# Self-similar solution for fractional Laplacian in cones

Krzysztof Bogdan1,∗, Piotr Knosalla2, Łukasz Leżaj1,3,† and Dominika
Pilarczyk1,‡ 1Wrocław University of Science and Technology, Faculty of Pure
and Applied Mathematics, wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>2University of Opole, Institute of Physics, ul.
Oleska 48, 45-052 Opole, Poland
<EMAIL_ADDRESS>3Department of Mathematics and Statistics,
University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland

###### Abstract.

We construct a self-similar solution of the heat equation for the fractional
Laplacian with Dirichlet boundary conditions in every fat cone. Furthermore,
we give the entrance law from the vertex and the Yaglom limit for the
corresponding killed isotropic stable Lévy process and precise large-time
asymptotics for solutions of the Cauchy problem in the cone.

###### Key words and phrases:

Dirichlet heat kernel; Martin kernel; entrance law; Yaglom limit; stable
process; cone

###### 2020 Mathematics Subject Classification:

Primary: 60G18, 60J35; secondary: 60G51, 60J50.

Krzysztof Bogdan was partially supported by the National Science Centre
(Poland): grant 2017/27/B/ST1/01339. Łukasz Leżaj was partially supported by
the National Science Centre (Poland): grant 2021/41/N/ST1/04139.

##  1\. Introduction

Let $d\in\mathbb{N}:=\\{1,2,\ldots\\}$. Consider an arbitrary non-empty open
cone $\Gamma\subseteq{\mathbb{R}^{d}}$. Thus, $ry\in\Gamma$ whenever
$y\in\Gamma$ and $r>0$. Let $p_{t}^{\Gamma}(x,y)$, $t>0$, $x,y\in\Gamma$, be
the Dirichlet heat kernel of the cone for the fractional Laplacian. More
specifically, for $\alpha\in(0,2)$, $p^{\Gamma}$ is the transition density of
the isotropic $\alpha$-stable Lévy process in ${\mathbb{R}^{d}}$ killed upon
leaving $\Gamma$; see, e.g., Bogdan and Grzywny [9]. Let
$M_{\Gamma}\colon{\mathbb{R}^{d}}\to[0,\infty)$ be the Martin kernel of
$\Gamma$ with the pole at infinity (for definitions, see Section 2). The
function is homogeneous (or self-similar) of some degree $\beta\in[0,\alpha)$.
Our first result captures the asymptotics of $p_{t}^{\Gamma}$ at the vertex
$0$ of $\Gamma$, as follows.

######  Theorem 1.1.

If the cone $\Gamma$ is fat, then for $s,t>0$, $x\in\Gamma$, we have

$\Psi_{t}(x):=\lim\limits_{\Gamma\in y\to
0}p_{t}^{\Gamma}(x,y)/M_{\Gamma}(y)\in(0,\infty),$ (1.1)
$\Psi_{t}(x)=t^{-(d+\beta)/\alpha}\Psi_{1}\big{(}t^{-1/\alpha}x\big{)},$

and

(1.2)
$\int_{\Gamma}\Psi_{s}(y)p^{\Gamma}_{t}(x,y)\,{\textrm{d}}y=\Psi_{s+t}(x).$

The proof of Theorem 1.1 is given in Section 3. In view of (1.1) and (1.2),
$\Psi_{t}(x)$ is a self-similar semigroup solution of the heat equation for
the fractional Laplacian with Dirichlet conditions. By (1.2), $\Psi_{t}$ is
also an entrance law for $p^{\Gamma}$ at the origin, see, e.g., Blumenthal [6,
p. 104], Haas and Rivero [23] or Bañuelos et al. [2].

The result is the next step in the program for the boundary potential theory
sketched in the Introduction of Bogdan at al. [10], building on the boundary
Harnack principle, Green function, and Dirichlet heat kernel estimates. In
Theorems 3.11 and 3.12 below, we make further applications to Probability by
obtaining the so-called Yaglom limit for $\Gamma$. We also give applications
to Functional Analysis and Partial Differential Equations, as follows.

Let $1\leqslant q\leqslant\infty$ and
$L^{q}(\Gamma):=L^{q}(\Gamma,{\textrm{d}}x)$. For a weight function $w>0$, we
denote $L^{q}(w):=L^{q}(\Gamma,w(x)\,{\textrm{d}}x)$. For instance,
$L^{1}(M_{\Gamma})=\\{f/M_{\Gamma}\colon f\in L^{1}\\}$. Then, for $1\leqslant
q<\infty$, we define

$\|f\|_{q,M_{\Gamma}}:=\|f/M_{\Gamma}\|_{L^{q}(M_{\Gamma}^{2})}=\bigg{(}\int_{\Gamma}|f(x)|^{q}M_{\Gamma}^{2-q}(x)\,{\textrm{d}}x\bigg{)}^{\frac{1}{q}}=\|f\|_{L^{q}(M_{\Gamma}^{2-q})},$

and, for $q=\infty$, we let

$\|f\|_{\infty,M_{\Gamma}}:=\operatorname*{ess\,sup}_{x\in\Gamma}|f(x)|/M_{\Gamma}(x).$

Of course, $\|f\|_{1,M_{\Gamma}}=\|f\|_{L^{1}(M_{\Gamma})}$. For a non-
negative or integrable function $f$ we let

$P_{t}^{\Gamma}f(x):=\int_{\Gamma}p_{t}^{\Gamma}(x,y)f(y)\,{\textrm{d}}y,\quad
t>0,\,x\in\Gamma.$

We say that the cone $\Gamma$ is _smooth_ if its boundary is $C^{1,1}$ outside
of origin, to wit, there is $r>0$ such that at every boundary point of
$\Gamma$ on the unit sphere $\mathbb{S}^{d-1}$, there exist inner and outer
tangent balls for $\Gamma$, with radii $r$. Put differently, for $d\geqslant
2$, the spherical cap, $\Gamma\cap\mathbb{S}^{d-1}$, is a $C^{1,1}$ subset of
$\mathbb{S}^{d-1}$. For instance, the right-circular cones (see Section 2) are
smooth.

The second result describes the large-time asymptotic behavior of the
semigroup $P_{t}^{\Gamma}$.

######  Theorem 1.2.

Let $q\in[1,\infty)$. Assume that the cone $\Gamma$ is smooth with
$\beta\geqslant\alpha/2$. Then for every $f\in L^{1}(M_{\Gamma})$ and
$A=\int_{\Gamma}f(x)M_{\Gamma}(x)\,\mathrm{d}x$ we have

(1.3)
$\lim_{t\to\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f-A\Psi_{t}\|_{q,M_{\Gamma}}=0.$

As stated in Lemma 4.9, the condition $\beta\geqslant\alpha/2$ is sharp for
smooth cones and $d\geqslant 2$; see also Example 4.12. Theorem 1.2 follows
from the more general Theorem 4.6, by means of Corollary 4.10.

Let us comment on previous developments in the literature and our methods. If
$\Gamma={\mathbb{R}^{d}}$, then $\beta=0$, $M_{\Gamma}=1$ and
$p^{\Gamma}_{t}(x,y)=p_{t}(x,y)$ is the transition density of the fractional
Laplacian on ${\mathbb{R}^{d}}$ (see below). In this case,
$\Psi_{t}(x)=p_{t}(0,y)$ and Theorem 1.2 was resolved by Vázquez [36], see
also Bogdan et al. [13] with $\kappa=0$ in [13, Eq. (1.1)]; see Example 4.4
below, too.

For general cones $\Gamma$, the behavior of $p_{t}^{\Gamma}$ is intrinsically
connected to properties of $M_{\Gamma}$, see, e.g., [9], [15], or Kyprianou et
al. [28]. The identification of the Martin kernel $M_{\Gamma}$ was
accomplished by Bañuelos and Bogdan [3]. Its crucial property is the
homogeneity of order $\beta\in[0,\alpha)$, which is also reflected in the
behavior of the Green function studied by Kulczycki [25] and Michalik [29,
Lemma 3.3], at least when $\Gamma$ is a right-circular cone. As we see in
Theorems 1.1 and 1.2, the exponent $\beta$ determines the self-similarity of
the semigroup solution and the asymptotic behavior of the semigroup
$P_{t}^{\Gamma}$, too. For more information on $\beta$ we refer to [3] and
Bogdan et al. [17].

If $\Gamma$ is a Lipschitz cone, then Theorem 1.1 follows from [15, Corollary
3.2 and Theorem 3.3]. However, the method presented in [15] does not apply to
general fat cones, in particular, to $\Gamma=\mathbb{R}\setminus\\{0\\}$ or
$\Gamma=\mathbb{R}^{2}\setminus([0,\infty)\times\\{0\\})$, which are of
interest for $\alpha\in(1,2)$.

Instead, in this work we develop an approach suggested by [13], where the
authors employ a stationary density of an Ornstein-Uhlenbeck type semigroup
corresponding to a homogeneous (self-similar) heat kernel in a different
setting (on the whole of ${\mathbb{R}^{d}}$). Another key tool in their
analysis is the so-called Doob conditioning using an invariant function or the
heat kernel. Due to Theorem 3.1 below, the Martin kernel $M_{\Gamma}$ is
invariant with respect to $P_{t}^{\Gamma}$, which indeed allows for Doob
conditioning. Then we form the corresponding Ornstein-Uhlenbeck semigroup and
prove existence of a stationary density $\varphi$ in Theorem 3.4 by using the
Schauder-Tychonoff fixed-point theorem. As we shall see in the proof of
Theorem 1.1, the self-similar semigroup solution $\Psi_{t}$ is directly
expressed by $\varphi$ and $M_{\Gamma}$.

The remaining developments in our paper are as follows. In Subsection 3.3 we
obtain an asymptotic relation between the Martin kernel and the survival
probability near the vertex of the cone (see Corollary 3.10). We also obtain a
Yaglom limit (quasi-stationary distribution) in Theorem 3.11, which describes
the behavior of the stable process starting from a fixed point $x\in\Gamma$
and conditioned to stay in a cone, generalizing Theorem 1.1 of [15]. In
Theorem 3.12 we considerably extend both results by allowing arbitrary initial
distributions with finite moment of order $\alpha$.

We note that the Yaglom limit for random walks in cones is discussed by
Denisov and Wachtel [27], but generally there are very few results in
literature on unbounded sets. For a broad survey on quasi-stationary
distributions, we refer to van Doorn and Pollet [35]; see also Champagnat and
Villemonais [19]. Self-similar solutions for general homogeneous semigroups
are discussed in Cholewa and Rodriguez-Bernal [21]. Patie and Savov [30]
discuss generalized Ornstein-Uhlenbeck semigroups, which they call generalized
Laguerre semigroups. Results related to Theorem 1.2, but for fractal Burgers
equation and fractional $p$-Laplacian can be found in Biler et al. [5, Theorem
2.2] and Vázquez [37, Theorem 1.2], respectively. For an approach to entrance
laws based on fluctuation theory of Markov additive processes, we refer to
[27], see also Chaumont et al. [20].

This brief review shows that the asymptotics of the heat kernel is in a busy
intersection between Probability, Partial Differential Equations and
Functional Analysis even though the fields do not always communicate. Our
approach should apply to rather general self-similar operator semigroup
kernels, at least when they enjoy an invariant function and suitable upper and
lower bounds.

Here is a simple example to illustrate our findings.

######  Example 1.3.

If $d=1$, $\alpha\in(0,2)$, and $\Gamma=(0,\infty)$, then $\beta=\alpha/2$,
$M_{\Gamma}(x)=(0\vee x)^{\alpha/2}$ for $x\in\mathbb{R}$, so, by Theorem 4.6,
$\lim_{t\to\infty}t^{(1+\alpha)/(2\alpha)}\|P_{t}^{\Gamma}f\|_{2}=0$ if
$\int_{0}^{\infty}f(x)x^{\alpha/2}\,{\textrm{d}}x=0$; see Example 4.11.
Furthermore, by Remark 3.10 and (4.10), the survival probability is
$\mathbb{P}_{x}(\tau_{\Gamma}>t)\approx(1\wedge t^{-1/\alpha}x)^{\alpha/2}$
and the heat kernel satisfies
$p_{1}^{\Gamma}(x,y)\approx(1+|x-y|)^{-1-\alpha}(1\wedge x)^{\alpha/2}(1\wedge
y)^{\alpha/2}$, so $\Psi_{t}(x)\approx(t^{1/\alpha}\vee
x)^{-1-\alpha}(t^{1/\alpha}\wedge x)^{\alpha/2}$. Here and below $x,y,t>0$. By
Lemma 3.6, the stationary density of the corresponding Ornstein-Uhlenbeck
semigroup is $\varphi(x)\approx(1+x)^{-1-3\alpha/2}$ and the Yaglom limit has
density
$\varphi(x)x^{\alpha/2}/\int_{0}^{\infty}\varphi(y)y^{\alpha/2}\,{\textrm{d}}y\approx
x^{\alpha/2}(1+x)^{-1-3\alpha/2}\approx\Psi_{1}(x)$; see Theorem 3.11. We
refer to [23, Example 5] for an exact but less explicit expression for the
Yaglom limit by means of exponential functionals. See also Example 4.11 for
the case of $\Gamma=\mathbb{R}\setminus\\{0\\}$ and Example 4.12 for
$d$-dimensional extensions.

### Acknowledgments

Part of the research for this work was conducted during Łukasz Leżaj’s post-
doctoral stay at the University of Jyväskylä from January to June 2022. He
expresses his gratitude to the University for their hospitality and to
Professor Stefan Geiss for his warm welcome, guidance, and support during the
stay.

##  2\. Preliminaries

For $x,z\in{\mathbb{R}^{d}}$, the standard scalar product of is denoted by
$x\cdot z$ and $|z|$ is the Euclidean norm. For $x\in{\mathbb{R}^{d}}$ and
$r\in(0,\infty)$, we let $B(x,r)=\\{y\in{\mathbb{R}^{d}}\colon|x-y|<r\\}$, the
ball centered at $x$ with radius $r$, and write $B_{r}:=B(0,r)$. All the
considered sets, functions and measures are Borel. For non-negative functions
$f,g$, we write $f\approx g$ if there is a number $c\in(0,\infty)$, i.e., a
constant, such that $c^{-1}f\leqslant g\leqslant cf$, and write $f\lesssim g$
if there is a constant $c$ such that $f\leqslant cg$. As usual, for two real
numbers $a,b\in\mathbb{R}$, we write $a\wedge b:=\min\\{a,b\\}$ and $a\vee
b:=\max\\{a,b\\}$.

Recall that $\alpha\in(0,2)$ and let

$\nu(z)=c_{d,\alpha}|z|^{-d-\alpha},\quad z\in{\mathbb{R}^{d}},$

where the constant $c_{d,\alpha}$ is such that

$\int_{{\mathbb{R}^{d}}}\big{(}1-\cos(\xi\cdot
z)\big{)}\nu(z)\,{\textrm{d}}z=|\xi|^{\alpha},\quad\xi\in{\mathbb{R}^{d}}.$

For $t>0$ we let

(2.1)
$p_{t}(x):=(2\pi)^{-d}\int_{{\mathbb{R}^{d}}}e^{-t|\xi|^{\alpha}}e^{-i\xi\cdot
x}\,{\textrm{d}}\xi,\quad x\in{\mathbb{R}^{d}}.$

By the Lévy-Khintchine formula, $p_{t}$ is a probability density function and

$\int_{{\mathbb{R}^{d}}}e^{i\xi\cdot
x}\,p_{t}(x)\,{\textrm{d}}x=e^{-t|\xi|^{\alpha}},\quad\xi\in{\mathbb{R}^{d}},\;t>0.$

We consider the isotropic $\alpha$-stable Lévy process
$\mathbf{X}=(X_{t},t\geqslant 0)$ in ${\mathbb{R}^{d}}$, with

$p_{t}(x,y):=p_{t}(y-x),\quad x,y\in{\mathbb{R}^{d}},\ t>0,$

as transition density. Thus,

$\mathbb{E}_{x}e^{i\xi\cdot X_{t}}=\int_{{\mathbb{R}^{d}}}e^{i\xi\cdot
y}\,p_{t}(x,y)\,{\textrm{d}}y=e^{i\xi\cdot
x-t|\xi|^{\alpha}},\quad\xi\in{\mathbb{R}^{d}},\,x\in{\mathbb{R}^{d}},\;t>0.$

The Lévy-Khintchine exponent of $\mathbf{X}$ is, of course, $|\xi|^{\alpha}$
and $\nu$ is the intensity of jumps. By (2.1),

(2.2) $\displaystyle p_{t}(x,y)$
$\displaystyle=t^{-d/\alpha}p_{1}\big{(}t^{-1/\alpha}x,t^{-1/\alpha}y\big{)},\quad
x,y\in{\mathbb{R}^{d}},\;t>0,$ and (2.3) $\displaystyle
p_{t}\left(Tx,Ty\right)$ $\displaystyle=p_{t}(x,y),\quad
x,y\in{\mathbb{R}^{d}},\;t>0,$

for every isometry $T$ on ${\mathbb{R}^{d}}$. It is well known that

(2.4) $p_{t}(x,y)\approx t^{-d/\alpha}\wedge t|y-x|^{-d-\alpha},\quad
x,y\in{\mathbb{R}^{d}},\;t>0,$

see, e.g., [7]. We then consider the time of the first exit of $\mathbf{X}$
from the cone $\Gamma$,

$\tau_{\Gamma}:=\inf\\{t\geqslant 0\colon X_{t}\notin\Gamma\\},$

and we define the Dirichlet heat kernel for $\Gamma$,

$p_{t}^{\Gamma}(x,y):=p_{t}(x,y)-\mathbb{E}_{x}\left[\tau_{\Gamma}<t;p_{t-\tau_{\Gamma}}\left(X_{\tau_{\Gamma}},y\right)\right],\quad
x,y\in\Gamma,\;t>0,$

see [9, 14, 22]. It immediately follows that $p_{t}^{\Gamma}(x,y)\leqslant
p_{t}(x,y)$ for all $x,y\in\Gamma$ and $t>0$. The Dirichlet heat kernel is
non-negative, and symmetric: $p_{t}^{\Gamma}(x,y)=p_{t}^{\Gamma}(y,x)$ for
$x,y\in\Gamma$, $t>0$. It satisfies the Chapman-Kolmogorov equations:

(2.5)
$p_{t+s}^{\Gamma}(x,y)=\int_{\Gamma}p_{t}^{\Gamma}(x,z)p_{s}^{\Gamma}(z,y)\,{\textrm{d}}z,\quad
x,y\in\Gamma,\;s,t>0.$

For nonnegative or integrable functions $f$ we define the killed semigroup by

$P_{t}^{\Gamma}f(x):=\mathbb{E}_{x}\left[\tau_{\Gamma}>t;f(X_{t})\right]=\int_{\Gamma}p_{t}^{\Gamma}(x,y)f(y)\,{\textrm{d}}y,\quad
x\in\Gamma,\;t>0.$

In particular, for $f\equiv 1$ we obtain the survival probability:

(2.6)
$\mathbb{P}_{x}(\tau_{\Gamma}>t)=\int_{\Gamma}p_{t}^{\Gamma}(x,y)\,{\textrm{d}}y,\quad
x\in\Gamma,\,t>0,$

see [11, Remark 1.9]. Since $t^{-1/\alpha}\Gamma=\Gamma$, the scaling (2.2)
extends to the Dirichlet heat kernel:

$p_{t}^{\Gamma}(x,y)=t^{-d/\alpha}p_{1}^{\Gamma}\big{(}t^{-1/\alpha}x,t^{-1/\alpha}y\big{)},\quad
x,y\in\Gamma,\;t>0.$

As a consequence,

(2.7)
$\mathbb{P}_{x}(\tau_{\Gamma}>t)=\mathbb{P}_{t^{-1/\alpha}x}(\tau_{\Gamma}>1),\quad
x\in\Gamma,\;t>0.$

Furthermore, by (2.3),

(2.8) $p_{t}^{T\Gamma}(Tx,Ty)=p_{t}^{\Gamma}(x,y),\quad x,y\in\Gamma,\;t>0.$

The operators $P_{t}^{\Gamma}$ and the kernel $p_{t}^{\Gamma}(x,y)$ are the
main subject of the paper. In view of (2.8), without loss of generality we may
assume that ${\bf 1}:=(0,\ldots,0,1)\in\Gamma$. By [3, Theorem 3.2], there is
a unique non-negative function $M_{\Gamma}$ on ${\mathbb{R}^{d}}$ such that
$M_{\Gamma}({\bf 1})=1$, $M_{\Gamma}=0$ on $\Gamma^{c}$, and for every open
bounded set $B\subseteq\Gamma$,

(2.9) $M_{\Gamma}(x)=\mathbb{E}_{x}M_{\Gamma}(X_{\tau_{B}}),\quad
x\in{\mathbb{R}^{d}}.$

Moreover, $M_{\Gamma}$ is locally bounded on ${\mathbb{R}^{d}}$ and
homogeneous of some order $\beta\in[0,\alpha)$, i.e.,

(2.10) $M_{\Gamma}(x)=|x|^{\beta}M_{\Gamma}(x/|x|),\quad x\in\Gamma.$

We call $M_{\Gamma}$ the Martin kernel of $\Gamma$ with the pole at infinity.

######  Example 2.1.

By [3], $\beta=\alpha/2$ if $\Gamma$ is a half-space and $\beta=\alpha-1$ if
$\Gamma=\mathbb{R}\setminus\\{0\\}$ and $1<\alpha<2$. By [12],
$\beta=(\alpha-1)/2$ if
$\Gamma=\mathbb{R}^{2}\setminus([0,\infty)\times\\{0\\})$ and $1<\alpha<2$.

Below we often assume that $\Gamma$ is _fat_ , i.e., $\kappa\in(0,1)$ exists
such that for all $Q\in\overline{\Gamma}$ and $r\in(0,\infty)$, there is a
point $A=A_{r}(Q)\in\Gamma\cap B(Q,r)$ such that $B(A,\kappa
r)\subseteq\Gamma\cap B(Q,r)$, see [10, Definition 1]. Recall that $\Gamma$ is
_smooth_ if $d=1$ or $d\geqslant 2$ and $\Gamma\cap\mathbb{S}^{d-1}$ is a
$C^{1,1}$ subset of $\mathbb{S}^{d-1}$. Furthermore, a cone $\Gamma$ is called
_right-circular_ , if
$\Gamma=\\{x=(x_{1},\ldots,x_{d})\in{\mathbb{R}^{d}}\setminus\\{0\\}:x_{d}>|x|\cos\eta\\}$,
with $\eta\in(0,\pi)$ called the angle of the cone. Of course, every right-
circular cone is smooth, and every smooth cone is fat.

By [10, Theorem 1], the following approximate factorization holds true for fat
cones:

(2.11)
$p_{t}^{\Gamma}(x,y)\approx\mathbb{P}_{x}(\tau_{\Gamma}>t)p_{t}(x,y)\mathbb{P}_{y}(\tau_{\Gamma}>t),\quad
x,y\in\Gamma,\;t>0.$

For $R\in(0,\infty)$, we let $\Gamma_{R}:=\Gamma\cap B_{R}$, the truncated
cone.

##  3\. Doob conditioning

The Martin kernel $M_{\Gamma}$ is invariant for the semigroup
$P_{t}^{\Gamma}$, as follows.

######  Theorem 3.1.

For all $x\in\Gamma$ and $t>0$, we have
$P_{t}^{\Gamma}M_{\Gamma}(x)=M_{\Gamma}(x)$.

###### Proof.

Fix $t>0$ and $x\in\Gamma$. We have

(3.1)
$P_{t}^{\Gamma}M_{\Gamma}(x)=\mathbb{E}_{x}\Big{[}\tau_{\Gamma}>t;\,M_{\Gamma}(X_{t})\Big{]}.$

Let $R>0$ and $\tau_{R}:=\tau_{\Gamma_{R}}$. By (2.9) and the strong Markov
property,

(3.2)
$M_{\Gamma}(x)=\mathbb{E}_{x}M_{\Gamma}(X_{\tau_{\Gamma_{R}}})=\mathbb{E}_{x}M_{\Gamma}\big{(}X_{t\wedge\tau_{R}}\big{)}=\mathbb{E}_{x}\Big{[}X_{t\wedge\tau_{R}}\in\Gamma;\,M_{\Gamma}\big{(}X_{t\wedge\tau_{R}}\big{)}\Big{]},$

where the last equality follows from the fact that $M_{\Gamma}=0$ outside
$\Gamma$. We note that $\mathbb{P}_{x}-a.s.$, $\tau_{R}\to\tau_{\Gamma}$ as
$R\to\infty$ (see, e.g., [1, proof of Proposition A.1]). We consider two
scenarios. On $\\{\tau_{\Gamma}=\infty\\}$, for $R$ large enough, we have:
$\tau_{R}>t$,
$\mathds{1}_{X_{t\wedge\tau_{R}}\in\Gamma}=1=\mathds{1}_{t<\tau_{\Gamma}}$,
and

$M_{\Gamma}\big{(}X_{t\wedge\tau_{R}}\big{)}\mathds{1}_{X_{t\wedge\tau_{R}}\in\Gamma}=M_{\Gamma}(X_{t})=M_{\Gamma}(X_{t})\mathds{1}_{t<\tau_{\Gamma}}.$

On $\\{\tau_{\Gamma}<\infty\\}$, for $R$ large enough we have:
$\tau_{R}=\tau_{\Gamma}$,
$\mathds{1}_{X_{t\wedge\tau_{R}}\in\Gamma}=\mathds{1}_{t<\tau_{\Gamma}}$, and

$M_{\Gamma}\big{(}X_{t\wedge\tau_{R}}\big{)}\mathds{1}_{X_{t\wedge\tau_{R}}\in\Gamma}=M_{\Gamma}(X_{t})\mathds{1}_{t<\tau_{\Gamma}},$

too. In both cases, the integrand on the right-hand side of (3.2) converges
$a.s.$ to the integrand on the right-hand side of (3.1) as $R\to\infty$. By
the local boundedness of $M_{\Gamma}$ and (2.10),

$\Big{|}M_{\Gamma}\big{(}X_{t\wedge\tau_{R}}\big{)}\mathds{1}_{X_{t\wedge\tau_{R}}\in\Gamma}\Big{|}\leqslant
c\big{|}X_{t\wedge\tau_{R}}\big{|}^{\beta}\leqslant c(X_{t}^{*})^{\beta},$

where

$X_{t}^{*}:=\sup_{0\leqslant s\leqslant t}|X_{s}|.$

Using [4, Theorem 2.1] and the fact that $\beta\in[0,\alpha$), we conclude
that $\mathbb{E}_{x}(X_{t}^{*})^{\beta}<\infty$. An application of the
dominated convergence theorem ends the proof. ∎

###  3.1. Renormalized kernel

We define the renormalized (Doob-conditioned) kernel

(3.3)
$\rho_{t}(x,y)=\frac{p_{t}^{\Gamma}(x,y)}{M_{\Gamma}(x)M_{\Gamma}(y)},\quad
x,y\in\Gamma,\;t>0.$

Note that $\rho$ is jointly continuous. By Theorem 3.1,

(3.4) $\int_{\Gamma}\rho_{t}(x,y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=1,\quad
x\in\Gamma,\;t>0,$

and by (2.5),

(3.5)
$\int_{\Gamma}\rho_{t}(x,y)\rho_{s}(y,z)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=\rho_{t+s}(x,z),\quad
x,y\in\Gamma,\;s,t>0.$

In other words, $\rho_{t}$ is a symmetric transition probability density on
$\Gamma$ with respect to the measure $M^{2}_{\Gamma}(y)\,{\textrm{d}}y$.
Furthermore, the following scaling property holds true: for all $x,y\in\Gamma$
and all $t>0$,

(3.6)
$\rho_{t}(x,y)=\frac{t^{-d/\alpha}p_{1}^{\Gamma}(t^{-1/\alpha}x,t^{-1/\alpha}y)}{t^{2\beta/\alpha}M_{\Gamma}(t^{-1/\alpha}x)M_{\Gamma}(t^{-1/\alpha}y)}=t^{-(d+2\beta)/\alpha}\rho_{1}(t^{-1/\alpha}x,t^{-1/\alpha}y).$

Therefore,

(3.7)
$\rho_{st}(t^{1/\alpha}x,t^{1/\alpha}y)=t^{-(d+2\beta)/\alpha}\rho_{s}(x,y),\quad
x,y\in\Gamma,\;s,t>0.$

By (2.11), for fat cones we have

(3.8)
$\rho_{t}(x,y)\approx\frac{\mathbb{P}_{x}(\tau_{\Gamma}>t)}{M_{\Gamma}(x)}p_{t}(x,y)\frac{\mathbb{P}_{y}(\tau_{\Gamma}>t)}{M_{\Gamma}(y)},\quad
x,y\in\Gamma,\,t>0.$

The boundary behavior of $\mathbb{P}_{x}(\tau_{\Gamma}>t)/M_{\Gamma}(x)$ is
important due to (3.8), but it is rather elusive. The next lemma strengthens
the upper bound from [3, Lemma 4.2].

######  Lemma 3.2.

There exists a constant $c$ depending only on $\alpha$ and $\Gamma$, such that

(3.9) $\mathbb{P}_{x}(\tau_{\Gamma}>t)\leqslant
c\Big{(}t^{-\beta/\alpha}+t^{-1}|x|^{\alpha-\beta}\Big{)}M_{\Gamma}(x),\quad
t>0,\,x\in\Gamma.$

######  Remark 3.3.

(1) For $t=1$, (3.9) reads as follows,

(3.10) $\mathbb{P}_{x}(\tau_{\Gamma}>1)\leqslant
c(1+|x|^{\alpha-\beta})M_{\Gamma}(x),\quad x\in\Gamma.$

(2) The estimate (3.9) applies to arbitrary cones and arguments $t,x$,
however, it is not optimal. For example, for the right-circular cones, we can
confront (3.8) with

$M_{\Gamma}(x)\approx\delta_{\Gamma}(x)^{\alpha/2}|x|^{\beta-\alpha/2},\quad
x\in\Gamma,$

and

$\mathbb{P}_{x}(\tau_{\Gamma}>1)\approx\big{(}1\wedge\delta_{\Gamma}(x)\big{)}^{\alpha/2}\big{(}1\wedge|x|\big{)}^{\beta-\alpha/2},\quad
x\in\Gamma,$

as provided by [29, Lemma 3.3] and [10, Example 7].
(3) For the right-circular cones, the ratio

$\frac{\mathbb{P}_{x}(\tau_{\Gamma}>1)}{M_{\Gamma}(x)}\approx\frac{\big{(}1+\delta_{\Gamma}(x))^{-\alpha/2}}{(1+|x|)^{\beta-\alpha/2}},\quad
x\in\Gamma,$

is bounded if and only if $\beta\geqslant\alpha/2$.

###### Proof of Lemma 3.2.

We slightly modify the proof of [3, Lemma 4.2]. First, suppose that $t=1$. The
case $x\in\Gamma_{1}$ in (3.10) is resolved by [3, Lemma 4.2], so we assume
that $x\in\Gamma\setminus\Gamma_{1}$. For every
$z\in{\mathbb{R}^{d}}\setminus\\{0\\}$ we define its projection on the unit
sphere $\tilde{z}:=z/|z|$. By (2.7),

$\mathbb{P}_{x}(\tau_{\Gamma}>1)=\mathbb{P}_{\tilde{x}}(\tau_{\Gamma}>|x|^{-\alpha}).$

Then we have

$\mathbb{P}_{\tilde{x}}(\tau_{\Gamma}>|x|^{-\alpha})\leqslant\mathbb{P}_{\tilde{x}}(\tau_{\Gamma_{2}}>|x|^{-\alpha})+\mathbb{P}_{\tilde{x}}(\tau_{\Gamma_{2}}<\tau_{\Gamma}).$

By the boundary Harnack principle (BHP), see Song and Wu [34, Theorem 3.1],
and the homogeneity of $M_{\Gamma}$ (2.10),

(3.11)
$\mathbb{P}_{\tilde{x}}(\tau_{\Gamma_{2}}<\tau_{\Gamma})\leqslant\mathbb{P}_{{\bf
1}}(\tau_{\Gamma_{2}}<\tau_{\Gamma})M_{\Gamma}(\tilde{x})=c_{1}|x|^{-\beta}M_{\Gamma}(x).$

We let

$c_{2}=\inf_{y\in\Gamma_{2}}\int_{\Gamma\setminus\Gamma_{2}}\nu(y-z)\,{\textrm{d}}z.$

Clearly, $c_{2}>0$. We recall the Ikeda-Watanabe formula:

$\displaystyle\mathbb{P}_{x}[\tau_{D}\in I,\;Y_{\tau_{D}-}\in
A,\;Y_{\tau_{D}}\in
B]=\int_{I}\int_{A}\int_{B}\,p_{s}^{D}(x,v)\nu(v,z)\,{\textrm{d}}z\,{\textrm{d}}v\,{\textrm{d}}s,$

where $x\in D$, $I\subseteq[0,\infty)$, $A\subseteq D$ and $B\subseteq D^{c}$,
see, e.g., Bogdan et al. [16, Section 4.2]. By Markov inequality and BHP,

$\displaystyle\mathbb{P}_{\tilde{x}}\big{(}\tau_{\Gamma_{2}}>|x|^{-\alpha}\big{)}$
$\displaystyle\leqslant|x|^{\alpha}\mathbb{E}_{\tilde{x}}\tau_{\Gamma_{2}}=|x|^{\alpha}\int_{\Gamma_{2}}G_{\Gamma_{2}}(\tilde{x},y)\,{\textrm{d}}y$
$\displaystyle\leqslant
c_{2}^{-1}|x|^{\alpha}\int_{\Gamma\setminus\Gamma_{2}}\int_{\Gamma_{2}}G_{\Gamma_{2}}(\tilde{x},y)\nu(y-z)\,{\textrm{d}}y\,{\textrm{d}}z$
$\displaystyle\leqslant
c_{2}^{-1}|x|^{\alpha}\mathbb{P}_{\tilde{x}}\big{(}X_{\tau_{\Gamma_{2}}}\in\Gamma\big{)}$
$\displaystyle\leqslant c_{1}c_{2}^{-1}|x|^{\alpha}\mathbb{P}_{{\bf
1}}\big{(}X_{\tau_{\Gamma_{2}}}\in\Gamma\big{)}M_{\Gamma}(\tilde{x})$
$\displaystyle=c_{1}c_{2}^{-1}c|x|^{\alpha-\beta}M_{\Gamma}(x).$

By (3.11), we get (3.10) when $x\in\Gamma\setminus\Gamma_{1}$. For arbitrary
$t>0$, we use (2.7) and (3.10):

$\displaystyle\mathbb{P}_{x}(\tau_{\Gamma}>t)$
$\displaystyle=\mathbb{P}_{t^{-1/\alpha}x}(\tau_{\Gamma}>1)$
$\displaystyle\leqslant
c\left(1+\left(t^{-1/\alpha}|x|\right)^{\alpha-\beta}\right)M_{\Gamma}(t^{-1/\alpha}x)$
$\displaystyle=c\left(t^{-\beta/\alpha}+t^{-1}|x|^{\alpha-\beta}\right)M_{\Gamma}(x).$

∎

By the proof of [3, Lemma 4.2], for every $R\in(0,\infty)$ there exists a
constant $c$, depending only on $\alpha$, $\Gamma$ and $R$, such that

$c^{-1}M_{\Gamma}(x)t^{-\beta/\alpha}\leqslant\mathbb{P}_{x}(\tau_{\Gamma}>t)\leqslant
cM_{\Gamma}(x)t^{-\beta/\alpha},\quad x\in\Gamma_{Rt^{1/\alpha}},\;t>0.$

In particular, for fat cones, in view of (2.4) and (3.8),

(3.12)
$\rho_{1}(x,y)\approx(1+|y|)^{-d-\alpha}\frac{\mathbb{P}_{y}(\tau_{\Gamma}>1)}{M_{\Gamma}(y)},\quad
x\in\Gamma_{R},\;y\in\Gamma,$

with comparability constant depending only on $\alpha$, $\Gamma$ and $R$.
Using Lemma 3.2 we also conclude that for every $R\geqslant 1$ there is a
constant $c$ depending only on $R$, $\alpha$ and $\Gamma$, such that

(3.13) $\rho_{1}(x,y)\leqslant c(1+|y|)^{-d-\beta},\quad
x\in\Gamma_{R},\;y\in\Gamma.$

###  3.2. Ornstein-Uhlenbeck kernel

Encouraged by [13], we let

(3.14) $\ell_{t}(x,y):=\rho_{1-e^{-t}}(e^{-t/\alpha}x,y),\quad
x,y\in\Gamma,\;t>0,$

and, by (3.7), we get the Chapman-Kolmogorov property for $\ell_{t}$:

$\int_{\Gamma}\ell_{t}(x,y)\ell_{s}(y,z)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=\ell_{t+s}(x,z),\quad
x,z\in\Gamma,\;s,t>0.$

By (3.4),

$\int_{\Gamma}\ell_{t}(x,y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=1,\quad
x\in\Gamma,\;t>0.$

Thus, $\ell_{t}$ is a transition probability density on $\Gamma$ with respect
to $M_{\Gamma}^{2}(y)\,{\textrm{d}}y$. We define the corresponding Ornstein-
Uhlenbeck semigroup:

$L_{t}f(y)=\int_{\Gamma}\ell_{t}(x,y)f(x)M_{\Gamma}^{2}(x)\,{\textrm{d}}x,\quad
x\in\Gamma,t>0.$

We easily see that the operators are bounded on
$L^{1}(M^{2}_{\Gamma}(y)\,{\textrm{d}}y)$. In fact, they preserve densities,
i.e., functions $f\geqslant 0$ such that
$\int_{\Gamma}f(x)M^{2}_{\Gamma}(x)\,{\textrm{d}}x=1$.

Before we immerse into details, let us note that the relations (3.12) and
(3.13) will be crucial in what follows. Both of them rely on the factorization
of the Dirichlet heat kernel (2.11), which is valid for fat sets. So, below in
this section we assume (sometimes tacitly) that $\Gamma$ is a fat cone.

######  Theorem 3.4.

Assume $\Gamma$ is a fat cone. Then there is a unique stationary density
$\varphi$ for the operators $L_{t}$, $t>0$.

###### Proof.

Fix $t>0$ and consider the family $F$ of non-negative functions on $\Gamma$
that have the form

$f(y)=\int_{\Gamma_{1}}\rho_{t}(x,y)\,\mu({\textrm{d}}x),\quad y\in\Gamma,$

for some sub-probability measure $\mu$ concentrated on $\Gamma_{1}$. By (3.4),
$F\subseteq L^{1}(M_{\Gamma}^{2}(y))$. By the scaling (3.7) and the same
reasoning as in the proof of [13, Theorem 3.2], $L_{t}F\subseteq F$. Since
$L_{t}$ is continuous, we also have $L_{t}\overline{F}\subseteq\overline{F}$,
where $\overline{F}$ is the closure of $F$ in the norm topology of
$L^{1}(\Gamma,M^{2}_{\Gamma}(y)\,{\textrm{d}}y)$. Next, we observe that $F$ is
convex, therefore by [18, Theorem 3.7], $\overline{F}$ is equal to the closure
of $F$ in the weak topology. In view of (3.12),

(3.15)
$f(y)\lesssim(1+|y|)^{-d-\alpha}\frac{\mathbb{P}_{y}(\tau_{\Gamma}>1)}{M_{\Gamma}(y)},\quad
y\in\Gamma,$

uniformly for $f\in F$. Moreover, (3.4) and (3.12) show that the right-hand
side of (3.15) is integrable with respect to
$M_{\Gamma}^{2}(y)\,{\textrm{d}}y$. Therefore, the family $F$ is uniformly
integrable with respect to $M^{2}_{\Gamma}(y)\,{\textrm{d}}y$. By [8, Theorem
4.7.20], $F$ is weakly pre-compact in $L^{1}(M^{2}_{\Gamma}(y))$, so
$\overline{F}$ is weakly compact. Furthermore, we invoke [18, Theorem 3.10] to
conclude that $L_{t}$ is weakly continuous. By the Schauder-Tychonoff fixed
point theorem [32, Theorem 5.28], there is a density $\varphi\in\overline{F}$
satisfying $L_{t}\varphi=\varphi$. It is unique by the strict positivity of
the kernel $\ell_{t}$, and the same for every $t>0$, see the proof of [13,
Theorem 3.2]. ∎

Let us note that by Theorem 3.4 and [26, Theorem 1 and Remark 2], the
following stability result for kernels $l_{t}$ in
$L^{1}(M_{\Gamma}^{2}(y)\,{\textrm{d}}y)$ holds true for every $x\in\Gamma$:

(3.16)
$\int_{\Gamma}\big{|}\ell_{t}(x,y)-\varphi(y)\big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty.$

We claim that the convergence in (3.16) is in fact uniform for $x$ in any
bounded subset $A\subseteq\Gamma$. Indeed, let $x,x_{0}\in A$. In view of
(3.14) and (3.8) we may write

$\displaystyle\int_{\Gamma}\big{|}\ell_{1+t}(x,y)-\varphi(y)\big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y$
$\displaystyle=\int_{\Gamma}\bigg{|}\int_{\Gamma}\ell_{1}(x,z)\big{(}\ell_{t}(z,y)-\varphi(y)\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z\bigg{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y$
(3.17) $\displaystyle\leqslant
c\int_{\Gamma}\ell_{1}(x_{0},z)\int_{\Gamma}\big{|}\ell_{t}(z,y)-\varphi(y)\big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\,M_{\Gamma}^{2}(z)\,{\textrm{d}}z.$

By (3.16), for every $z\in\Gamma$,

$I_{t}(z):=\int_{\Gamma}\big{|}\ell_{t}(z,y)-\varphi(y)\big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty.$

Moreover,
$I_{t}(z)\leqslant\int_{\Gamma}\big{(}\ell_{t}(z,y)+\varphi(y)\big{)}M_{\Gamma}^{2}(y)\,{\textrm{d}}y=2$.
Since

$\int_{\Gamma}2\ell_{1}(x_{0},z)M_{\Gamma}^{2}(z)\,{\textrm{d}}z=2<\infty,$

by the dominated convergence theorem the iterated integral in (3.17) tends to
$0$ as $t\to\infty$, so the convergence in (3.16) is uniform for all $x\in A$,
as claimed. By rewriting (3.16) in terms of $\rho$, we get that, uniformly for
$x\in A$,

(3.18)
$\int_{\Gamma}\Big{|}\rho_{1-e^{-t}}\big{(}e^{-t/\alpha}x,y\big{)}-\varphi(y)\Big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty.$

This leads to the following spacial asymptotics for $\rho_{1}$.

######  Corollary 3.5.

Let $\Gamma$ be a fat cone. If $\Gamma\ni x\to 0$ then
$\int_{\Gamma}\big{|}\rho_{1}(x,y)-\varphi(y)\big{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0$.

###### Proof.

By the scalings (3.6) and (3.7),

$\rho_{1-e^{-t}}\big{(}e^{-t/\alpha}x,y\big{)}=\big{(}1-e^{-t}\big{)}^{-(d+2\beta)/\alpha}\rho_{1}\Big{(}\big{(}e^{t}-1\big{)}^{-1/\alpha}x,\big{(}1-e^{-t}\big{)}^{-1/\alpha}y\Big{)},$

thus, in view of (3.18),

(3.19)
$\int_{\Gamma}\bigg{|}\rho_{1}\Big{(}\big{(}e^{t}-1\big{)}^{-1/\alpha}x,\big{(}1-e^{-t}\big{)}^{-1/\alpha}y\Big{)}-\varphi(y)\bigg{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty.$

By the continuity of dilations in $L^{1}({\mathbb{R}^{d}})$,

$\int_{\Gamma}\bigg{|}\varphi\Big{(}\big{(}1-e^{-t}\big{)}^{1/\alpha}y\Big{)}M_{\Gamma}^{2}(y)-\varphi(y)M_{\Gamma}^{2}(y)\bigg{|}\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty.$

Thus, by a change of variables in (3.19) and the triangle inequality, we
conclude that

$\int_{\Gamma}\bigg{|}\rho_{1}\Big{(}\big{(}e^{t}-1\big{)}^{-1/\alpha}z,y\Big{)}-\varphi(y)\bigg{|}M_{\Gamma}^{2}(y)\,{\textrm{d}}y\to
0\quad\text{as}\quad t\to\infty$

uniformly for all $z\in A$. To end the proof, we take $A=B_{1}$ and
$x=\big{(}e^{t}-1\big{)}^{-1/\alpha}z$, where
$t=\ln\big{(}1+|x|^{-\alpha}\big{)}$ and $z=x/|x|\in A$. . ∎

######  Lemma 3.6.

After a modification on set of Lebesgue measure $0$, $\varphi$ is continuous
on $\Gamma$ and

$\varphi(y)\approx(1+|y|)^{-d-\alpha}\frac{\mathbb{P}_{y}(\tau_{\Gamma}>1)}{M_{\Gamma}(y)},\quad
y\in\Gamma.$

###### Proof.

By Corollary 3.5 and (3.12),

$\varphi(y)\approx(1+|y|)^{-d-\alpha}\frac{\mathbb{P}_{y}(\tau_{\Gamma}>1)}{M_{\Gamma}(y)}$

on $\Gamma$ less a set of Lebesgue measure zero. Theorem 3.4 entails that
$\varphi=L_{1}\varphi$ $a.e.$, so it suffices to verify that $L_{1}\varphi$ is
continuous on $\Gamma$. To this end we note that $\ell_{1}(x,y)$ is continuous
in $x,y\in\Gamma$. Next, by (3.14) and (3.8),

$\displaystyle\ell_{1}(x,y)\approx\frac{\mathbb{P}_{e^{-1/\alpha}x}\big{(}\tau_{\Gamma}>1-e^{-1}\big{)}}{M_{\Gamma}\big{(}e^{-1/\alpha}x\big{)}}p_{1-e^{-1}}\big{(}e^{-1/\alpha}x,y\big{)}\frac{\mathbb{P}_{y}\big{(}\tau_{\Gamma}>1-e^{-1}\big{)}}{M_{\Gamma}(y)},\quad
x,y\in\Gamma.$

Let $R>1$. By (2.4) and (3.9), [10, Remark 3] and (2.7), and the homogeneity
(2.10) of $M_{\Gamma}$,

$\ell_{1}(x,y)\lesssim(1+|x|)^{-d-\alpha}\frac{\mathbb{P}_{x}(\tau_{\Gamma}>1)}{M_{\Gamma}(x)},\quad
x\in\Gamma,y\in\Gamma_{R}.$

By the dominated convergence theorem, $L_{1}\varphi$ is continuous on
$\Gamma_{R}$. ∎

In what follows, $\varphi$ denotes the continuous modification from Lemma 3.6.

######  Theorem 3.7.

Let $\Gamma$ be a fat cone. For every $t>0$, uniformly in $y\in\Gamma$ we have

$\rho_{t}(0,y):=\lim_{\Gamma\ni x\to
0}\rho_{t}(x,y)=t^{-(d+2\beta)/\alpha}\varphi\big{(}t^{-1/\alpha}y\big{)}.$

###### Proof.

If $\beta=0$ then $\rho_{t}(x,y)=p_{t}(x,y)$ and the claim is simply the
continuity property of the heat kernel $p_{t}$. Thus, we assume that
$\beta>0$.

We only prove the claim for $t=1$; the extension to arbitrary $t$ is a
consequence of the scaling (3.6). By (3.7) and the Chapman-Kolmogorov
property, for $x,y\in\Gamma$,

$\displaystyle\rho_{1}(x,y)$
$\displaystyle=2^{(d+2\beta)/\alpha}\rho_{2}\big{(}2^{1/\alpha}x,2^{1/\alpha}y\big{)}$
$\displaystyle=2^{(d+2\beta)/\alpha}\int_{\Gamma}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z.$

We will prove that, uniformly in $y\in\Gamma$,

(3.20)
$\int_{\Gamma}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z\to\int_{\Gamma}\varphi(z)\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z$

as $\Gamma\ni x\to 0$. To this end we first claim that there is
$c\in(0,\infty)$ dependent only on $\alpha$ and $\Gamma$, such that for all
$x\in\Gamma_{1}$ and $y\in\Gamma$,

(3.21)
$\int_{\Gamma}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z\leqslant
c(1+|y|)^{-\beta}.$

Indeed, denote $\tilde{y}=2^{1/\alpha}y$. By (3.8), Lemma 3.6 and (3.10),
there is $c>0$ such that for all $z,y\in\Gamma$ and $x\in\Gamma_{1}$,

$\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,\widetilde{y}\big{)}M_{\Gamma}^{2}(z)\lesssim(1+|z|)^{-d-\alpha}(1+|z-\widetilde{y}|)^{-d-\alpha}(1+|y|)^{\alpha-\beta}.$

We split the integral in (3.21) into two integrals. For $z\in
A:=B(\widetilde{y},|\widetilde{y}|/2)$ we use the fact that
$|z|\approx|\widetilde{y}|\approx|y|$ and $1+|z-\widetilde{y}|\geqslant 1$,
therefore

(3.22)
$\displaystyle\int_{A}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,\widetilde{y}\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z$
$\displaystyle\lesssim|y|^{d}(1+|y|)^{-d-\beta}\leqslant(1+|y|)^{-\beta}.$

For $z\in\Gamma\setminus A$ we simply have $1+|z|\geqslant 1$, thus,

$\displaystyle\int_{\Gamma\setminus
A}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,\widetilde{y}\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z$
$\displaystyle\lesssim(1+|y|)^{\alpha-\beta}\int_{\Gamma\setminus
A}(1+|z-\widetilde{y}|)^{-d-\alpha}\,{\textrm{d}}z$
$\displaystyle\lesssim(1+|y|)^{\alpha-\beta}\int_{|\widetilde{y}|/2}^{\infty}(1+r)^{-1-\alpha}\,{\textrm{d}}r$
$\displaystyle\lesssim(1+|y|)^{-\beta}.$

Combining it with (3.22), we arrive at (3.21), as claimed.

Let $\varepsilon>0$. In view of (3.21) and the fact that $\beta>0$, there is
$R\in(0,\infty)$ depending only on $\alpha$, $\beta$, $\Gamma$ and
$\varepsilon$ such that

(3.23)
$\int_{\Gamma}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z<\varepsilon,$

provided that $y\in\Gamma\setminus\Gamma_{R}$. For $y\in\Gamma_{R}$, by (3.13)
we get

$\int_{\Gamma}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z\lesssim\int_{\Gamma}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}M_{\Gamma}^{2}(z)\,{\textrm{d}}z,$

with the implied constant dependent only on $\alpha$, $\beta$, $\Gamma$ and
$R$, but not otherwise dependent of $y$. Thus, by Corollary 3.5,

(3.24)
$\int_{\Gamma}\big{|}\rho_{1}\big{(}2^{1/\alpha}x,z\big{)}-\varphi(z)\big{|}\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z<\varepsilon$

for all $y\in\Gamma_{R}$ and $x\in\Gamma_{1}$ small enough. Putting (3.24)
together with (3.23) we arrive at (3.20). Using the scaling property (3.7) and
Theorem 3.4,

$\displaystyle\lim_{\Gamma\ni x\to 0}\rho_{1}(x,y)$
$\displaystyle=2^{(d+2\beta)/\alpha}\int_{\Gamma}\varphi(z)\rho_{1}\big{(}z,2^{1/\alpha}y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z$
$\displaystyle=\int_{\Gamma}\varphi(z)\rho_{1/2}\big{(}2^{-1/\alpha}z,y\big{)}M_{\Gamma}^{2}(z)\,{\textrm{d}}z=L_{\ln
2}\varphi(y)=\varphi(y).$

The proof is complete. ∎

Note that by the symmetry of $\rho_{t}$, for $x\in\Gamma$,

$\rho_{t}(x,0):=\lim_{\Gamma\ni y\to
0}\rho_{t}(x,y)=\rho_{t}(0,x)=t^{-(d+2\beta)/\alpha}\varphi\big{(}t^{-1/\alpha}x\big{)}.$

Recall also that by (3.12) and (3.4),

$\rho_{1}(x,y)\approx(1+|y|)^{-d-\alpha}\frac{\mathbb{P}_{y}(\tau_{\Gamma}>1)}{M_{\Gamma}(y)}\in
L^{1}(M_{\Gamma}^{2}(y)\,{\textrm{d}}y).$

Thus, by Theorem 3.7 and the dominated convergence theorem,

(3.25) $\int_{\Gamma}\varphi(x)M_{\Gamma}^{2}(x)\,\,{\textrm{d}}x=1.$

Let us summarize the results of this section in one statement.

######  Theorem 3.8.

Assume $\Gamma$ is a fat cone. Then the function $\rho$ has a continuous
extension to $(0,\infty)\times(\Gamma\cup\\{0\\})\times(\Gamma\cup\\{0\\})$
and

(3.26) $\rho_{t}(0,y):=\lim_{\Gamma\ni x\to 0}\rho_{t}(x,y)\in(0,\infty),\quad
t>0,\,y\in\Gamma,$

satisfies

(3.27) $\rho_{t}(0,y)=t^{-(d+2\beta)/\alpha}\rho_{1}(0,t^{-1/\alpha}y),\quad
t>0,\,y\in\Gamma,$

and

(3.28)
$\int_{\Gamma}\rho_{t}(0,y)\rho_{s}(y,z)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=\rho_{t+s}(0,z),\quad
s,t>0,\,z\in\Gamma.$

###### Proof.

The existence of the limit (3.26) and the scaling property (3.27) are proved
in Theorem 3.7, see also Lemma 3.6. For the proof of (3.28) we employ (3.5) to
write

$\displaystyle\rho_{t+s}(0,z)=\lim_{\Gamma\ni y\to
0}\rho_{t+s}(y,z)=\lim_{\Gamma\ni y\to
0}\int_{\Gamma}\rho_{t}(y,w)\rho_{s}(w,z)M_{\Gamma}^{2}(w)\,{\textrm{d}}w,$

and use (3.4), (3.6), (3.13), and the dominated convergence theorem.

Thus, it remains to prove the continuity of $\rho$ on
$(0,\infty)\times(\Gamma\cup\\{0\\})\times(\Gamma\cup\\{0\\})$. By symmetry
and the Chapman-Kolmogorov property (3.5) of $\rho_{1}$,

(3.29)
$\rho_{1}(x,y)=\int_{\Gamma}\rho_{1/2}(x,z)\rho_{1/2}(y,z)M_{\Gamma}^{2}(z)\,{\textrm{d}}z,\quad
x,y\in\Gamma.$

The continuity of $\rho_{1}$ on $\Gamma\times\Gamma$ together with Theorem
3.7, for every $x_{0},y_{0}\in\Gamma\cup\\{0\\}$ we have
$\rho_{1/2}(x,z)\to\rho_{1/2}(x_{0},z)$ and
$\rho_{1/2}(y,z)\to\rho_{1/2}(y_{0},z)$ as $x\to x_{0}$ and $y\to y_{0}$.
Moreover, (3.12) entails that

$\rho_{1/2}(x,z)\rho_{1/2}(y,z)M_{\Gamma}^{2}(z)\leqslant
c(1+|z|)^{-2d-2\alpha},$

with the constant $c$ possibly dependent on $x_{0}$ and $y_{0}$. It follows by
the dominated convergence theorem that

$\rho_{1}(x,y)\to\int_{\Gamma}\rho_{1/2}(x_{0},z)\rho_{1/2}(y_{0},z)M_{\Gamma}^{2}(z)\,{\textrm{d}}z,$

as $x\to x_{0}$ and $y\to y_{0}$ and in view of (3.29), it is an extension of
$\rho_{1}$ to $(\Gamma\cup\\{0\\})\times(\Gamma\cup\\{0\\})$, which will be
denoted by the same symbol. It follows now from (3.6) that

$t^{-(d+2\beta)/\alpha}\rho_{1}\big{(}t^{-1/\alpha}x,t^{-1/\alpha}y\big{)},\quad
x,y\in\Gamma\cup\\{0\\},\,t>0,$

is a finite continuous extension of $\rho_{t}$ for every $t>0$. It remains to
observe that the extension is unique and jointly continuous in
$(t,x,y)\in\mathbb{R}_{+}\times(\Gamma\cup\\{0\\})\times(\Gamma\cup\\{0\\}$. ∎

######  Corollary 3.9.

We have $\rho_{1}(0,0)=\lim_{\Gamma\ni x,y\to 0}\rho_{1}(x,y)\in(0,\infty).$

###### Proof.

By Theorem 3.8, $\rho_{1}(0,0)=\lim_{\Gamma\ni y\to 0}\varphi(y)=:\varphi(0)$.
Thus, the claim follows by Lemma 3.6 and [3, Lemma 4.2]. ∎

###### Proof of Theorem 1.1.

By (3.26), $\Psi_{t}(x)=\rho_{t}(0,x)M_{\Gamma}(x)$, $t>0$, $x\in\Gamma$.
Thus, the existence of $\Psi_{t}$ is just a reformulation of (3.26). The
scaling property (1.1) follows immediately from (3.27) and the homogeneity of
the Martin kernel (2.10), and (1.2) is equivalent to (3.28). ∎

We conclude this part by rephrasing (3.25) in terms of $\Psi_{t}$:

(3.30) $\int_{\Gamma}\Psi_{t}(x)M_{\Gamma}(x)\,{\textrm{d}}x=1,\quad t>0.$

###  3.3. Yaglom limit

The above results quickly lead to calculation of the Yaglom limit for the
stable process (conditioned to stay in a cone). Note that our proof is
different from that in [15]. We also cover more general cones, e.g.,
$\mathbb{R}\setminus\\{0\\}$ and
$\mathbb{R}^{2}\setminus([0,\infty)\times\\{0\\})$. First, we obtain the
following extension of [15, Theorem 3.1].

######  Corollary 3.10.

Let $\Gamma$ be a fat cone. For every $t>0$,

$\lim_{\Gamma\ni x\to
0}\frac{\mathbb{P}_{x}(\tau_{\Gamma}>t)}{M_{\Gamma}(x)}=C_{1}t^{-\beta/\alpha}\quad\mbox{where}\quad
C_{1}=\int_{\Gamma}\varphi(z)M_{\Gamma}(z)\,{\textrm{d}}z\in(0,\infty).$

###### Proof.

It is enough to prove the claim for $t=1$; the general case follows by the
scalings (2.7) and (2.10). We have

$\frac{\mathbb{P}_{x}(\tau_{\Gamma}>1)}{M_{\Gamma}(x)}=\int_{\Gamma}\frac{p_{1}^{\Gamma}(x,y)}{M_{\Gamma}(x)}\,{\textrm{d}}y=\int_{\Gamma}\rho_{1}(x,y)M_{\Gamma}(y)\,{\textrm{d}}y,\quad
x\in\Gamma.$

We use (3.12), the dominated convergence theorem, and Theorem 3.7 to get the
conclusion. ∎

The first identity below is the Yaglom limit.

######  Theorem 3.11.

Assume $\Gamma$ is a fat cone and $B$ is a bounded subset of $\Gamma$. Then,

$\lim_{t\to\infty}\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)=\mu(A),\quad A\subseteq\Gamma,$

uniformly in $x\in B$, where

$\mu(A):=\frac{1}{C_{1}}\int_{A}\varphi(y)M_{\Gamma}(y)\,{\textrm{d}}y,\quad
A\subseteq\Gamma.$

###### Proof.

By (2.6) and the scaling property (2.7),

$\displaystyle\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)$
$\displaystyle=\frac{\mathbb{P}_{x}\left(\tau_{\Gamma}>t,t^{-1/\alpha}X_{t}\in
A\right)}{\mathbb{P}_{x}(\tau_{\Gamma}>t)}$
$\displaystyle=\frac{\mathbb{P}_{t^{-1/\alpha}x}\left(\tau_{\Gamma}>1,X_{1}\in
A\right)}{\mathbb{P}_{t^{-1/\alpha}x}(\tau_{\Gamma}>1)}$
$\displaystyle=\int_{A}\frac{p_{1}^{\Gamma}\big{(}t^{-1/\alpha}x,y\big{)}}{M_{\Gamma}\big{(}t^{-1/\alpha}x\big{)}}\,{\textrm{d}}y\cdot\frac{M_{\Gamma}(t^{-1/\alpha}x)}{\mathbb{P}_{t^{-1/\alpha}x}(\tau_{\Gamma}>1)}.$

The claim follows by Corollary 3.10, (3.12), and the dominated convergence
theorem. ∎

######  Theorem 3.12.

If $\Gamma$ is a fat cone and $\gamma$ is a probability measure on $\Gamma$
with $\int_{\Gamma}(1+|y|)^{\alpha}\,\gamma({\rm d}y)<\infty$, then

$\lim_{t\to\infty}\mathbb{P}_{\gamma}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)=\mu(A),\quad A\subseteq\Gamma.$

###### Proof.

Let $t\geqslant 1$. In view of [24, Lemma 8.7], we may write

$\displaystyle\mathbb{P}_{\gamma}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)$
$\displaystyle=\frac{\mathbb{P}_{\gamma}\left(t^{-1/\alpha}X_{t}\in
A,\tau_{\Gamma}>t\right)}{\mathbb{P}_{\gamma}\left(\tau_{\Gamma}>t\right)}$
$\displaystyle=\int_{\Gamma}\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)\frac{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{\gamma}\left(\tau_{\Gamma}>t\right)}\,\gamma({\textrm{d}}x).$

We first prove that for all $x\in\Gamma$,

(3.31)
$\frac{\mathbb{P}_{\gamma}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}:=\int_{\Gamma}\frac{\mathbb{P}_{y}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}\,\gamma(\,{\textrm{d}}y)\to\int_{\Gamma}\frac{M_{\Gamma}(y)}{M_{\Gamma}(x)}\,\gamma({\textrm{d}}y),$

as $t\to\infty$. Indeed, fix $x\in\Gamma$. First we note that by local
boundedness of $M_{\Gamma}$ and (2.10),

$\int_{\Gamma}M_{\Gamma}(y)\,\gamma({\rm d}y)\leqslant
c\int_{\Gamma}(1+|y|)^{\beta}\,\gamma({\rm d}y)<\infty,$

so the right-hand side of (3.31) is finite. Next, by Corollary 3.10, (2.7),
and (2.10),

$\displaystyle\lim_{t\to\infty}\frac{\mathbb{P}_{y}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}=\lim_{t\to\infty}\frac{\mathbb{P}_{t^{-1/\alpha}y}\left(\tau_{\Gamma}>1\right)M_{\Gamma}(t^{-1/\alpha}x)}{\mathbb{P}_{t^{-1/\alpha}x}\left(\tau_{\Gamma}>1\right)M_{\Gamma}(t^{-1/\alpha}y)}\frac{M_{\Gamma}(y)}{M_{\Gamma}(x)}=\frac{M_{\Gamma}(y)}{M_{\Gamma}(x)},\quad
x,y\in\Gamma.$

Moreover, since $x$ is fixed, we may assume that $t\geqslant
1\vee|x|^{\alpha}$. Thus, by [3, Lemma 4.2], Lemma 3.2, the local boundedness
of $M_{\Gamma}$ and (2.10),

$\frac{\mathbb{P}_{y}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}\leqslant
c\frac{\big{(}t^{-\beta/\alpha}+t^{-1}|y|^{\alpha-\beta}\big{)}M_{\Gamma}(y)}{t^{-\beta/\alpha}M_{\Gamma}(x)}\leqslant
c\frac{\big{(}1+|y|^{\alpha-\beta}\big{)}M_{\Gamma}(y)}{M_{\Gamma}(x)}\leqslant
c\frac{(1+|y|)^{\alpha}}{M_{\Gamma}(x)}.$

Thus, the dominated convergence theorem yields (3.31), as desired.

Next, we consider a family $\mathcal{F}_{1}$ of functions $f_{t}$ of the form

$f_{t}(x)=\frac{\mathbb{P}_{x}(\tau_{\Gamma}>t)}{\mathbb{P}_{\gamma}(\tau_{\Gamma}>t)},\quad
x\in\Gamma,\,t\geqslant 1.$

Denote

$f(x)=\frac{M_{\Gamma}(x)}{\int_{\Gamma}M_{\Gamma}(y)\,\gamma({\rm d}y)},\quad
x\in\Gamma.$

By virtue of (3.31), $f_{t}\to f$ everywhere in $\Gamma$ as $t\to\infty$.
Thus, $f_{t}\to f$ in measure $\gamma$ as $t\to\infty$, see [33, Definition
22.2]. Moreover, we have

$\int_{\Gamma}f(x)\,\gamma({\rm
d}x)=1=\lim_{t\to\infty}1=\lim_{t\to\infty}\int_{\Gamma}f_{t}(x)\,\gamma({\rm
d}x).$

Therefore, by Vitali’s theorem [33, Theorem 22.7], the family
$\mathcal{F}_{1}$ is uniformly integrable. If we now consider the family
$\mathcal{F}_{2}$ of functions $\tilde{f}_{t}$ of the form

$\tilde{f}_{t}(x)=\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)f_{t}(x),\quad x\in\Gamma,\,t\geqslant 1,$

then a trivial bound $\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)\leqslant 1$ shows that $\mathcal{F}_{2}$
is uniformly integrable as well (see, e.g., [33, Theorem 22.9]). By Theorem
3.11, (3.31) and [33, Theorem 22.7],

$\lim_{t\to\infty}\int_{\Gamma}\mathbb{P}_{x}\left(t^{-1/\alpha}X_{t}\in
A\;\middle|\;\tau_{\Gamma}>t\right)\frac{\mathbb{P}_{x}\left(\tau_{\Gamma}>t\right)}{\mathbb{P}_{\gamma}\left(\tau_{\Gamma}>t\right)}\,\gamma({\textrm{d}}x)=\int_{\Gamma}\mu(A)\frac{M_{\Gamma}(x)}{\int_{\Gamma}M_{\Gamma}(y)\,\gamma({\rm
d}y)}\,\gamma({\rm d}x)=\mu(A).$

The proof is complete. ∎

######  Example 3.13.

Note that $\beta=0$ if and only if $\Gamma^{c}$ is a polar set and then
$M_{\Gamma}(x)=1$ for all $x\in\Gamma$, see [3, Theorem 3.2]. Consequently, we
have $p_{t}^{\Gamma}(x,y)=p_{t}(x,y)$ and $\mathbb{P}_{x}(\tau_{\Gamma}>t)=1$
for all $x,y\in\Gamma$ and all $t>0$. It follows that
$\rho_{t}(x,y)=p_{t}(x,y)$ and a direct calculation using the Chapman-
Kolmogorov property entails that $\varphi(y)=p_{1}(0,y)$ is the stationary
density for the (classical) $\alpha$-stable Ornstein-Uhlenbeck semigroup, see
(3.14) and Theorem 3.4. The statement of Theorem 3.7 thus reduces to
continuity of the heat kernel of the isotropic $\alpha$-stable Lévy process.
Theorems 3.11 and 3.12 trivialize in a similar way. Incidentally, in this case
the moment condition on $\gamma$ in Theorem 3.12 is superfluous. Further
examples are given in Section 4.

##  4\. Asymptotic behavior for the killed semigroup

This section is devoted to examples and applications in Functional Analysis
and Partial Differential Equations. Note that in Lemmas 4.1 and 4.2 we do not
assume that $\Gamma$ is fat.

######  Lemma 4.1.

$\\{P_{t}^{\Gamma}\\}_{t>0}$ is a strongly continuous contraction semigroup on
$L^{1}(M_{\Gamma})$ and

(4.1)
$\int_{\Gamma}P_{t}^{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x=\int_{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x,\quad
t>0,\quad f\in L^{1}(M_{\Gamma}).$

###### Proof.

Let $f\geqslant 0$. By the Fubini-Tonelli theorem, the symmetry of
$p_{t}^{\Gamma}$ and Theorem 3.1,

(4.2)
$\int_{\Gamma}P_{t}^{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x=\int_{\Gamma}\int_{\Gamma}p_{t}^{\Gamma}(x,y)f(y)M_{\Gamma}(y)\,{\textrm{d}}y\,{\textrm{d}}x=\int_{\Gamma}f(y)M_{\Gamma}(y)\,{\textrm{d}}y.$

Since $|P_{t}^{\Gamma}f|\leqslant P_{t}^{\Gamma}|f|$, the contractivity
follows. Furthermore, for arbitrary $f\in L^{1}(M_{\Gamma})$ we write
$f=f_{+}-f_{-}$ and use (4.2) to prove (4.1). The semigroup property follows
from (2.5).

To prove the strong continuity, we fix $f\in L^{1}(M_{\Gamma})$ and let
$G:=fM_{\Gamma}\in L^{1}(\Gamma)$. There is a sequence $g_{n}\in
C_{c}^{\infty}(\Gamma)$ such that $\|g_{n}-G\|_{L^{1}(\Gamma)}\to 0$ as
$n\to\infty$. For $f_{n}:=g_{n}/M_{\Gamma}$ we get $f_{n}\in
C_{c}^{\infty}(\Gamma)$ and
$\|f_{n}-f\|_{L^{1}(M_{\Gamma})}=\|g_{n}-G\|_{L^{1}(\Gamma)}\to 0$. By the
first part of the proof,

$\displaystyle\|P_{t}^{\Gamma}f-f\|_{L^{1}(M_{\Gamma})}$
$\displaystyle\leqslant\|P_{t}^{\Gamma}f-P_{t}^{\Gamma}f_{n}\|_{L^{1}(M_{\Gamma})}+\|P_{t}^{\Gamma}f_{n}-f_{n}\|_{L^{1}(M_{\Gamma})}+\|f_{n}-f\|_{L^{1}(M_{\Gamma})}$
$\displaystyle\leqslant
2\|f_{n}-f\|_{L^{1}(M_{\Gamma})}+\|P_{t}^{\Gamma}f_{n}-f_{n}\|_{L^{1}(M_{\Gamma})}.$

It remains to prove that $\|P_{t}^{\Gamma}f-f\|_{L^{1}(M_{\Gamma})}\to 0$ as
$t\to 0^{+}$ for every $f\in C_{c}^{\infty}(\Gamma)$. To this end we let
$\varepsilon>0$ and choose $R>0$ such that $\operatorname{supp}f\in B_{R}$ and
$\int_{\Gamma\setminus\Gamma_{R}}P_{t}^{\Gamma}|f|(x)M_{\Gamma}(x)\,{\textrm{d}}x<\varepsilon$.
Then,

(4.3)
$\|P_{t}^{\Gamma}f-f\|_{L^{1}(M_{\Gamma})}<\int_{\Gamma_{R}}\big{|}P_{t}^{\Gamma}f(x)-f(x)\big{|}M_{\Gamma}(x)\,{\textrm{d}}x+\varepsilon.$

Considering the integrand in (4.3), for all $x\in\Gamma_{R}$ we have

(4.4)
$\big{|}P_{t}^{\Gamma}f(x)-f(x)\big{|}\leqslant\int_{\Gamma}p_{t}^{\Gamma}(x,y)|f(y)-f(x)|\,{\textrm{d}}y+|f(x)|\mathbb{P}_{x}(\tau_{\Gamma}\leqslant
t).$

Since $P_{t}f\to f$ uniformly as $t\to 0^{+}$, for $t>0$ small enough we get

$\displaystyle\int_{\Gamma}p_{t}^{\Gamma}(x,y)|f(y)-f(x)|\,{\textrm{d}}y\leqslant\int_{{\mathbb{R}^{d}}}p_{t}(x,y)|f(y)-f(x)|\,{\textrm{d}}y<\varepsilon.$

On the other hand,

$|f(x)|\mathbb{P}_{x}(\tau_{\Gamma}\leqslant
t)\leqslant\|f\|_{\infty}\sup_{x\in K}\mathbb{P}_{x}(\tau_{\Gamma}\leqslant
t),$

where $K:=\operatorname{supp}f$. We have
$r:=\operatorname{dist}(K,\Gamma^{c})>0$, so

$\mathbb{P}_{x}(\tau_{\Gamma}\leqslant
t)\leqslant\mathbb{P}_{x}(\tau_{B(x,r)}\leqslant
t)=\mathbb{P}_{0}(\tau_{B_{r}}\leqslant t)\leqslant
ctr^{-\alpha}<\varepsilon,$

for $t$ small enough, see, e.g., [31]. By (4.3) and (4.4) we get, as required,

$\|P_{t}^{\Gamma}f-f\|_{L^{1}(M_{\Gamma})}<\varepsilon+(\varepsilon+\|f\|_{\infty}\varepsilon)|\Gamma_{R}|\sup_{\Gamma_{R}}M_{\Gamma}.$

∎

Recall that

$\|f\|_{q,M_{\Gamma}}:=\|f/M_{\Gamma}\|_{L^{q}(M_{\Gamma}^{2})}=\bigg{(}\int_{\Gamma}|f(x)|^{q}M_{\Gamma}^{2-q}(x)\,{\textrm{d}}x\bigg{)}^{\frac{1}{q}}=\|f\|_{L^{q}(M_{\Gamma}^{2-q})},$

if $1\leqslant q<\infty$, and

$\|f\|_{\infty,M_{\Gamma}}:={\rm ess}\sup_{x\in\Gamma}|f(x)|/M_{\Gamma}(x).$

The following characterization of hypercontractivity of $P^{\Gamma}_{t}$ is
crucial for the proof of (1.3).

######  Lemma 4.2.

Let $q\in[1,\infty)$. We have

(4.5) $\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}\leqslant
Ct^{-\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|f\|_{1,M_{\Gamma}}$

for all $t>0$ and all non-negative functions $f$ on ${\mathbb{R}^{d}}$ if and
only if

(4.6)
$\sup_{y\in\Gamma}\int_{\Gamma}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x<\infty.$

###### Proof.

Assume (4.6). Let $f\geqslant 0$. With the notation $F:=f/M_{\Gamma}$ we get

$\|P_{1}^{\Gamma}f\|_{q,M_{\Gamma}}=\Bigg{(}\int_{\Gamma}\bigg{(}\int_{\Gamma}\rho_{1}(x,y)F(y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y\bigg{)}^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x\Bigg{)}^{1/q}.$

Let $c$ be the supremum in (4.6). By Minkowski integral inequality,

$\displaystyle\Bigg{(}\int_{\Gamma}\bigg{(}\int_{\Gamma}\rho_{1}(x,y)F(y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y\bigg{)}^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x\Bigg{)}^{1/q}$
$\displaystyle\leqslant\int_{\Gamma}\bigg{(}\int_{\Gamma}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x\bigg{)}^{1/q}F(y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y$
$\displaystyle\leqslant
c\int_{\Gamma}F(y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y=c\|f\|_{1,M_{\Gamma}}.$

For $t>0$, by scaling we get (4.5) as follows:

$\displaystyle\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}$
$\displaystyle=t^{\frac{d+\beta(2-q)}{\alpha
q}}\|P_{1}^{\Gamma}f(t^{1/\alpha}\,\cdot\,)\|_{q,M_{\Gamma}}\leqslant
ct^{\frac{d+\beta(2-q)}{\alpha q}}\|f(t^{1/\alpha}\,\cdot\,)\|_{1,M_{\Gamma}}$
$\displaystyle=ct^{\frac{d+\beta(2-q)}{\alpha
q}}t^{-\frac{d+\beta}{\alpha}}\|f\|_{1,M_{\Gamma}}=ct^{-\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|f\|_{1,M_{\Gamma}}.$

Conversely, assume (4.5). Let $y\in\Gamma$. Let $g_{n}\geqslant 0$,
$n\in\mathbb{N}$, be functions in $C^{\infty}_{c}(\Gamma)$ approximating
$\delta_{y}$, the Dirac measure at $y$, as follows:

$\int_{\Gamma}g_{n}(x)\,{\textrm{d}}x=1,\qquad\text{and}\qquad\lim_{n\to\infty}\int_{\Gamma}h(x)g_{n}(x)\,{\textrm{d}}x=h(y),$

for every function $h$ continuous near $y$. For $f_{n}:=g_{n}/M_{\Gamma}$,
$\|f_{n}\|_{1,M_{\Gamma}}=\|g_{n}\|_{1}=1$ and

$P_{1}^{\Gamma}f_{n}(x)=\int_{\Gamma}p_{1}^{\Gamma}(x,z)\frac{g_{n}(z)}{M_{\Gamma}(z)}\,{\textrm{d}}z\to\frac{p_{1}^{\Gamma}(x,y)}{M_{\Gamma}(y)},$

as $n\to\infty$. By (4.5) and Fatou’s lemma,

$\displaystyle C^{q}$
$\displaystyle\geqslant\liminf_{n\to\infty}\|P_{1}^{\Gamma}f_{n}\|_{q,M_{\Gamma}}^{q}$
$\displaystyle=\liminf_{n\to\infty}\int_{\Gamma}\bigg{|}\int_{\Gamma}\frac{p_{1}^{\Gamma}(x,z)}{M_{\Gamma}(z)}g_{n}(z)\,{\textrm{d}}z\bigg{|}^{q}M_{\Gamma}^{2-q}(x)\,{\textrm{d}}x$
$\displaystyle\geqslant\int_{\Gamma}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x.$

Since $y\in\Gamma$ was arbitrary, we obtain (4.6). ∎

######  Remark 4.3.

Of course, (4.5) extends to arbitrary $f\in L^{1}(M_{\Gamma})$.

######  Example 4.4.

As in Example 3.13, we assume that $\beta=0$. In fact, to simplify notation,
let $\Gamma={\mathbb{R}^{d}}$. Then (4.6) is trivially satisfied for every
$q\in[1,\infty)$, because $\rho_{1}(x,y)=p_{1}(x,y)$ is bounded. Therefore, by
(4.5), for each $f\in L^{1}$,

$\|P_{t}f\|_{q}\leqslant Ct^{-\frac{d}{\alpha}\frac{q-1}{q}}\|f\|_{1}.$

This agrees with [36], see also [13].

Here is a refinement of Lemma 4.2.

######  Lemma 4.5.

Let $q\in[1,\infty)$, assume (4.6) and suppose $\Gamma$ is fat. If $f\in
L^{1}(M_{\Gamma})$, $\int_{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x=0$ then

(4.7)
$\lim_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}=0.$

If, additionally, $f$ has compact support, then (4.7) is true for $q=\infty$,
too.

###### Proof.

Let $\omega>0$. First, we prove prove (4.7) for a compactly supported function
$f\in L^{1}(M_{\Gamma})$ satisfying

(4.8) $\int_{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x=0.$

Step 1. Case $q=\infty$.
For $t>0$ we let

$\displaystyle I(t)$
$\displaystyle:=t^{\frac{d+2\beta}{\alpha}}\|P_{t}^{\Gamma}f\|_{\infty,M_{\Gamma}}=t^{\frac{d+2\beta}{\alpha}}\sup_{x\in\Gamma}\left|\int_{\Gamma}\rho_{t}(x,y)M_{\Gamma}(y)f(y)\,{\textrm{d}}y\right|.$

By (4.8),

$\displaystyle
I(t)=t^{\frac{d+2\beta}{\alpha}}\sup_{x\in\Gamma}\left|\int_{\Gamma}\left(\rho_{t}(x,y)-\rho_{t}(x,0)\right)M_{\Gamma}(y)f(y)\,{\textrm{d}}y\right|.$

Since $f$ has compact support, for sufficiently large $t>0$ we have

$\displaystyle I(t)$
$\displaystyle=t^{\frac{d+2\beta}{\alpha}}\sup_{x\in\Gamma}\left|\int_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}\left(\rho_{t}(x,y)-\rho_{t}(x,0)\right)M_{\Gamma}(y)f(y)\,{\textrm{d}}y\right|$
$\displaystyle\leqslant
t^{\frac{d+2\beta}{\alpha}}\sup_{\begin{subarray}{c}x\in\Gamma\\\ |y|\leqslant
t^{\frac{1}{\alpha}}\omega\end{subarray}}\big{|}\rho_{t}(x,y)-\rho_{t}(x,0)\big{|}\int_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}M_{\Gamma}(y)|f(y)|\,{\textrm{d}}y$
$\displaystyle=\sup_{\begin{subarray}{c}x\in\Gamma\\\
|y|\leqslant\omega\end{subarray}}\left|\rho_{1}\left(x,y\right)-\rho_{1}\left(x,0\right)\right|\int_{\Gamma}M_{\Gamma}(y)|f(y)|\,{\textrm{d}}y,$

where in the last line we used scaling (3.6) of $\rho$. By Theorem 3.7, we can
make it arbitrary small by choosing small $\omega$, and (4.7) follows in this
case.
Step 2. Case $q=1$.
For $t>0$ we let

$J(t):=\|P_{t}^{\Gamma}f\|_{1,M_{\Gamma}}=\int_{\Gamma}\left|\int_{\Gamma}p^{\Gamma}_{t}(x,y)M_{\Gamma}(x)f(y)\,{\textrm{d}}y\right|\,{\textrm{d}}x=\int_{\Gamma}\Big{|}\int_{\Gamma}\rho_{t}(x,y)M_{\Gamma}^{2}(x)f(y)M_{\Gamma}(y)\,{\textrm{d}}y\Big{|}\,{\textrm{d}}x.$

Applying (4.8), we get

$J(t)\leqslant\int_{\Gamma}\int_{\Gamma}\big{|}\rho_{t}(x,y)-\rho_{t}(x,0)\big{|}M_{\Gamma}^{2}(x)|f(y)|M_{\Gamma}(y)\,{\textrm{d}}y\,{\textrm{d}}x.$

Since $f$ has compact support,

$\displaystyle J(t)$ $\displaystyle\leqslant\int_{\Gamma}\int_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}\big{|}\rho_{t}(x,y)-\rho_{t}(x,0)\big{|}M_{\Gamma}^{2}(x)|f(y)|M_{\Gamma}(y)\,{\textrm{d}}y\,{\textrm{d}}x$
$\displaystyle\leqslant\sup_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}\|\rho_{t}(\cdot,y)-\rho_{t}(\cdot,0)\|_{L^{1}(M_{\Gamma}^{2})}\int_{\Gamma}M_{\Gamma}(y)|f(y)|\,{\textrm{d}}y,$

for sufficiently large $t$. In view of (2.10) and (3.6), by changing variables
$t^{-1/\alpha}x\to x$ and $t^{-1/\alpha}y\to y$ we obtain

$\displaystyle\sup_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}\|\rho_{t}(\cdot,y)-\rho_{t}(\cdot,0)\|_{L^{1}(M_{\Gamma}^{2})}$
$\displaystyle=t^{-\frac{d+2\beta}{\alpha}}\sup_{|y|\leqslant
t^{\frac{1}{\alpha}}\omega}\int\left|\rho_{1}\left(t^{-{1}/{\alpha}}x,t^{-{1}/{\alpha}}y\right)-\rho_{1}\left(t^{-{1}/{\alpha}}x,0\right)\right|M_{\Gamma}^{2}(x)\,{\textrm{d}}x$
$\displaystyle=\sup_{|y|\leqslant\omega}\|\rho_{1}(\cdot,y)-\rho_{1}(\cdot,0)\|_{L^{1}(M_{\Gamma}^{2})}.$

By Corollary 3.5, we can make it arbitrary small by choosing small $\omega$,
so (4.7) is true.
Step 3. Case $q\in(1,\infty)$.
By Hölder inequality we get that, as $t\to\infty$,

$\displaystyle
t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}$
$\displaystyle=t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\bigg{(}\int_{\Gamma}\big{|}P_{t}^{\Gamma}f(x)/M_{\Gamma}(x)\big{|}^{q-1}\big{|}P_{t}^{\Gamma}f(x)M_{\Gamma}(x)\big{|}\,{\textrm{d}}x\bigg{)}^{\frac{1}{q}}$
$\displaystyle\leqslant\left(t^{\frac{d+2\beta}{\alpha}}\|P_{t}^{\Gamma}f\|_{\infty,M_{\Gamma}}\right)^{\frac{q-1}{q}}\|P_{t}^{\Gamma}f\|_{1,M_{\Gamma}}^{\frac{1}{q}}\to
0,$

since both factors converge to zero as $t\rightarrow\infty$ by Step 1. and
Step 2.

Finally, consider arbitrary $f\in L^{1}(M_{\Gamma})$ with
$\int_{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x=0$. Let $R>0$ and
$f_{R}(x)=(f(x)-c_{R})\mathbf{1}_{|x|\leqslant R}$, where
$c_{R}=\int_{|x|\leqslant
R}f(x)M_{\Gamma}(x)\,{\textrm{d}}x/\int_{|x|\leqslant
R}M_{\Gamma}(x)\,{\textrm{d}}x$. Of course,

(4.9) $\int_{\Gamma}M_{\Gamma}(x)f_{R}(x)\,{\textrm{d}}x=0,$

and $f_{R}$ is compactly supported. Furthermore, due to our assumptions,

$\displaystyle\|f-f_{R}\|_{L^{1}(M_{\Gamma})}$
$\displaystyle=|c_{R}|\int_{|x|\leqslant
R}M_{\Gamma}(x)\,{\textrm{d}}x+\int_{|x|>R}M_{\Gamma}(x)|f(x)|\,{\textrm{d}}x$
$\displaystyle=\Big{|}\int_{|x|\leqslant
R}M_{\Gamma}(x)f(x)\,{\textrm{d}}x\Big{|}+\int_{|x|>R}M_{\Gamma}(x)|f(x)|\,{\textrm{d}}x\to
0$

as $R\rightarrow\infty$. Let ${\varepsilon}>0$ and choose $R>0$ so large that

$\|f-f_{R}\|_{1,M_{\Gamma}}<{\varepsilon}.$

For $q=1$, by using the triangle inequality and Lemma 4.1, we get

$\displaystyle\|P_{t}^{\Gamma}f\|_{1,M_{\Gamma}}$
$\displaystyle\leqslant\|P_{t}^{\Gamma}f_{R}\|_{1,M_{\Gamma}}+\|P_{t}^{\Gamma}(f-f_{R})\|_{1,M_{\Gamma}}$
$\displaystyle\leqslant\|P_{t}^{\Gamma}f_{R}\|_{1,M_{\Gamma}}+\|f-f_{R}\|_{1,M_{\Gamma}},$

and Step 2. yields

$\limsup_{t\rightarrow\infty}\|P_{t}^{\Gamma}f\|_{1,M_{\Gamma}}\leqslant{\varepsilon},$

which proves (4.7) in this case.

If $1<q<\infty$, then using the triangle inequality and Lemma 4.2, we obtain

$\displaystyle
t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}$
$\displaystyle\leqslant
t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f_{R}\|_{q,M_{\Gamma}}+t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}(f-f_{R})\|_{q,M_{\Gamma}}$
$\displaystyle\leqslant
t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f_{R}\|_{q,M_{\Gamma}}+C\|f-f_{R}\|_{1,M_{\Gamma}}.$

By (4.9) and Step 3.,

$\limsup_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f\|_{q,M_{\Gamma}}\leqslant
2C{\varepsilon}.$

This completes the proof of (4.7) for $q\in(1,\infty)$. ∎

######  Theorem 4.6.

Let $q\in[1,\infty)$, assume (4.6) and suppose $\Gamma$ is fat. Then for $f\in
L^{1}(M_{\Gamma})$ and $A=\int_{\Gamma}f(x)M_{\Gamma}(x)\,{\textrm{d}}x$,

$\lim_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f-A\Psi_{t}\|_{q,M_{\Gamma}}=0.$

######  Remark 4.7.

In view of (3.30) and Lemma 4.1, the constant $A$ in Theorem 4.6 satisfies

$\int_{\Gamma}\big{(}P_{t}^{\Gamma}f(x)-A\Psi_{s}(x)\big{)}M_{\Gamma}(x)\,{\textrm{d}}x=0,\quad
s,t>0.$

###### Proof of Theorem 4.6.

By (2.5), (1.2), Remark 4.7 and Lemma 4.5,

$\displaystyle\lim_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}f-A\Psi_{t}\|_{q,M_{\Gamma}}$
$\displaystyle=\lim_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t+1}^{\Gamma}f-A\Psi_{t+1}\|_{q,M_{\Gamma}}$
$\displaystyle=\lim_{t\rightarrow\infty}t^{\frac{d+2\beta}{\alpha}\frac{q-1}{q}}\|P_{t}^{\Gamma}\big{(}P_{1}^{\Gamma}f-A\Psi_{1}\big{)}\|_{q,M_{\Gamma}}=0.$

∎

###  4.1. Applications

We conclude the article by providing several applications and examples which
apply to our results. In particular, we draw the reader’s attention to Lemma
4.9, which provides sharp distinction between cones bigger and smaller than
the half-space
$\mathbb{R}_{+}^{d}:=\\{x=(x_{1},\ldots,x_{d})\in{\mathbb{R}^{d}}\colon
x_{d}>0\\}$. The same behavior is displayed by the bigger class of smooth
cones, as we assert in Corollary 4.10. First, we note a simple observation.

######  Example 4.8.

Let $q=1$. By (3.4), the condition (4.6) holds for every fat cone $\Gamma$.

######  Lemma 4.9.

Let $q\in(1,\infty)$ and suppose $\Gamma$ is a right-circular cone. Then (4.6)
holds if $\beta\geqslant\alpha/2$. Conversely, if $d\geqslant 2$ and
$\beta<\alpha/2$, then (4.6) does not hold.

###### Proof.

Recall that by [10, Theorem 2 and Eq. (3)],

(4.10) $p_{1}^{\Gamma}(x,y)\approx
p_{1}(x,y)\frac{\big{(}1\wedge\delta_{\Gamma}(x)\big{)}^{\alpha/2}\big{(}1\wedge\delta_{\Gamma}(y)\big{)}^{\alpha/2}}{\big{(}1\wedge|x|)^{\alpha/2-\beta}\big{(}1\wedge|y|)^{\alpha/2-\beta}},\quad
x,y\in\Gamma.$

Moreover, [29, Lemma 3.3] entails that

$M_{\Gamma}(x)\approx\delta_{\Gamma}(x)^{\alpha/2}|x|^{\beta-\alpha/2},\quad
x\in{\mathbb{R}^{d}}.$

Using this together with (4.10) and (2.4), we infer that, for $x,y\in\Gamma$,

(4.11) $\displaystyle\begin{aligned} \rho_{1}(x,y)&\approx
p_{1}(x,y)\frac{\big{(}1\wedge\delta_{\Gamma}(x)\big{)}^{\alpha/2}\big{(}1\wedge\delta_{\Gamma}(y)\big{)}^{\alpha/2}}{\big{(}1\wedge|x|)^{\alpha/2-\beta}\big{(}1\wedge|y|)^{\alpha/2-\beta}\delta_{\Gamma}(x)^{\alpha/2}|x|^{\beta-\alpha/2}\delta_{\Gamma}(y)^{\alpha/2}|y|^{\beta-\alpha/2}}\\\
&\approx\big{(}1+|x-y|\big{)}^{-d-\alpha}\frac{\big{(}1+\delta_{\Gamma}(x))^{-\alpha/2}\big{(}1+\delta_{\Gamma}(y))^{-\alpha/2}}{(1+|x|)^{\beta-\alpha/2}(1+|y|)^{\beta-\alpha/2}}.\end{aligned}$

Let $q\in(1,\infty)$ and assume $\beta\geqslant\alpha/2$. Then it follows from
(4.11) that $\rho_{1}(x,y)\lesssim 1$ for $x,y\in\Gamma$, and (3.4) entails
that

(4.12) $\displaystyle\begin{aligned}
\int_{\Gamma}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x&\leqslant\|\rho_{1}(x,\,\cdot\,)\|_{\infty}^{q-1}\int_{\Gamma}\rho_{1}(x,y)M_{\Gamma}^{2}(y)\,{\textrm{d}}y\\\
&=\|\rho_{1}(x,\,\cdot\,)\|_{\infty}^{q-1}\lesssim 1.\end{aligned}$

Thus, we get (4.6) as claimed.

Now assume that $d\geqslant 2$ and $\beta<\alpha/2$. Let $y\in\Gamma$ be such
that $\delta_{\Gamma}(y)=2$, so that $A:=B(y,1)\subseteq\Gamma$. Then for
$x\in A$ one clearly has that $1+|x-y|\approx 1$ and
$\delta_{\Gamma}(x)\approx 1$. Then it follows from (4.11) that

$\displaystyle\int_{\Gamma}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x$
$\displaystyle\geqslant\int_{A}\rho_{1}(x,y)^{q}M_{\Gamma}^{2}(x)\,{\textrm{d}}x$
$\displaystyle\approx\int_{A}(1+|x|)^{q(\alpha/2-\beta)}(1+|y|)^{q(\alpha/2-\beta)}\delta_{\Gamma}(x)^{\alpha}|x|^{2\beta-\alpha}\,{\textrm{d}}x$
$\displaystyle\approx|y|^{(q-1)(\alpha-2\beta)}.$

Since $\alpha-2\beta>0$ and $q>1$, by taking $|y|\to\infty$ we see that (4.6)
cannot hold in this case. ∎

######  Corollary 4.10.

For $d\geqslant 2$ and smooth cone $\Gamma$, (4.6) holds if and only if
$\beta\geqslant\alpha/2$.

###### Proof.

Recall that $\Gamma$ is open and $C^{1,1}$ outside of the origin. From the
harmonicity and homogeneity of $M_{\Gamma}$, by the boundary Harnack principle
we get, as in [29, Lemma 3.3], that

$M_{\Gamma}(x)\approx\delta_{\Gamma}(x)^{\alpha/2}|x|^{\beta-\alpha/2},\quad
x\in\Gamma.$

Moreover, since a smooth cone is fat, its Dirichlet heat kernel satisfies
(4.10). Thus, one can directly repeat the proof of Lemma 4.9 to conclude the
claim. ∎

######  Example 4.11.

For $d=1$, either $\Gamma=(0,\infty)$ or $\Gamma=\mathbb{R}\setminus\\{0\\}$
and both cases are (trivially) smooth cones, with $\delta_{\Gamma}(y)=|y|$ for
$y\in\Gamma$.

When $\Gamma=(0,\infty)$, then (4.11) yields the boundedness of $\rho_{1}$ and
(4.6) holds through (4.12), since this $\Gamma$ is right-circular. Recall that
here one has $\beta=\alpha/2$ and $M_{\Gamma}(x)=(0\vee x)^{\alpha/2}$ by [3,
Example 3.2], and we refer to Example 1.3 for the rest of the summary of this
case.

If $\alpha\in(1,2)$ and $\Gamma=\mathbb{R}\setminus\\{0\\}$, then
$\Gamma^{c}=\\{0\\}$ is a non-polar set, $\beta=\alpha-1$, and
$M_{\Gamma}(x)=|x|^{\alpha-1}$; see [3, Example 3.3]. Note that in this
example, $\Gamma$ is not right-circular anymore. Nevertheless, by [10, Example
8], the survival probability is
$\mathbb{P}_{x}(\tau_{\Gamma}>t)\approx(1\wedge t^{-1/\alpha}|x|)^{\alpha-1}$
and

$p_{1}^{\Gamma}(x,y)\approx(1+|x-y|)^{-1-\alpha}(1\wedge|x|)^{\alpha-1}(1\wedge|y|)^{\alpha-1},\quad
x,y\in\Gamma.$

Thus,

$\rho_{1}(x,y)\approx(1+|x-y|)^{-1-\alpha}(1+|x|)^{1-\alpha}(1+|y|)^{1-\alpha}\lesssim
1,\quad x,y\in\Gamma,$

and one may apply (4.12) to get (4.6), too. It then follows from Theorem 4.6
with $q=2$ that
$\lim_{t\to\infty}t^{(2\alpha-1)/(2\alpha)}\|P_{t}^{\Gamma}f\|_{2}=0$ if
$\int_{\mathbb{R}}f(x)|x|^{\alpha-1}\,{\textrm{d}}x=0$. Accordingly, by Lemma
3.6, the stationary density $\varphi$ of Theorem 3.4 satisfies
$\varphi(x)\approx(1+|x|)^{-2\alpha}$. Here and below, $x\in\Gamma$ and $t>0$.
Then, $\Psi_{t}(x)\approx t^{-1}(1+t^{-1/\alpha}|x|)^{-1-\alpha}(1\wedge
t^{-1/\alpha}|x|)^{\alpha-1}$ and the density of the Yaglom limit is
comparable with
$\varphi(x)|x|^{\alpha-1}\approx(1+|x|)^{-2\alpha}|x|^{\alpha-1}\approx\Psi_{1}(x)$.

######  Example 4.12.

Let $d\geqslant 2$ and $\Gamma$ be a right-circular cone which is a subset of
the half-space $\mathbb{R}_{+}^{d}$. Then by [3, Example 3.2 and Lemma 3.3],
we have $\beta\geqslant\alpha/2$ and Lemma 4.9 gives (4.6). On the other hand,
if $\Gamma$ is such that $\mathbb{R}_{+}^{d}\subsetneq\Gamma$, then
$\beta<\alpha/2$ by [3, Lemma 3.3], and Lemma 4.9 asserts that (4.6) does not
hold.

The following extends Example 1.3. Let $\Gamma=\mathbb{R}_{+}^{d}$, so that
$M_{\Gamma}(x)=(0\vee x_{d})^{\alpha/2}$. By Theorem 4.6,
$\lim_{t\to\infty}t^{(d+\alpha)/(2\alpha)}\|P_{t}^{\Gamma}f\|_{2}=0$ if
$\int_{\mathbb{R}^{d}_{+}}f(x)x_{d}^{\alpha/2}\,{\textrm{d}}x=0$. Furthermore,
the survival probability is $\mathbb{P}_{x}(\tau_{\Gamma}>t)\approx 1\wedge
t^{-1/2}(0\vee x_{d})^{\alpha/2}$ and

$p_{1}^{\Gamma}(x,y)\approx(1+|x-y|)^{-d-\alpha}(1\wedge
x_{d})^{\alpha/2}(1\wedge y_{d})^{\alpha/2},$

see [10, Example 2]. Here and below, $x,y\in\Gamma$. So,
$\Psi_{t}(x)\approx(t^{1/\alpha}\vee|x|)^{-d-\alpha}(t^{1/\alpha}\wedge
x_{d})^{\alpha/2}$, $t>0$. The stationary density $\varphi$ is comparable to
$(1+|x|)^{-d-\alpha}(1+x_{d})^{-\alpha/2}$ and the density of the Yaglom
distribution is
$\varphi(x)x_{d}^{\alpha/2}/\int_{\mathbb{R}^{d}_{+}}\varphi(y)y_{d}^{\alpha/2}\,{\textrm{d}}y\approx(1+|x|)^{-d-\alpha}(1\wedge
x_{d})^{\alpha/2}\approx\Psi_{1}(x)$.

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