\int _{D_n}u(z,s){\mathbb {P}}^y(\tau _{D_n}>1)\nu (y,z)\, dy\, dz\, ds\ \\&\approx \int _{t-\varepsilon }^t\int _{D\setminus D_n}u(z,s)\int _{D_n}p_{t+\theta -s}^{D_n}(x,y)\nu (y,z)\, dy\, dz\, ds\\&=\int _{t-\varepsilon }^t\int _{D\setminus D_n}u(z,s)J^{D_n} (t+\theta ,x,s,z)\, dz\, ds. \end{aligned}$$

Note that

$$\begin{aligned}&\,\int _{t-\varepsilon }^t\int _{D\setminus D_n}u(z,s)J^{D_n}(t+ \theta ,x,s,z)\, dz\, ds\\ =&\, u(t+\theta ,x) - \int _{t}^{t+ \theta }\int _{D\setminus D_n}u(z,s)J^{D_n}(t+\theta ,x,s,z)\, dz\, ds - P_{\theta + \varepsilon }^{D_n}u(t-\varepsilon )(x)\\ =&\, P_{\theta }^{D_n}u(t)(x) - P_{\theta + \varepsilon }^{D_n}u(t-\varepsilon )(x)\ \\ =&\,P_{\theta }^{D_n}(u(t) - P_\varepsilon ^{D_n}u(t-\varepsilon ))(x). \end{aligned}$$

We note that by the caloricity of u, the monotone convergence theorem, and the fact that $p_t^{D_n}\nearrow p_t^D$ pointwise, we have $P_ \tau ^{D_n}u(t)(y)\nearrow P_\tau ^D u(t)(y) \le u(\tau +t,y) < \infty$, for $\tau =\theta ,\varepsilon$. Furthermore, we have $0\le u(t,y) - P_\varepsilon ^{D_n}u(t-\varepsilon )(y)\le u(t,y)$ for all $y\in D$, so by the dominated convergence theorem,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } P_{\theta }^{D_n}(u(t) - P_\varepsilon ^{D_n}u(t- \varepsilon ))(x) = P_{\theta }^{D}(u(t) - P_\varepsilon ^{D}u(t-\varepsilon ))(x). \end{aligned}$$

Now, it suffices to show that $u(t) - P_\varepsilon ^D u(t- \varepsilon )$ converges to 0 pointwise in D. To this end, we let $y\in D$ and take n such that $y\in D_n$. Then,

$$\begin{aligned}&u(t,y) - P_\varepsilon ^D u(t-\varepsilon )(y) \le u(t,y) - P_\varepsilon ^{D_n} u(t- \varepsilon )(y) \\ {}&\quad = \int _{t-\varepsilon }^t \int _{D\setminus D_n} u(s,z)J^{D_n} (t,y,s,z)\, dz\, ds, \end{aligned}$$

and the last expression converges to 0 as $\varepsilon \rightarrow 0^+$ by the dominated convergence theorem. This proves (6.3). $\square$