#  Heat kernel estimates for Dirichlet forms degenerate at the boundary

Soobin Cho, Panki Kim, Renming Song and Zoran Vondraček  Department of
Mathematical Sciences, Seoul National University, Seoul 08826, Republic of
Korea<EMAIL_ADDRESS>Department of Mathematical Sciences and Research
Institute of Mathematics, Seoul National University, Seoul 08826, Republic of
Korea<EMAIL_ADDRESS>Department of Mathematics, University of Illinois,
Urbana, IL 61801, USA<EMAIL_ADDRESS>Department of Mathematics, Faculty
of Science, University of Zagreb, Zagreb, Croatia, and Department of
Mathematical Sciences, Seoul National University, Seoul, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

The goal of this paper is to prove the existence of heat kernels for two types
of purely discontinuous symmetric Markov processes in the upper half-space of
${\mathbb{R}}^{d}$ with jump kernels degenerate at the boundary, and to
establish sharp two-sided estimates on the heat kernels. The jump kernels are
of the form $J(x,y)={\cal B}(x,y)|x-y|^{-\alpha-d}$, $\alpha\in(0,2)$, where
the function ${\cal B}$ depends on four parameters and may vanish at the
boundary. Our results are the first sharp two-sided estimates for the heat
kernels of non-local operators with jump kernels degenerate at the boundary.

The first type of processes we consider are conservative Markov processes on
the closed half-space with jump kernel $J(x,y)$. Depending on the regions
where the parameters belong, the heat kernels estimates have three different
forms, two of them are qualitatively different from all previously known heat
kernel estimates.

The second type of processes we consider are the processes above killed either
by a critical potential or upon hitting the boundary of the half-space. We
establish that their heat kernel estimates have the approximate factorization
property with survival probabilities decaying as a power of the distance to
the boundary, where the power depends on the constant in the critical
potential.

Finally, by using the heat kernel estimates, we obtain sharp two-sided Green
function estimates that cover the main result of [31]. This alternative proof
provides an explanation of the anomalous form of the estimates.

This research is supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIP) (No. 2021R1A4A1027378).

Research supported in part by a grant from the Simons Foundation (#960480,
Renming Song)

Research supported in part by the Croatian Science Foundation under the
project 4197. (Zoran Vondraček)

AMS 2020 Mathematics Subject Classification: Primary 60J35, 60J45; Secondary
60J50, 60J76, 31C25, 35K08

Keywords and phrases: Markov processes, Dirichlet forms, jump kernel, killing
potential, heat kernel, heat kernel estimates, Green function

###### Contents

  1. 1 Introduction and main results
  2. 2 Nash inequality and the existence of the heat kernel
  3. 3 Parabolic Hölder regularity and consequences
  4. 4 Parabolic Harnack inequality and preliminary lower bound of heat kernels
  5. 5 Preliminary upper bounds of heat kernels
  6. 6 Sharp lower bounds of heat kernels
  7. 7 Sharp upper bounds of heat kernels
  8. 8 Proofs of Theorems 1.1 and 1.2
  9. 9 Green function estimates
  10. 10 Appendix

##  1\. Introduction and main results

In this paper, we study both conservative and non-conservative purely
discontinuous (self-similar) Markov processes in the upper half-space of
${\mathbb{R}}^{d}$ with jump kernels of the form
$J(x,y)=|x-y|^{-d-\alpha}{\cal B}(x,y)$, $\alpha\in(0,2)$. The function ${\cal
B}(x,y)$ may tend to 0 when $x$ or $y$ tends to boundary of the half-space,
and so the jump kernel may be degenerate. The main focus of the paper is to
show the existence and continuity of the transition densities of the processes
(or the heat kernels of the corresponding non-local operators), and to
establish their sharp two-sided estimates. Heat kernel estimates for non-local
operators have been the subject of many papers in the last twenty years, see
[2, 4, 5, 10, 12, 13, 15, 16, 17, 19, 24, 26] and the references therein. In
all of the papers mentioned above, the function ${\cal B}$ is assumed to be
bounded between two positive constants, which can be viewed as a uniform
ellipticity condition for non-local operators. To the best of our knowledge,
the current paper is the first one to study sharp two-sided heat kernel
estimates when the jump kernel is degenerate. Boundary Harnack principle and
sharp two-sided Green function estimates for purely discontinuous Markov
processes with degenerate jump kernels, which can be viewed as the elliptic
counterpart of results of the current paper, have been recently obtained in
[30, 31, 32].

The first type of processes we look at are conservative jump processes on
$\overline{{\mathbb{R}}}^{d}_{+}:=\\{x=(\widetilde{x},x_{d}):\widetilde{x}\in{\mathbb{R}}^{d-1},x_{d}\geq
0\\}$ with jump kernel $J(x,y)=|x-y|^{-d-\alpha}{\cal B}(x,y)$, where the
function ${\cal B}(x,y)$ is symmetric, homogeneous, horizontally translation
invariant, and is allowed to approach 0 at the boundary at arbitrary fixed
polynomial rate in terms of some non-negative parameters
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}$. The generator of such a process is
the non-local operator

$L^{{\cal
B}}_{\alpha}f(x)=\mathrm{p.v.}\int_{{{\mathbb{R}}}^{d}_{+}}(f(y)-f(x))|x-y|^{-d-\alpha}{\cal
B}(x,y)\,dy.$

When ${\cal B}(x,y)$ is bounded between two positive constants, the heat
kernel estimates for such processes are of the form
$\min\\{t^{-d/\alpha},tJ(x,y)\\}$. This was first established in the
pioneering work [15], even for the case of metric measure spaces. This form of
the estimates reflects the fact that the main contribution to the heat kernel
comes from one (big) jump from $x$ to $y$. This feature has been observed in
all subsequent studies, see [2, 16, 10, 17, 19, 24] and the references
therein. In our setting of jump kernel degenerate at the boundary, there are
two novel features in the heat kernel estimates. The first one appears in the
case when the involved parameters satisfy $\beta_{2}<\alpha+\beta_{1}$, in
which case the heat kernel is comparable to
$\min\\{t^{-d/\alpha},tJ(x+t^{1/\alpha}\mathbf{e}_{d},y+t^{1/\alpha}\mathbf{e}_{d})\\}$,
where $\mathbf{e}_{d}=(\widetilde{0},1)$. In words, the form of the heat
kernel estimates shows that the main contribution to the heat kernel at time
$t$ comes from one jump from the point $t^{1/\alpha}$ units above $x$ to the
point $t^{1/\alpha}$ units above $y$. Due to the fact the jump kernel vanishes
at the boundary, it is very unlikely that the process will make one (big) jump
from (or to) a point very close to the boundary. The second feature is more
striking and indicates a sort of a phase-transition at the level
$\beta_{2}=\alpha+\beta_{1}$: When the parameters satisfy
$\beta_{2}\geq\alpha+\beta_{1}$, in addition to the already mentioned part
$\min\\{t^{-d/\alpha},tJ(x+t^{1/\alpha}\mathbf{e}_{d},y+t^{1/\alpha}\mathbf{e}_{d})\\}$,
the sharp heat kernel estimates include a part which reflects a significant
contribution to the heat kernel coming from two jumps connecting $x$ and $y$.
The precise description is given in Theorem 1.1.

The second type of processes we look at are the ones described in the previous
paragraph but killed either by a critical potential $\kappa x_{d}^{-\alpha}$
or upon hitting the boundary of
${\mathbb{R}}^{d}_{+}:=\\{x=(\widetilde{x},x_{d}):\widetilde{x}\in{\mathbb{R}}^{d-1},x_{d}>0\\}$
(the latter happens only when $\alpha\in(1,2)$). The generator of such a
process is the non-local operator $L^{{\cal B}}f(x)=L^{{\cal
B}}_{\alpha}f(x)-\kappa x_{d}^{-\alpha}f(x)$. We study the effect of such
killings on the heat kernel. When ${\cal B}(x,y)$ is bounded between two
positive constants, the effect of killing on the heat kernel of the process
is, after intensive research during the last fifteen years, fairly well
understood. In most cases, the heat kernel of the killed process has the so
called approximate factorization property: It is comparable to the product of
the heat kernel of non-killed (original) process and the survival
probabilities starting from points $x$ and $y$. In case of smooth open sets,
the survival probability can be expressed in terms of the distance between the
point and the boundary, see [4, 5, 8, 12, 13, 21, 27] and the references
therein. In our setting of jump kernel degenerate at the boundary, we
establish the same property: The heat kernel of the killed process enjoys the
approximate factorization property with survival probabilities decaying as the
$q$-th power of the distance to the boundary, where $q$ is in one-to-one
correspondence with the constant $\kappa\geq 0$, see Theorem 1.2. When
$\kappa=0$, this theorem generalizes [14] (for half spaces) where the
factorization property for censored stable process is established. Due to the
quite complicated form of the heat kernel estimates in Theorem 1.1, obtaining
the factorization property in Theorem 1.2 is a formidable task.

We now introduce the precise setup and state the main results of this paper.
This setup was first introduced in [30] and was motivated by the results of
[28, 29] on subordinate killed Lévy processes. In fact, subordinate killed
Lévy processes, whose analytical counterparts are fractional powers of
Dirichlet fractional Laplacians, are the main natural examples of Markov
processes with jump kernels satisfying the assumptions below.

Let $d\geq 1$ and $0<\alpha<2$. Recall that
${\mathbb{R}}^{d}_{+}=\\{(\widetilde{x},x_{d})\in{\mathbb{R}}^{d}:x_{d}>0\\}$
and
$\overline{\mathbb{R}}^{d}_{+}=\\{(\widetilde{x},x_{d})\in{\mathbb{R}}^{d}:x_{d}\geq
0\\}$. Here and below, $a\wedge b:=\min\\{a,b\\}$, $a\vee b:=\max\\{a,b\\}$,
and $a\asymp b$ means that $c\leq b/a\leq c^{-1}$ for some $c\in(0,1)$. For
$b_{1},b_{2},b_{3},b_{4}\geq 0$, let

$\displaystyle B_{b_{1},b_{2},b_{3},b_{4}}(x,y)$
$\displaystyle:=\left(\frac{x_{d}\wedge y_{d}}{|x-y|}\wedge
1\right)^{b_{1}}\left(\frac{x_{d}\vee y_{d}}{|x-y|}\wedge 1\right)^{b_{2}}$
$\displaystyle\qquad\times\log^{b_{3}}\left(e+\frac{(x_{d}\vee
y_{d})\wedge|x-y|}{x_{d}\wedge
y_{d}\wedge|x-y|}\right)\,\log^{b_{4}}\left(e+\frac{|x-y|}{(x_{d}\vee
y_{d})\wedge|x-y|}\right).$ (1.1)

Let $J(x,y)=|x-y|^{-d-\alpha}{\cal B}(x,y)$, where the function ${\cal
B}:\overline{{\mathbb{R}}}^{d}_{+}\times\overline{{\mathbb{R}}}^{d}_{+}\to[0,\infty)$
will be assumed to satisfy some or all of the following hypotheses:

(A1) ${\cal B}(x,y)={\cal B}(y,x)$ for all
$x,y\in\overline{{\mathbb{R}}}^{d}_{+}$.

(A2) If $\alpha\geq 1$, then there exist $\theta>\alpha-1$ and $C_{1}>0$ such
that

$|{\cal B}(x,x)-{\cal B}(x,y)|\leq C_{1}\left(\frac{|x-y|}{x_{d}\wedge
y_{d}}\right)^{\theta},\quad x,y\in{\mathbb{R}}^{d}_{+}.$

(A3)(I) There exist $C_{2}\geq 1$ and
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}\geq 0$, with $\beta_{1}>0$ if
$\beta_{3}>0$, and $\beta_{2}>0$ if $\beta_{4}>0$, such that

$C_{2}^{-1}B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x,y)\leq{\cal
B}(x,y)\leq C_{2},\quad x,y\in\overline{{\mathbb{R}}}^{d}_{+}.$

(II) There exists $C_{3}>0$ such that

${\cal B}(x,y)\leq C_{3}B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x,y),\quad
x,y\in\overline{{\mathbb{R}}}^{d}_{+},$

where $\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ are the same constants as in
(I).

(A4) For all $x,y\in\overline{{\mathbb{R}}}^{d}_{+}$ and $a>0$, ${\cal
B}(ax,ay)={\cal B}(x,y)$. In case $d\geq 2$, for all
$x,y\in\overline{{\mathbb{R}}}^{d}_{+}$ and
$\widetilde{z}\in{\mathbb{R}}^{d-1}$, ${\cal
B}(x+(\widetilde{z},0),y+(\widetilde{z},0))={\cal B}(x,y)$.

These four hypotheses were introduced in [30], and with the same notation as
above repeated in [31, 32]. Condition (A2) is not needed in Theorems 1.1 and
1.3, while in Theorems 1.2 and 1.4 it is used through several results from
[30, 31, 32].

Throughout this paper, we always assume that ${\cal B}(x,y)$ satisfies (A1),
(A3)(I) and (A4).

Consider the following symmetric form

${\cal
E}^{0}(u,v):=\frac{1}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))(v(x)-v(y))J(x,y)dydx.$

Since ${\cal B}(x,y)$ is bounded,
$C_{c}^{\infty}({\overline{\mathbb{R}}}^{d}_{+})$ is closable in
$L^{2}({\overline{\mathbb{R}}}^{d}_{+},dx)=L^{2}({\mathbb{R}}^{d}_{+},dx)$ by
Fatou’s lemma. Let ${\overline{\cal F}}$ be the closure of
$C_{c}^{\infty}({\overline{\mathbb{R}}}^{d}_{+})$ in
$L^{2}({\mathbb{R}}^{d}_{+},dx)$ under the norm ${\cal E}^{0}_{1}:={\cal
E}^{0}+(\cdot,\cdot)_{L^{2}({\mathbb{R}}^{d}_{+},dx)}$. Then $({\cal
E}^{0},{\overline{\cal F}})$ is a regular Dirichlet form on
$L^{2}({\mathbb{R}}^{d}_{+},dx)$. Let $\overline{Y}=(\overline{Y}_{t},t\geq
0;{\mathbb{P}}_{x},x\in\overline{\mathbb{R}}^{d}_{+}\setminus{\cal
N}^{\prime})$ be the Hunt process associated with $({\cal
E}^{0},{\overline{\cal F}})$, where ${\cal N}^{\prime}$ is an exceptional set.

Here is our first main result. The heat kernel estimates are expressed in
different but equivalent forms, each providing a different viewpoint. Recall
that ${\mathbf{e}}_{d}=(\widetilde{0},1)\in{\mathbb{R}}^{d}$.

######  Theorem 1.1.

Suppose that (A1), (A3) and (A4) hold. Then the process $\overline{Y}$ can be
refined to be a conservative Feller process with strong Feller property
starting from every point in ${\overline{\mathbb{R}}}^{d}_{+}$ and has a
jointly continuous heat kernel
$\overline{p}:(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}\to(0,\infty)$.
Moreover, the heat kernel $\overline{p}$ has the following estimates: For all
$(t,x,y)\in(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$,

$\displaystyle\overline{p}(t,x,y)\asymp
t^{-d/\alpha}\wedge\bigg{[}tJ(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})$
$\displaystyle\,+{\bf
1}_{\\{\beta_{2}\geq\alpha+\beta_{1}\\}}t^{2}\int_{(x_{d}\vee y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!J(x+t^{1/\alpha}{\mathbf{e}}_{d},x+r{\mathbf{e}}_{d})\,J(x+r{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})r^{d-1}dr\bigg{]}.$
(1.2)

Furthermore the heat kernel estimates in (1.1) can be rewritten in terms of
$B_{b_{1},b_{2},b_{3},b_{4}}$ explicitly by considering three cases
separately:

(i) If $\beta_{2}<\alpha+\beta_{1}$, then for all
$(t,x,y)\in(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$,

$\displaystyle\overline{p}(t,x,y)$
$\displaystyle\asymp\bigg{(}t^{-d/\alpha}\wedge\frac{tB_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})}{|x-y|^{d+\alpha}}\bigg{)}$
$\displaystyle\asymp\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d}).$
(1.3)

(ii) If $\beta_{2}>\alpha+\beta_{1}$, then for all
$(t,x,y)\in(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$,

$\displaystyle\overline{p}(t,x,y)\asymp\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})$
$\displaystyle+\\!\bigg{(}\\!1\wedge\frac{t}{|x-y|^{\alpha}}\\!\bigg{)}B_{\beta_{1},\beta_{1},0,\beta_{3}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\\!\bigg{(}\\!e\\!+\\!\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)}\\!\bigg{]}.$ (1.4)

(iii) If $\beta_{2}=\alpha+\beta_{1}$, then for all
$(t,x,y)\in(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$,

$\displaystyle\overline{p}(t,x,y)\asymp\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})$
$\displaystyle\\!+\\!\bigg{(}\\!1\wedge\frac{t}{|x-y|^{\alpha}}\\!\bigg{)}B_{\beta_{1},\beta_{1},0,\beta_{3}+\beta_{4}+1}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\\!\bigg{(}\\!e\\!+\\!\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)}\bigg{]}.$ (1.5)

Note that if $x$ or $y$ is close to the boundary, then
$J(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})$ is not
comparable to $J(x,y)$ in our setting. Thus, even in case
$\beta_{2}<\alpha+\beta_{1}$, the form of the heat kernel is different from
the usual form. The appearance of the two terms in the brackets on the right-
hand sides of (1.4)–(1.1) reflects the fact that the dominant contribution to
the heat kernel may come from either one jump or two jumps. Moreover, it is
easy to see that neither of these two terms dominates the other one for all
$(t,x,y)$. Below we illustrate this feature when $\beta_{2}>\alpha+\beta_{1}$.

Let $\beta_{2}>\alpha+\beta_{1}$. When $|x-y|>6t^{1/\alpha}$, the second term
in the brackets in (1.4), i.e.,

$\displaystyle\bigg{(}\\!1\wedge\frac{t}{|x-y|^{\alpha}}\\!\bigg{)}B_{\beta_{1},\beta_{1},0,\beta_{3}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\\!\bigg{(}\\!e\\!+\\!\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)}$ (1.6)

is comparable to

$\displaystyle
t|x-y|^{d+\alpha}\int_{B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)}J(x+t^{1/\alpha}{\mathbf{e}}_{d},z)\,J(z,y+t^{1/\alpha}{\mathbf{e}}_{d})dz,$
(1.7)

see Remark 6.5 below. In the special case when $\beta_{3}=\beta_{4}=0$ (and
$\beta_{2}>\alpha+\beta_{1}$), Figure 1(b), where the comparability of (1.6)
and (1.7) is used, illustrates the regions where one jump and two jumps
dominate.

The different forms of the heat kernel estimates in Theorem 1.1 are
consequences of the relationship among $\beta_{1}$, $\beta_{2}$ and $\alpha$
only, the values of $\beta_{3}$ and $\beta_{4}$ do not play any role in this.
To reduce technicalities, the reader, on a first reading, may assume
$\beta_{3}=\beta_{4}=0$, without loosing the essential features of this paper.
Note that, even in this special case, a logarithmic term appears in the
estimates when $\beta_{2}=\alpha+\beta_{1}$. We need the logarithmic terms in
the assumptions to cover the important examples in [21, 29].

(a) One jump regime

(b) Two jumps regime

Figure 1. Dominant path from $x$ to $y$ (with $x_{d}\leq y_{d}$) at time
$t<(|x-y|/6)^{\alpha}$ when $\beta_{2}>\alpha+\beta_{1}$,
$\beta_{3}=\beta_{4}=0$

.

Let ${\cal F}^{0}$ be the closure of $C_{c}^{\infty}({\mathbb{R}}^{d}_{+})$ in
$L^{2}({\mathbb{R}}^{d}_{+},dx)$ under the norm ${\cal E}^{0}_{1}$. Then
$({\cal E}^{0},{\cal F}^{0})$ is also a regular Dirichlet form on
$L^{2}({\mathbb{R}}^{d}_{+},dx)$ and it is the part form of $({\cal
E}^{0},{\overline{\cal F}})$ on ${\mathbb{R}}^{d}_{+}$. Let
$Y^{0}=(Y^{0}_{t},t\geq
0;{\mathbb{P}}_{x},x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}_{0})$ be the Hunt
process associated with $({\cal E}^{0},{\cal F}^{0})$ where ${\cal N}_{0}$ is
an exceptional set. $Y^{0}$ can be regarded as the part process of
$\overline{Y}$ killed upon exiting ${\mathbb{R}}^{d}_{+}$.

For $\kappa\in(0,\infty)$, define

$\displaystyle{\cal E}^{\kappa}(u,v)$ $\displaystyle:={\cal
E}^{0}(u,v)+\int_{{\mathbb{R}}^{d}_{+}}u(x)v(x)\kappa x_{d}^{-\alpha}dx,$
$\displaystyle{\cal F}^{\kappa}$ $\displaystyle:=\widetilde{\cal F}^{0}\cap
L^{2}({\mathbb{R}}^{d}_{+},\kappa x_{d}^{-\alpha}dx),$

where $\widetilde{\cal F}^{0}$ is the family of all ${\cal E}^{0}_{1}$-quasi-
continuous functions in ${\cal F}^{0}$. Then $({\cal E}^{\kappa},{\cal
F}^{\kappa})$ is a regular Dirichlet form on $L^{2}({\mathbb{R}}^{d}_{+},dx)$
with $C_{c}^{\infty}({\mathbb{R}}^{d}_{+})$ as a special standard core, see
[23, Theorems 6.1.1 and 6.1.2]. Let $Y^{\kappa}=(Y^{\kappa}_{t},t\geq
0;{\mathbb{P}}_{x},x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}_{\kappa})$ be the
Hunt process associated with $({\cal E}^{\kappa},{\cal F}^{\kappa})$ where
${\cal N}_{\kappa}$ is an exceptional set. For $\kappa\in[0,\infty)$, we
denote by $\zeta^{\kappa}$ the lifetime of $Y^{\kappa}$. Define
$Y^{\kappa}_{t}=\partial$ for $t\geq\zeta^{\kappa}$, where $\partial$ is a
cemetery point added to the state space ${\mathbb{R}}^{d}_{+}$.

We now associate with the constant $\kappa$ a positive parameter $q$ which
plays an important role in the paper. For $q\in(-1,\alpha+\beta_{1})$, set

$\displaystyle C(\alpha,q,{\cal
B})\\!=\\!\begin{cases}\displaystyle\int_{{\mathbb{R}}^{d-1}}\\!\frac{1}{(|\widetilde{u}|^{2}+1)^{(d+\alpha)/2}}\\!\int_{0}^{1}\frac{(s^{q}-1)(1-s^{\alpha-q-1})}{(1-s)^{1+\alpha}}{\cal
B}\big{(}((1-s)\widetilde{u},1),s{\mathbf{e}}_{d}\big{)}ds\,d\widetilde{u}\\!&\mbox{if
}d\geq 2,\\\\[14.22636pt]
\displaystyle\int_{0}^{1}\frac{(s^{q}-1)(1-s^{\alpha-q-1})}{(1-s)^{1+\alpha}}{\cal
B}\big{(}1,s\big{)}ds&\mbox{if }d=1.\end{cases}$

If we additionally assume that (A3)(II) holds, then the constant
$C(\alpha,q,{\cal B})$ is well defined and finite for every
$q\in(-1,\alpha+\beta_{1})$, $C(\alpha,q,{\cal B})=0$ if and only if
$q\in\\{0,\alpha-1\\}$, and $\lim_{q\to-1}C(\alpha,q,{\cal
B})=\lim_{q\to\alpha+\beta_{1}}C(\alpha,q,{\cal B})=\infty$ (see [30, Lemma
5.4 and Remark 5.5]). Note that for every $s\in(0,1)$,
$q\mapsto(s^{q}-1)(1-s^{\alpha-q-1})$ is strictly decreasing on
$(-1,(\alpha-1)/2)$ and strictly increasing on
$((\alpha-1)/2,\alpha+\beta_{1})$. Thus, the shape of the map $q\mapsto
C(\alpha,q,{\cal B})$ is given as follows.

$q$ | $-1$ | $\cdots$ | $(\alpha-1)\wedge 0$ | $\cdots$ | $\frac{1}{2}(\alpha-1)$ | $\cdots$ | $(\alpha-1)\vee 0$ | $\cdots$ | $\alpha+\beta_{1}$
---|---|---|---|---|---|---|---|---|---
$C(\alpha,q,{\cal B})$ | $\infty$ | $\searrow$ | 0 | $\searrow$ | $\text{minimum}\leq 0$ | $\nearrow$ | 0 | $\nearrow$ | $\infty$

Consequently, for every $\kappa\geq 0$, there exists a unique
$q_{\kappa}\in[(\alpha-1)_{+},\alpha+\beta_{1})$ such that

$\kappa=C(\alpha,q_{\kappa},{\cal B}).$ (1.8)

######  Theorem 1.2.

Suppose that (A1)–(A4) and (1.8) hold with
$q_{\kappa}\in[(\alpha-1)_{+},\alpha+\beta_{1})$. Then the process
$Y^{\kappa}$ can be refined to start from every point in
${\mathbb{R}}^{d}_{+}$ and has a jointly continuous heat kernel
$p^{\kappa}:(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}\to(0,\infty)$.
Moreover, the following approximate factorization holds for all
$(t,x,y)\in(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$:

$\displaystyle
p^{\kappa}(t,x,y)\asymp\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\overline{p}(t,x,y)\asymp{\mathbb{P}}_{x}(\zeta^{\kappa}>t){\mathbb{P}}_{y}(\zeta^{\kappa}>t)\,\overline{p}(t,x,y),$
(1.9)

where $\overline{p}(t,x,y)$ is the heat kernel of $\overline{Y}$.

As a consequence of Lemma 2.3 below, we will see that, when $\alpha\leq 1$,
the process $\overline{Y}$ started from ${\mathbb{R}}^{d}_{+}$ will never hit
$\partial{\mathbb{R}}^{d}_{+}$ and is equal to $Y^{0}$. Thus for
$x,y\in{\mathbb{R}}^{d}_{+}$ we have that $p^{0}(t,x,y)=\overline{p}(t,x,y)$,
implying that the non-trivial content of Theorem 1.2 is for $\kappa>0$ when
$\alpha\leq 1$.

Let

$\overline{G}(x,y)=\int_{0}^{\infty}\overline{p}(t,x,y)dt\quad\text{and}\quad
G^{\kappa}(x,y)=\int_{0}^{\infty}p^{\kappa}(t,x,y)dt.$ (1.10)

When $\overline{G}(\cdot,\cdot)$ is not identically infinite, it is called the
Green function of $\overline{Y}$, and when $G^{\kappa}(\cdot,\cdot)$ is not
identically infinite, it is called the Green function of $Y^{\kappa}$.

As a consequence of the heat kernel estimates, we get the Green function
estimates. The following theorem says that the Green function of
$\overline{Y}$ is comparable to that of the isotropic $\alpha$-stable process
in ${\mathbb{R}}^{d}$ even though the jump kernel of $\overline{Y}$ may be
degenerate.

######  Theorem 1.3.

Suppose that (A1), (A3)(I) and (A4) hold. If $d>\alpha$, then

$\overline{G}(x,y)\asymp\frac{1}{|x-y|^{d-\alpha}},\quad
x,y\in{\overline{\mathbb{R}}}_{+}^{d}.$ (1.11)

If $d\leq\alpha$, then $\overline{G}(x,y)=\infty$ for all
$x,y\in{\overline{\mathbb{R}}}_{+}^{d}$.

When $d>(\alpha+\beta_{1}+\beta_{2})\wedge 2$, sharp two-sided estimates on
$G^{\kappa}(x,y)$ were obtained in [31, 32]. In the following theorem, we
extend those results by removing the restriction on $d$ and give another proof
using the heat kernel estimates. The advantage of the new proof is that it
explains the reason for the phase transition in the Green function estimates
for $d\geq 2$.

Let

$\displaystyle H_{q}(x,y)=\\!\begin{cases}\displaystyle 1&\mbox{if
}q<\alpha+\frac{1}{2}(\beta_{1}+\beta_{2}),\\\\[6.0pt]
\displaystyle\log^{\beta_{4}+1}\left(e+\frac{|x-y|}{(x_{d}\vee
y_{d})\wedge|x-y|}\right)&\mbox{if
}q=\alpha+\frac{1}{2}(\beta_{1}+\beta_{2}),\\\\[9.0pt]
\displaystyle\left(\frac{x_{d}\vee y_{d}}{|x-y|}\wedge
1\right)^{2\alpha+\beta_{1}+\beta_{2}-2q}\log^{\beta_{4}}\left(e+\frac{|x-y|}{(x_{d}\vee
y_{d})\wedge|x-y|}\right)&\mbox{if
}q>\alpha+\frac{1}{2}(\beta_{1}+\beta_{2}).\end{cases}$

######  Theorem 1.4.

Suppose that (A1)–(A4) and (1.8) hold with
$q_{\kappa}\in[(\alpha-1)_{+},\alpha+\beta_{1})$. When $\alpha\leq 1$, suppose
also that $q_{\kappa}>0$ (or, equivalently, $\kappa>0$). Then $G^{\kappa}$ has
the following estimates:

(i) If $d\geq 2$, then for all $x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle G^{\kappa}(x,y)$
$\displaystyle\asymp\frac{H_{q_{\kappa}}(x,y)}{|x-y|^{d-\alpha}}\left(\frac{x_{d}\wedge
y_{d}}{|x-y|}\wedge 1\right)^{q_{\kappa}}\left(\frac{x_{d}\vee
y_{d}}{|x-y|}\wedge 1\right)^{q_{\kappa}}.$

(ii) If $d=1$, then for all $x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle
G^{\kappa}(x,y)\asymp\begin{cases}\displaystyle\frac{1}{|x-y|^{1-\alpha}}\left(\frac{x\wedge
y}{|x-y|}\wedge 1\right)^{q_{\kappa}}&\mbox{ if }\alpha<1,\\\\[9.0pt]
\displaystyle\left(\frac{x\wedge y}{|x-y|}\wedge
1\right)^{q_{\kappa}}\log\left(e+\frac{(x\wedge
y)\vee|x-y|}{|x-y|}\right)&\mbox{ if }\alpha=1,\\\\[9.0pt]
\displaystyle(x\wedge y)^{\alpha-1}\left(\frac{x\wedge y}{|x-y|}\wedge
1\right)^{q_{\kappa}-\alpha+1}&\mbox{ if }\alpha>1.\end{cases}$

Note that when $d\geq 2$, at the threshold
$q_{\kappa}=\alpha+\frac{1}{2}(\beta_{1}+\beta_{2})$, there is a transition
from the usual behavior of the Green function estimates to anomalous behavior.

We describe now the strategy for proving our main results and the organization
of the paper.

It is well known that an appropriate Nash-type inequality implies the
existence of the heat kernel (outside an exceptional set) and its
$\alpha$-stable-type upper bound. So we start in Section 2 with establishing a
Nash-type inequality, see Proposition 2.6. To this end, we consider a certain
Feller process in ${\mathbb{R}}^{d}_{+}$ with continuous paths, subordinate it
by an independent $\alpha/2$-stable subordinator, and show a Nash-type
inequality for the Dirichlet form of the subordinate process. In case
$\alpha<1$, one can estimate the Dirichlet form of the subordinate process
from above by ${\cal E}^{0}$ and thus prove a Nash-type inequality for ${\cal
E}^{0}$. In case $\alpha\geq 1$, we use the scaling property of the process
and a truncation of the jump kernel together with the already obtained
inequality for $\alpha<1$.

In Section 3, we prove parabolic Hölder regularity for both $\overline{Y}$ and
$Y^{\kappa}$ and use this to remove the exceptional sets and extend
$\overline{p}(t,x,y)$ and $p^{\kappa}(t,x,y)$ continuously to
$(0,\infty)\times\overline{{\mathbb{R}}}^{d}_{+}\times\overline{{\mathbb{R}}}^{d}_{+}$,
respectively $(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$.
To prove the parabolic Hölder regularity, we need an interior lower bound on
the heat kernels. The proof of this lower bound is based on an argument that
has already appeared in [30, Section 3] and is given here in Proposition 3.1.
Next, we prove several lower bounds on the heat kernels, mean exit times and
exit distributions of the underlying process and its killed version that allow
us to apply the standard arguments for establishing the parabolic Hölder
regularity. We end Section 3 with the elementary Lemma 3.15 which will be used
repeatedly in deriving the heat kernel upper bounds.

Following the arguments from [9, 21], in Section 4 we establish the parabolic
Harnack inequality, see Theorem 4.3, and use it to establish the important
preliminary off-diagonal lower bound $p^{\kappa}(t,x,y)\geq ctJ(x,y)$ for $t$
small compared to $|x-y|$, $x_{d}$ and $y_{d}$, see Proposition 4.5.

In Section 5 we prove the following preliminary upper bound:

$p^{\kappa}(t,x,y)\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),\quad
t>0,\ x,y\in{\mathbb{R}}^{d}_{+},$

for all $q_{\kappa}\in[(\alpha-1)_{+},\alpha+\beta_{1})$, see Proposition 5.1.
Proposition 5.1 is proved in two steps: Using Lemma 3.15 and several results
from [30, 31, 32] (see Lemmas 5.10-5.11) we first prove Proposition 5.1 in
case $q_{\kappa}<\alpha$. The real challenge is to extend it to the full range
of $q_{\kappa}$ by covering the case $q_{\kappa}\geq\alpha$. For this we use
the upper bound on Green potentials of powers of the distance to the boundary.
In case $d\geq 2$, such bound was proved in [31, 32], and here we prove the
corresponding result for $d=1$. The restriction $q_{\kappa}<\alpha$ is removed
in Lemma 5.13 by using a bootstrap (induction) argument.

Section 6 is devoted to proving sharp heat kernel lower bounds. The estimates
are given in terms of a function $A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)$, a
space-time version of the function $B_{b_{1},b_{2},b_{3},b_{4}}(x,y)$. We
first prove a preliminary lower bound, Proposition 6.2, by using the semigroup
property and some interior lower bound on the heat kernel. This bound turns
out to be sufficient in case $\beta_{2}<\alpha+\beta_{1}$. For the case
$\beta_{2}\geq\alpha+\beta_{1}$ we need a sharper lower bound obtained in the
key Lemma 6.4. There we apply the semigroup property (and the preliminary
lower bound) on a carefully chosen interior set on which we have good control
of the terms appearing in the preliminary lower bound. By combining
Proposition 6.2 and Lemma 6.4, we obtain in Proposition 6.6 the sharp lower
bound in cases $\beta_{2}\neq\alpha+\beta_{1}$. The remaining case
$\beta_{2}=\alpha+\beta_{1}$ is more delicate and is covered in Proposition
6.9.

Sharp heat kernel upper bound is more difficult to establish and Section 7 is
devoted to this task. The first step is to establish in Lemma 7.2 an upper
bound which includes the function $A_{\beta_{1},0,\beta_{3},0}$. The proof
uses Lemma 3.15 and an induction argument which consecutively increases the
first parameter of the function $A$ until reaching $\beta_{1}$, thus making
the decay successively sharper. The sharp upper bound in case
$\beta_{2}<\alpha+\beta_{1}$, respectively $\beta_{2}\geq\alpha+\beta_{1}$, is
given in Theorem 7.5 and Corollary 7.9, respectively Theorem 7.10. The proofs
are based on Lemma 3.15, Lemma 7.2, and a number of delicate technical lemmas
involving multiple space-time integrals of the preliminary heat kernel
estimates.

In Section 8, we combine the upper bounds obtained in Section 7 with the lower
bounds from Section 6 and give the proofs of Theorems 1.1–1.2.

It is well known that the Green function is the integral over time of the heat
kernel. In Section 9 we first use this observation together with the estimates
of $\overline{p}(t,x,y)$ obtained in Proposition 2.7 and Lemma 3.7 (see also
Remark 3.12) to prove Theorem 1.3. By using the same method of integrating the
heat kernel estimates of $p^{\kappa}(t,x,y)$ over time we establish Theorem
1.4, thus reproving and extending the main result of [31]. This new proof
sheds more light on the anomalous behavior of these Green function estimates.
Lemma 9.1 clearly shows that they are consequences of the different forms of
the small time heat kernel estimates.

The paper ends with an appendix which contains a number of technical results
not depending on the preliminary estimates of the heat kernel.

Throughout this paper, the constants $\beta_{1}$, $\beta_{2}$, $\beta_{3}$,
$\beta_{4}$ will remain the same, and $\kappa$ always stands for a non-
negative number. The notation $C=C(a,b,\ldots)$ indicates that the constant
$C$ depends on $a,b,\ldots$. The dependence on $\kappa,d$ and $\alpha$ may not
be mentioned explicitly. Lower case letters $c_{i},i=1,2,\dots$ are used to
denote the constants in the proofs and the labeling of these constants starts
anew in each proof. We denote by $m_{d}$ the Lebesgue measure on
${\mathbb{R}}^{d}$. For Borel subset $D\subset{\mathbb{R}}^{d}$,
$\delta_{D}(x)$ denotes the distance between $x$ and the boundary $\partial
D$.

##  2\. Nash inequality and the existence of the heat kernel

The goal of this section is (i) to prove a Nash type inequality (Proposition
2.6); (ii) to deduce the existence of the heat kernels of $\overline{Y}$ and
$Y^{\kappa}$; and (iii) to establish their preliminary upper bounds
(Proposition 2.7).

We begin the section by introducing the notation for the relevant semigroups
and establishing their scale invariance and horizontal translation invariance.

For $\kappa\geq 0$, we denote by $(\overline{P}_{t})_{t\geq 0}$ and
$(P_{t}^{\kappa})_{t\geq 0}$ the semigroups corresponding to $({\cal
E}^{0},\overline{\cal F})$ and $({\cal E}^{\kappa},{\cal F}^{\kappa})$
respectively. $(\overline{P}_{t})_{t\geq 0}$ and $(P_{t}^{\kappa})_{t\geq 0}$
define contraction semigroups on
$L^{p}({\overline{\mathbb{R}}}^{d}_{+},dx)=L^{p}({\mathbb{R}}^{d}_{+},dx)$ for
every $p\in[1,\infty]$, and when $p\in[1,\infty)$, these semigroups are
strongly continuous. For $t>0$ and $p,q\in[1,\infty]$, define

$\displaystyle\lVert\overline{P}_{t}\rVert_{p\to
q}=\sup\left\\{\lVert\overline{P}_{t}f\rVert_{L^{q}({\mathbb{R}}^{d}_{+},dx)}:f\in
L^{p}({\mathbb{R}}^{d}_{+},dx),\,\lVert
f\rVert_{L^{p}({\mathbb{R}}^{d}_{+},dx)}\leq 1\right\\}.$

For $f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and $r>0$, define $f^{(r)}(x)=f(rx)$.
The following scaling property of $(P_{t}^{\kappa})_{t\geq 0}$ comes from [30,
Lemma 5.1] and [32, Lemma 2.1]. By the same proof, $(\overline{P}_{t})_{t\geq
0}$ also has the same scaling property. We give the proof for the reader’s
convenience.

######  Lemma 2.1.

Let $p\in[1,\infty]$ and $\kappa\geq 0$. For any $f\in
L^{p}({\mathbb{R}}^{d}_{+},dx)$, $t>0$ and $r>0$, we have

$\overline{P}_{t}f(x)=\overline{P}_{r^{-\alpha}t}f^{(r)}(x/r)\quad\text{and}\quad
P^{\kappa}_{t}f(x)=P^{\kappa}_{r^{-\alpha}t}f^{(r)}(x/r)\quad\text{in
}\,L^{p}({\mathbb{R}}^{d}_{+},dx).$

In particular, we have

$\displaystyle\lVert\overline{P}_{t}\rVert_{1\to\infty}=t^{-d/\alpha}\lVert\overline{P}_{1}\rVert_{1\to\infty}\quad\text{for
all }\,t>0.$ (2.1)

Proof. Using (A4), changes of the variables and the correspondence between
Dirichlet forms and semigroups, we see that for any nice function $f$,

$\displaystyle\lim_{t\downarrow
0}\frac{1}{t}\int_{{\mathbb{R}}^{d}_{+}}\big{(}f^{(r)}(z/r)-P_{r^{-\alpha}t}^{\kappa}f^{(r)}(z/r)\big{)}f^{(r)}(z/r)dz$
$\displaystyle=r^{d-\alpha}\lim_{t\downarrow
0}\frac{1}{r^{-\alpha}t}\int_{{\mathbb{R}}^{d}_{+}}\big{(}f^{(r)}(x)-P_{r^{-\alpha}t}^{\kappa}f^{(r)}(x)\big{)}f^{(r)}(x)dx$
$\displaystyle=\frac{r^{d-\alpha}}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(f^{(r)}(x)-f^{(r)}(y))^{2}\frac{{\cal
B}(x,y)}{|x-y|^{d+\alpha}}dxdy+\kappa
r^{d-\alpha}\int_{{\mathbb{R}}^{d}_{+}}f^{(r)}(x)^{2}x_{d}^{-\alpha}dx$
$\displaystyle=\frac{r^{2d}}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(f(rx)-f(ry))^{2}\frac{{\cal
B}(rx,ry)}{|rx-ry|^{d+\alpha}}dxdy+\kappa
r^{d}\int_{{\mathbb{R}}^{d}_{+}}f(rx)^{2}(rx_{d})^{-\alpha}dx$
$\displaystyle=\frac{1}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(f(x)-f(y))^{2}\frac{{\cal
B}(x,y)}{|x-y|^{d+\alpha}}dxdy+\int_{{\mathbb{R}}^{d}_{+}}f(x)^{2}\kappa
x_{d}^{-\alpha}dx={\cal E}^{\kappa}(f,f).$

Thus, by the uniqueness of the corresponding semigroup, we deduce the scaling
property of $(P^{\kappa}_{t})_{t\geq 0}$. By a similar proof, we also get the
scaling property of $(\overline{P}_{t})_{t\geq 0}$.

Now, we have that, for every $t>0$,

$\displaystyle\lVert\overline{P}_{t}\rVert_{1\to\infty}=t^{-d/\alpha}\sup\left\\{\lVert\overline{P}_{t}f\rVert_{L^{\infty}({\mathbb{R}}^{d}_{+},dx)}:\lVert
f\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq t^{d/\alpha}\right\\}$
$\displaystyle=t^{-d/\alpha}\sup\left\\{\lVert\overline{P}_{1}f^{(t^{1/\alpha})}\rVert_{L^{\infty}({\mathbb{R}}^{d}_{+},dx)}:\lVert
f\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq
1\right\\}=t^{-d/\alpha}\lVert\overline{P}_{1}\rVert_{1\to\infty},$

which proves (2.1). The proof is complete. $\Box$

Let $d\geq 2$. For $f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and
$\widetilde{z}\in{\mathbb{R}}^{d-1}$, define
$f_{z}(x)=f(x+(\widetilde{z},0))$. From (A4), we also get the following
horizontal translation invariance property of the semigroups
$(\overline{P}_{t})_{t\geq 0}$ and $(P_{t}^{\kappa})_{t\geq 0}$.

######  Lemma 2.2.

Let $d\geq 2$, $p\in[1,\infty]$ and $\kappa\geq 0$. For any $f\in
L^{p}({\mathbb{R}}^{d}_{+},dx)$, $t>0$ and
$\widetilde{z}\in{\mathbb{R}}^{d-1}$, we have

$\overline{P}_{t}f(x)=\overline{P}_{t}f_{z}(x-(\widetilde{z},0))\quad\text{and}\quad
P^{\kappa}_{t}f(x)=P^{\kappa}_{t}f_{z}(x-(\widetilde{z},0))\quad\text{in
}\,L^{p}({\mathbb{R}}^{d}_{+},dx).$

A consequence of the next lemma is that, in case when $\alpha\leq 1$, the
process $\overline{Y}$ started from ${\mathbb{R}}^{d}_{+}$ will never hit
$\partial{\mathbb{R}}^{d}_{+}$ and is equal to $Y^{0}$.

######  Lemma 2.3.

If $\alpha\leq 1$, then ${\cal F}^{0}={\overline{\cal F}}$.

Proof. Define

$\displaystyle\widetilde{\cal C}(u,v)$
$\displaystyle:=\frac{1}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+\alpha}}dydx,$
$\displaystyle{\cal D}(\widetilde{\cal C})$ $\displaystyle:=\text{closure of
$C_{c}^{\infty}(\overline{\mathbb{R}}^{d}_{+})$ in
$L^{2}({\mathbb{R}}^{d}_{+},dx)$ under $\widetilde{\cal
C}+(\cdot,\cdot)_{L^{2}({\mathbb{R}}^{d}_{+},dx)}$}.$

Then $(\widetilde{\cal C},{\cal D}(\widetilde{\cal C}))$ is a regular
Dirichlet form associated with the reflected $\alpha$-stable process in
$\overline{\mathbb{R}}^{d}_{+}$ in the sense of [6]. By (A3)(I), ${\cal
E}^{0}(u,u)\leq C_{2}\widetilde{\cal C}(u,u)$ for all $u\in
C_{c}^{\infty}(\overline{\mathbb{R}}^{d}_{+})$ and hence ${\cal
D}(\widetilde{\cal C})\subset{\overline{\cal F}}$. By [6, Theorem 2.5(i) and
Remark 2.2(1)], since $\alpha\leq 1$,
$\overline{\mathbb{R}}^{d}_{+}\setminus{\mathbb{R}}^{d}_{+}$ is
$(\widetilde{\cal C},{\cal D}(\widetilde{\cal C}))$-polar and hence is also
$({\cal E}^{0},{\overline{\cal F}})$-polar. Therefore, when starting from
${\mathbb{R}}^{d}_{+}$, $\overline{Y}$ will never exit ${\mathbb{R}}^{d}_{+}$.
Hence $\overline{Y}$ and $Y^{0}$ are the same when they start from
$x\in{\mathbb{R}}^{d}_{+}$ and thus ${\cal F}^{0}={\overline{\cal F}}$. $\Box$

In order to prove the Nash type inequality, we first consider a Brownian
motion on ${\mathbb{R}}^{d}_{+}$ killed with a critical potential and a
subordinate process obtained by time changing this killed Brownian motion with
an independent $\alpha/2$-stable subordinator. Then using results from [21],
the Hardy inequality in [32, Proposition 3.2] and comparing the Dirichlet form
corresponding to the subordinate process with $({\cal E}^{0},{\overline{\cal
F}})$, we deduce the desired result.

For any $\gamma\geq 0$, denote by $I_{\gamma}$ the modified Bessel function of
the first kind which is defined by

$I_{\gamma}(r)=\sum_{m=0}^{\infty}\frac{1}{m!\,\Gamma(\gamma+1+m)}\left(\frac{r}{2}\right)^{2m+\gamma},$

where $\Gamma(r):=\int_{0}^{\infty}u^{r-1}e^{-u}du$ is the Gamma function. It
is known that (see, e.g. [1, (9.6.7) and (9.7.1)])

$I_{\gamma}(r)\asymp(1\wedge r)^{\gamma+1/2}r^{-1/2}e^{r},\quad r>0.$ (2.2)

Define for $t>0$ and
$x=(x_{1},\dots,x_{d}),y=(y_{1},\dots,y_{d})\in{\mathbb{R}}^{d}_{+}$,

$q^{\gamma}(t,x,y)=\frac{\sqrt{x_{d}y_{d}}}{2t}I_{\gamma}\left(\frac{x_{d}y_{d}}{2t}\right)\exp\left(-\frac{x_{d}^{2}+y_{d}^{2}}{4t}\right)\prod_{i=1}^{d-1}\left[\frac{1}{\sqrt{4\pi
t}}\exp\left(-\frac{|x_{i}-y_{i}|^{2}}{4t}\right)\right].$

Note that by (2.2),

$q^{\gamma}(t,x,y)\asymp\left(1\wedge\frac{x_{d}y_{d}}{t}\right)^{\gamma+1/2}t^{-d/2}\exp\left(-\frac{|x-y|^{2}}{4t}\right).$
(2.3)

By [33, Lemma 4.1 and Theorem 4.9], $q^{\gamma}(t,x,y)$ is the transition
density of the Feller process (killed Brownian motion with critical potential)
$W^{\gamma}=(W^{\gamma}_{t})_{t\geq 0}$ on ${\mathbb{R}}^{d}_{+}$ associated
with the following regular Dirichlet form $({\cal Q}^{\gamma},{\cal D}({\cal
Q}^{\gamma}))$:

$\displaystyle{\cal Q}^{\gamma}(u,v)$
$\displaystyle:=\int_{{\mathbb{R}}^{d}_{+}}\nabla u(x)\cdot\nabla
v(x)dx+\Big{(}\gamma^{2}-\frac{1}{4}\Big{)}\int_{{\mathbb{R}}^{d}_{+}}u(x)v(x)x_{d}^{-2}dx,$
$\displaystyle{\cal D}({\cal Q}^{\gamma})$ $\displaystyle:=\text{closure of
$C_{c}^{\infty}({\mathbb{R}}^{d}_{+})$ in $L^{2}({\mathbb{R}}^{d}_{+},dx)$
under ${\cal Q}^{\gamma}_{1}={\cal
Q}^{\gamma}+(\cdot,\cdot)_{L^{2}({\mathbb{R}}^{d}_{+},dx)}$}.$

Let $S=(S_{t})_{t\geq 0}$ be an $\alpha/2$-stable subordinator independent of
$W^{\gamma}$, and let $X^{\gamma}=(X^{\gamma}_{t})_{t\geq 0}$ be the
subordinate process $X^{\gamma}_{t}:=W^{\gamma}_{S_{t}}$. Then $X^{\gamma}$ is
a Hunt process with no diffusion part. The transition density
$p^{\gamma}(t,x,y)$ of $X^{\gamma}_{t}$ exists and is given by

$p^{\gamma}(t,x,y)=\int_{0}^{\infty}q^{\gamma}(s,x,y)\frac{d}{ds}{\mathbb{P}}(S_{t}\leq
s).$

Also, the jump kernel $J^{\gamma}(dx,dy)$ and the killing measure
$\kappa^{\gamma}(dx)$ of $X^{\gamma}$ have densities $J^{\gamma}(x,y)$ and
$\kappa^{\gamma}(x)$ that are given by the following formulas (see, for
instance, [34, (2.8)–(2.9)]):

$\displaystyle J^{\gamma}(x,y)$
$\displaystyle=\int_{0}^{\infty}q^{\gamma}(t,x,y)\,\nu_{\alpha/2}(t)dt,\qquad\kappa^{\gamma}(x)=\int_{0}^{\infty}\bigg{(}1-\int_{{\mathbb{R}}^{d}_{+}}q^{\gamma}(t,x,y)dy\bigg{)}\nu_{\alpha/2}(t)dt,$

where $\nu_{\alpha/2}(t)=\frac{\alpha/2}{\Gamma(1-\alpha/2)}t^{-1-\alpha/2}$
is the Lévy density of the $\alpha/2$-stable subordinator $S$.

######  Lemma 2.4.

(i) There exists a constant $c_{\gamma,\alpha}>0$ such that
$\kappa^{\gamma}(x)=c_{\gamma,\alpha}x_{d}^{-\alpha}$ for every
$x\in{\mathbb{R}}^{d}_{+}$.

(ii) It holds that for any $x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle J^{\gamma}(x,y)$
$\displaystyle\asymp\left(1\wedge\frac{x_{d}}{|x-y|}\right)^{\gamma+1/2}\left(1\wedge\frac{y_{d}}{|x-y|}\right)^{\gamma+1/2}\frac{1}{|x-y|^{d+\alpha}}=\frac{B_{\gamma+1/2,\gamma+1/2,0,0}(x,y)}{|x-y|^{d+\alpha}}.$

(iii) There exists a constant $C>0$ such that

$p^{\gamma}(t,x,y)\leq
Ct^{-d/\alpha},\quad\,t>0,\;x,y\in{\mathbb{R}}^{d}_{+}.$

Proof. (i) Observe that the process $W^{\gamma}$ satisfies the scaling
property and horizontal translation invariance, namely, for any $t>0$,
$x,y\in{\mathbb{R}}^{d}_{+}$, $\lambda>0$ and
$\widetilde{z}\in{\mathbb{R}}^{d-1}$,

$q^{\gamma}(\lambda^{2}t,\lambda x,\lambda
y)=\lambda^{-d}q^{\gamma}(t,x,y)\quad\text{ and }\quad
q^{\gamma}(t,x+(\widetilde{z},0),y+(\widetilde{z},0))=q^{\gamma}(t,x,y).$

Therefore, for every $x\in{\mathbb{R}}^{d}_{+}$, using the change of variables
$z=(y-(\widetilde{x},0))/x_{d}$ in the second equality below and
$s=x_{d}^{-2}t$ in the third, we get that

$\displaystyle\kappa^{\gamma}(x)$
$\displaystyle=\frac{\alpha/2}{\Gamma(1-\alpha/2)}\int_{0}^{\infty}\bigg{(}1-\int_{{\mathbb{R}}^{d}_{+}}q^{\gamma}(t,(\widetilde{0},x_{d}),(\widetilde{y}-\widetilde{x},y_{d}))dy\bigg{)}t^{-\alpha/2-1}dt$
$\displaystyle=\frac{\alpha/2}{\Gamma(1-\alpha/2)}\int_{0}^{\infty}\bigg{(}1-x_{d}^{d}\int_{{\mathbb{R}}^{d}_{+}}q^{\gamma}(t,(\widetilde{0},x_{d}),x_{d}z)dz\bigg{)}t^{-\alpha/2-1}dt$
$\displaystyle=\frac{\alpha/2}{\Gamma(1-\alpha/2)}\int_{0}^{\infty}\bigg{(}1-\int_{{\mathbb{R}}^{d}_{+}}q^{\gamma}(x_{d}^{-2}t,(\widetilde{0},1),z)dz\bigg{)}t^{-\alpha/2-1}dt$
$\displaystyle=\frac{\alpha/2}{\Gamma(1-\alpha/2)}x_{d}^{-\alpha}\int_{0}^{\infty}\bigg{(}1-\int_{{\mathbb{R}}^{d}_{+}}q^{\gamma}(s,(\widetilde{0},1),z)dz\bigg{)}s^{-\alpha/2-1}ds=\kappa^{\gamma}((\widetilde{0},1))x_{d}^{-\alpha}.$

(ii) In view of (2.3), the condition $\mathbf{HK_{U}^{h}}$ in [21] holds with
$C_{0}=0$, $\Phi(r)=r^{2}$ and
$h(t,x,y)=(1\wedge(x_{d}y_{d}/t))^{\gamma+1/2}$. Note that the tail of the
Lévy measure of $S$ is given by
$\int_{r}^{\infty}\nu_{\alpha/2}(u)du=r^{-\alpha/2}/\Gamma(1-\alpha/2)$. Thus,
the condition (Poly-$\infty$) in [21] also holds and we obtain from [21,
Theorem 4.1] that for $x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle J^{\gamma}(x,y)$
$\displaystyle\asymp\left(1\wedge\frac{x_{d}y_{d}}{|x-y|^{2}}\right)^{\gamma+1/2}\\!\frac{1}{|x-y|^{d+\alpha}}\asymp\left(1\wedge\frac{x_{d}}{|x-y|}\right)^{\gamma+1/2}\\!\left(1\wedge\frac{y_{d}}{|x-y|}\right)^{\gamma+1/2}\\!\frac{1}{|x-y|^{d+\alpha}}.$

In the second comparison, we used the fact that
$(1\wedge\frac{x_{d}}{|x-y|})(1\wedge\frac{y_{d}}{|x-y|})\leq(1\wedge\frac{x_{d}y_{d}}{|x-y|^{2}})\leq
2(1\wedge\frac{x_{d}}{|x-y|})(1\wedge\frac{y_{d}}{|x-y|})$ for all
$x,y\in{\mathbb{R}}^{d}_{+}$, which can be proved elementarily by using
$y_{d}\leq x_{d}+|x-y|$.

(iii) Since the conditions (Poly-$\infty$) and $\mathbf{HK_{U}^{h}}$ in [21]
hold, the result follows from [21, Proposition 4.5(ii)]. $\Box$

Denote by $({\cal C}^{\gamma},{\cal D}({\cal C}^{\gamma}))$ the regular
Dirichlet form associated with the subordinate process $X^{\gamma}$. Then
since $X^{\gamma}$ has no diffusion part, we get from Lemma 2.4(i) that

${\cal
C}^{\gamma}(u,v)=\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))(v(x)-v(y))J^{\gamma}(x,y)dydx+c_{\gamma,\alpha}\int_{{\mathbb{R}}^{d}_{+}}u(x)v(x)x_{d}^{-\alpha}dx.$

Also, we have $C_{c}^{\infty}({\mathbb{R}}^{d}_{+})\subset{\cal D}({\cal
C}^{\gamma})$ since ${\cal D}({\cal Q}^{\gamma})\subset{\cal D}({\cal
C}^{\gamma})$, see [34].

######  Lemma 2.5.

There exists a constant $C>0$ such that

$\lVert u\rVert_{L^{2}({\mathbb{R}}^{d}_{+},dx)}^{2(1+\alpha/d)}\leq C{\cal
C}^{\gamma}(u,u)\quad\text{for every }\;u\in
C_{c}^{\infty}({\mathbb{R}}^{d}_{+})\text{ with }\lVert
u\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq 1.$

Proof. By [7, Theorem 2.1] (see also [11, Theorem 3.4] and [22, Theorem
II.5]), the result follows from Lemma 2.4(iii). $\Box$

######  Proposition 2.6.

There exists a constant $C>0$ such that

$\lVert u\rVert_{L^{2}({\mathbb{R}}^{d}_{+},dx)}^{2(1+\alpha/d)}\leq C{\cal
E}^{0}(u,u)\quad\text{for every }\;u\in{\overline{\cal F}}\text{ with }\lVert
u\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq 1.$ (2.4)

Proof. We first assume that $\alpha<1$. Let $\gamma=\beta_{1}\vee\beta_{2}$.
Using Lemmas 2.5 and 2.4(i)–(ii), the Hardy inequality in [32, Proposition
3.2] and (A3)(I), we get that for any $u\in
C_{c}^{\infty}({\mathbb{R}}^{d}_{+})$ with $\lVert
u\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq 1$,

$\displaystyle\lVert u\rVert_{L^{2}({\mathbb{R}}^{d}_{+},dx)}^{2(1+\alpha/d)}$
$\displaystyle\leq
c_{1}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))^{2}\frac{B_{\gamma+1/2,\gamma+1/2,0,0}(x,y)}{|x-y|^{d+\alpha}}dydx+c_{\gamma,\alpha}\int_{{\mathbb{R}}^{d}_{+}}u(x)^{2}x_{d}^{-\alpha}dx$
$\displaystyle\leq
c_{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))^{2}\frac{B_{\gamma+1/2,\gamma+1/2,0,0}(x,y)}{|x-y|^{d+\alpha}}dydx\leq
c_{3}{\cal E}^{0}(u,u),$

where $B_{\gamma+1/2,\gamma+1/2,0,0}$ is defined in (1). By Lemma 2.3,
${\overline{\cal F}}$ is the closure of $C_{c}^{\infty}({\mathbb{R}}^{d}_{+})$
under ${\cal E}^{0}_{1}$. Therefore, we conclude that (2.4) is true when
$\alpha<1$.

Now, we assume that $\alpha\geq 1$. Since (2.4) is valid when $\alpha<1$ and
$C_{c}^{\infty}(\overline{\mathbb{R}}^{d}_{+})\subset{\overline{\cal F}}$, we
get that for every $u\in C_{c}^{\infty}(\overline{\mathbb{R}}^{d}_{+})$ with
$\lVert u\rVert_{L^{1}({\mathbb{R}}^{d}_{+},dx)}\leq 1$,

$\displaystyle{\cal E}^{0}(u,u)$
$\displaystyle\geq\frac{1}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))^{2}\frac{{\cal
B}(x,y)}{|x-y|^{d+1/2}}{\bf 1}_{\\{|x-y|\leq 1\\}}dydx$
$\displaystyle=\frac{1}{2}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}(u(x)-u(y))^{2}\frac{{\cal
B}(x,y)}{|x-y|^{d+1/2}}(1-{\bf 1}_{\\{|x-y|>1\\}})dydx$ $\displaystyle\geq
c_{4}\lVert
u\rVert_{L^{2}({\mathbb{R}}^{d}_{+},dx)}^{2(1+1/(2d))}-2C_{2}\lVert
u\rVert^{2}_{L^{2}({\mathbb{R}}^{d}_{+},dx)}\sup_{x\in{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+},|x-y|>1}\frac{dy}{|x-y|^{d+1/2}}$
$\displaystyle\geq c_{4}\lVert
u\rVert_{L^{2}({\mathbb{R}}^{d}_{+},dx)}^{2(1+1/(2d))}-c_{5}\lVert
u\rVert^{2}_{L^{2}({\mathbb{R}}^{d}_{+},dx)}.$

Since ${\overline{\cal F}}$ is the closure of
$C_{c}^{\infty}(\overline{\mathbb{R}}^{d}_{+})$ under ${\cal E}^{0}_{1}$, [7,
Theorem 2.1] yields that $\lVert\overline{P}_{t}\rVert_{1\to\infty}\leq
c_{6}t^{-2d}e^{c_{5}t}$ for all $t>0$. By (2.1), it follows that
$\lVert\overline{P}_{t}\rVert_{1\to\infty}=t^{-d/\alpha}\lVert\overline{P}_{1}\rVert_{1\to\infty}\leq
c_{6}e^{c_{5}}t^{-d/\alpha}$ for all $t>0$. Using [7, Theorem 2.1] again, we
conclude that (2.4) holds for $\alpha\geq 1$ and finish the proof. $\Box$

As a consequence of the Nash-type inequality (2.4), we get the existence and a
priori upper bounds of the heat kernels $\overline{p}(t,x,y)$ and
$p^{\kappa}(t,x,y)$ of $\overline{Y}$ and $Y^{\kappa}$ respectively.

######  Proposition 2.7.

Let $\kappa\geq 0$. The processes $\overline{Y}$ and $Y^{\kappa}$ have heat
kernels $\overline{p}(t,x,y)$ and $p^{\kappa}(t,x,y)$ defined on
$(0,\infty)\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal
N})\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal N})$ and
$(0,\infty)\times({\mathbb{R}}^{d}_{+}\setminus{\cal
N})\times({\mathbb{R}}^{d}_{+}\setminus{\cal N})$ respectively, where ${\cal
N}\subset\overline{\mathbb{R}}^{d}_{+}$ is a properly exceptional set for
$\overline{Y}$. Moreover, there exists a constant $C>0$ such that

$p^{\kappa}(t,x,y)\leq\overline{p}(t,x,y),\quad
x,y\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$ (2.5)

and

$\overline{p}(t,x,y)\leq
C\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),\quad
t>0,\;x,y\in\overline{\mathbb{R}}^{d}_{+}\setminus{\cal N}.$ (2.6)

Proof. By our Nash-type inequality (2.4) and [3], $\overline{Y}$ has a heat
kernel $\overline{p}(t,x,y)$ on
$(0,\infty)\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal
N})\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal N})$ for a properly
exceptional set ${\cal N}$ and

$\overline{p}(t,x,y)\leq ct^{-d/\alpha},\quad
t>0,\;x,y\in\overline{\mathbb{R}}^{d}_{+}\setminus{\cal N}.$ (2.7)

Since ${\cal B}(x,y)$ is bounded above, using (2.7), we can follow the
arguments given in [11, Example 5.5] line by line and conclude that (2.6)
holds. According to [32] (see the discussion before Lemma 2.1 there),
$Y^{\kappa}$ can be realized as a subprocess of $\overline{Y}$. Thus,
$Y^{\kappa}$ has a heat kernel $p^{\kappa}(t,x,y)$ defined on
$(0,\infty)\times({\mathbb{R}}^{d}_{+}\setminus{\cal
N})\times({\mathbb{R}}^{d}_{+}\setminus{\cal N})$ and we also obtain (2.5).
$\Box$

For notational convenience, we extend the domain of $p^{\kappa}(t,x,y)$ to
$(0,\infty)\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal
N})\times(\overline{\mathbb{R}}^{d}_{+}\setminus{\cal N})$ by letting
$p^{\kappa}(t,x,y)=0$ if $x\in\partial{\mathbb{R}}^{d}_{+}\setminus{\cal N}$
or $y\in\partial{\mathbb{R}}^{d}_{+}\setminus{\cal N}$.

##  3\. Parabolic Hölder regularity and consequences

For $\kappa\geq 0$ and an open set $D\subset{\overline{\mathbb{R}}}^{d}_{+}$
relative to the topology on ${\overline{\mathbb{R}}}^{d}_{+}$, we denote by
$\overline{Y}^{D}$ and $\overline{P}^{D}_{t}$ the part of the process
$\overline{Y}$ killed upon exiting $D$ and its semigroup, and by
$Y^{\kappa,D}$ and $P_{t}^{\kappa,D}$ the part of the process $Y^{\kappa}$
killed upon exiting $D\cap{\mathbb{R}}^{d}_{+}$ and its semigroup,
respectively. The Dirichlet forms of $\overline{Y}^{D}$ and $Y^{\kappa,D}$ are
$({\cal E}^{0},\overline{{\cal F}}^{D})$ and $({\cal E}^{\kappa},{\cal
F}^{\kappa,D})$, where $\overline{{\cal F}}^{D}=\\{u\in\overline{{\cal
F}}:u=0\mbox{ q.\/e. on }{\overline{\mathbb{R}}}^{d}_{+}\setminus D\\}$ and
${\cal F}^{\kappa,D}=\\{u\in{\cal F}^{\kappa}:u=0\mbox{ q.\/e. on
}{\mathbb{R}}^{d}_{+}\setminus D\\}$ respectively. For $u,v\in{\cal
F}^{\kappa,D}$,

${\cal
E}^{\kappa}(u,v)=\frac{1}{2}\int_{D}\int_{D}((u(x)-u(y))(v(x)-v(y))J(x,y)dydx+\int_{D}u(x)v(x)\kappa_{D}(x)dx,$
(3.1)

where

$\kappa_{D}(x)=\int_{{\mathbb{R}}^{d}_{+}\setminus D}J(x,y)dy+\kappa
x_{d}^{-\alpha}.$ (3.2)

Let $\overline{\tau}_{D}=\inf\\{t>0:\overline{Y}_{t}\notin D\\}$ and
$\tau^{\kappa}_{D}=\inf\\{t>0:Y^{\kappa}_{t}\notin
D\cap{\mathbb{R}}^{d}_{+}\\}$. For $x,y\notin\mathcal{N}$, let

$\displaystyle\overline{p}^{D}(t,x,y)$
$\displaystyle=\overline{p}(t,x,y)-{\mathbb{E}}_{x}\big{[}\overline{p}(t-\overline{\tau}_{D},\overline{Y}_{\overline{\tau}_{D}},y);\,\overline{\tau}_{D}<t\big{]},$
$\displaystyle p^{\kappa,D}(t,x,y)$
$\displaystyle=p^{\kappa}(t,x,y)-{\mathbb{E}}_{x}\big{[}p^{\kappa}(t-\tau_{D}^{\kappa},Y^{\kappa}_{\tau^{\kappa}_{D}},y);\,\tau_{D}^{\kappa}<t\big{]}.$
(3.3)

By the strong Markov property, $\overline{p}^{D}(t,x,y)$ and
$p^{\kappa,D}(t,x,y)$ are the transition densities of $\overline{Y}^{D}$ and
$Y^{\kappa,D}$ respectively.

In case when $D$ is a relatively compact open subset of
${\mathbb{R}}^{d}_{+}$, one can show that the process $Y^{\kappa,D}$ can start
from every point in $D$ and provide an interior lower bound for its transition
density. We accomplish this by identifying the semigroup $P^{\kappa,D}_{t}$
with the Feynman-Kac semigroup of the part process on $D$ of an auxiliary
process. This idea has already been used in [30, Subsection 3.1]. For the
benefit of the reader, we repeat some of the details here. Note that, unlike
[30, Subsection 3.1], (A3)(II) is not assumed.

Recall that we denote by $m_{d}$ the Lebesgue measure on ${\mathbb{R}}^{d}$.

######  Proposition 3.1.

If $D$ is a relatively compact open subset of ${\mathbb{R}}^{d}_{+}$, then
$Y^{\kappa,D}$ has a transition density $p^{\kappa,D}(t,x,y)$ defined for any
$(t,x,y)\in(0,\infty)\times D\times D$. Furthermore, for any $T>0$ and
$b\in(0,1]$, there exists a constant $C=C(b,T,D)>0$ such that

$p^{\kappa,D}(t,x,y)\geq
C\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\quad
t\in(0,T),\;x,y\in D\textrm{ such that
}\delta_{D}(x)\wedge\delta_{D}(y)>bt^{1/\alpha}.$

Proof. For $\gamma>0$ let
$J_{\gamma}:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to[0,\infty]$ be defined by
$J_{\gamma}(x,y)=J(x,y)$ if $x,y\in D$, and
$J_{\gamma}(x,y)=\gamma|x-y|^{-\alpha-d}$ otherwise. It follows from (A3)(I)
and the relative compactness of $D$ that
$J_{\gamma}(x,y)\asymp|x-y|^{-\alpha-d}$. Hence by [16, Theorem 1.2], there
exists a Feller and strongly Feller process $Z$ (that can start from every
point in ${\mathbb{R}}^{d})$ with a continuous transition density
$\widetilde{q}(t,x,y)$ on
$(0,\infty)\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ satisfying

$c_{1}^{-1}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\leq\widetilde{q}(t,x,y)\leq
c_{1}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),\quad
t>0,\;x,y\in{\mathbb{R}}^{d},$ (3.4)

for some constant $c_{1}\geq 1$.

Denote the part of the process $Z$ killed upon exiting $D$ by $Z^{D}$. The
Dirichlet form of $Z^{D}$ is $({\cal C},\mathcal{D}_{D}({\cal C}))$, where for
$u,v\in{\cal D}_{D}({\cal C})$,

$\displaystyle{\cal C}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{D}\int_{D}(u(x)-u(y))(v(x)-v(y))J_{\gamma}(x,y)\,dy\,dx+\int_{D}u(x)v(x)\kappa^{Z}_{D}(x)\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{D}\int_{D}(u(x)-u(y))(v(x)-v(y))J(x,y)\,dy\,dx+\int_{D}u(x)v(x)\kappa^{Z}_{D}(x)\,dx$

with

$\kappa^{Z}_{D}(x)=\int_{{\mathbb{R}}^{d}\setminus
D}J_{\gamma}(x,y)\,dy=\gamma\int_{{\mathbb{R}}^{d}\setminus
D}|x-y|^{-d-\alpha}\,dy\,,\quad x\in D\,,$ (3.5)

and $\mathcal{D}_{D}({\cal C})=\\{u\in\mathcal{D}({\cal C}):\,u=0\textrm{
q.e.~{}on }{\mathbb{R}}^{d}\setminus D\\}$.

Let $\delta_{D}=\mathrm{dist}(D,\partial{\mathbb{R}}^{d}_{+})$ and let $V$ be
the $\delta_{D}/2$-neighborhood of $D$, that is
$V:=\\{x\in{\mathbb{R}}^{d}_{+}:\,\mathrm{dist}(x,D)<\delta_{D}/2\\}$. Then

$\kappa_{D}(x)=\int_{{\mathbb{R}}^{d}_{+}\setminus
V}J(x,y)\,dy+\int_{V\setminus D}J(x,y)\,dy+\kappa x_{d}^{-\alpha}\,.$

It follows from (A3)(I) and the relative compactness of $D$ that
$c_{2}|x-y|^{-d-\alpha}\leq J(x,y)\leq C_{2}|x-y|^{-d-\alpha}$ for all $x,y\in
V$ with $c_{2}:=c_{2}(D)>0$. It is easy to see that $\sup_{x\in
D}\int_{{\mathbb{R}}^{d}_{+}\setminus V}J(x,y)\,dy=:c_{3}\ <\infty$. Therefore

$c_{2}\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\leq\kappa_{D}(x)\leq
c_{3}+C_{2}\int_{V\setminus D}|x-y|^{-d-\alpha}dy+c_{4}\,,\quad x\in U\,,$

where $c_{4}:=\kappa\sup_{x\in D}x_{d}^{-\alpha}$. Since

$\inf_{x\in D}\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\geq m_{d}(V\setminus
D)\,\mathrm{diam}(V)^{-d-\alpha}=:c_{5}>0\,,$

we conclude that

$c_{2}\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\leq\kappa_{D}(x)\leq
c_{6}\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\,,\quad x\in D\,.$

Further, since

$\kappa_{D}^{Z}(x)=\gamma\bigg{(}\int_{{\mathbb{R}}^{d}\setminus
V}|x-y|^{-d-\alpha}\,dy+\int_{V\setminus
D}|x-y|^{-d-\alpha}\,dy\bigg{)}\,,\quad x\in D\,$

and $\sup_{x\in D}\int_{{\mathbb{R}}^{d}\setminus
V}|x-y|^{-d-\alpha}\,dy=:c_{7}<\infty$, we see that there is a constant
$c_{8}>0$ such

$\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\leq\gamma^{-1}\kappa_{D}^{Z}(x)\leq
c_{8}\int_{V\setminus D}|x-y|^{-d-\alpha}\,dy\,,\quad x\in D,.$

It follows that $c_{6}^{-1}\kappa_{D}(x)\leq\gamma^{-1}\kappa_{D}^{Z}(x)\leq
c_{8}c_{2}^{-1}\kappa_{D}(x)$ for all $x\in D$ with positive constants
$c_{2},c_{6},c_{8}$ not depending on $\gamma$. Now we choose $\gamma>0$ so
small that $\gamma c_{8}c_{2}^{-1}\leq 1$. With this choice we get that
$\kappa_{D}^{Z}(x)\leq\kappa_{D}(x)$ for all $x\in D$. In particular, with
$c_{9}:=\gamma c_{6}^{-1}$ we see that

$c_{9}\kappa_{D}(x)\leq\kappa_{D}^{Z}(x)\leq\kappa_{D}(x)\,,\qquad x\in D\,.$
(3.6)

It follows that for $u\in C_{c}^{\infty}(D)$,

$\displaystyle{\cal E}^{\kappa,D}_{1}(u,u)$ $\displaystyle=$
$\displaystyle{\cal E}^{\kappa,D}(u,u)+\int_{U}u(x)^{2}\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{D}\int_{D}(u(x)-u(y))^{2}J(x,y)\,dy\,dx+\int_{D}u(x)^{2}\kappa_{D}(x)\,dx+\int_{U}u(x)^{2}dx$
$\displaystyle\asymp$
$\displaystyle\frac{1}{2}\int_{D}\int_{D}(u(x)-u(y))^{2}J_{\gamma}(x,y)\,dy\,dx+\int_{D}u(x)^{2}\kappa_{D}^{Z}(x)\,dx+\int_{D}u(x)^{2}dx$
$\displaystyle=$ $\displaystyle{\cal C}(u,u)+\int_{D}u(x)^{2}dx={\cal
C}_{1}(u,u)\,.$

Since $C_{c}^{\infty}(D)$ is a core of both $({\cal E}^{\kappa},{\cal
F}^{\kappa,D})$ and $({\cal C},\mathcal{D}_{D}({\cal C}))$, we conclude that
${\cal F}^{\kappa,D}=\mathcal{D}_{D}({\cal C})$.

Define $\widetilde{\kappa}:D\to{\mathbb{R}}$ by
$\widetilde{\kappa}(x):=\kappa_{D}(x)-\kappa_{D}^{Z}(x)$, $x\in D.$ By the
choice of $\gamma$ we have that $\widetilde{\kappa}\geq 0$. On the other hand,
it follows from (3.5) and (3.6) that there is a constant $c_{10}>0$ such that

$\widetilde{\kappa}(x)\leq\kappa_{D}(x)\leq
c_{10}\delta_{D}(x)^{-\alpha}\,,\quad x\in D.$ (3.7)

Let $\mu(dx)=\widetilde{\kappa}(x)\,dx$ be a measure on $D$. Using (3.4) and
(3.7) one can check that $\mu\in{\bf K}_{1}(D)$, where the class ${\bf
K}_{1}(D)$ is defined in [20, Definition 2.12].

For any Borel function $f:D\to[0,\infty)$ let

$T^{D,\widetilde{\kappa}}_{t}f(x)={\mathbb{E}}_{x}\left[\exp\left(-\int_{0}^{t}\widetilde{\kappa}(Z^{D}_{s})ds\right)f(Z^{D}_{t})\right],\quad
t>0,\;x\in D.$ (3.8)

be the Feynman-Kac transform of the semigroup corresponding to the killed
process $Z^{D}$. By [20, Theorem 2.15], the semigroup
$(T^{D,\widetilde{\kappa}}_{t})_{t\geq 0}$ has a transition density
$p^{Z,D}(t,x,y)$ (with respect to the Lebesgue measure) such that for every
$T>0$ and $b\in(0,1]$ there exists a constant $c_{2}=c_{2}(b,T,D)>0$ such that

$p^{Z,D}(t,x,y)\geq c_{2}\widetilde{q}(t,x,y),\quad t\in(0,T),\;x,y\in
D\textrm{ such that }\delta_{D}(x)\wedge\delta_{D}(y)>bt^{1/\alpha}.$ (3.9)

Finally, by computing the Dirichlet forms of $Y^{\kappa,D}$ and $Z^{D}$ (for
the latter use [23, Theorem 6.1.2]), we conclude that they coincide. This
implies that $P^{\kappa.D}_{t}=T^{D,\widetilde{\kappa}}_{t}$. Combining the
lower bound in (3.4) with (3.9), the proof is complete. $\Box$

Recall that ${\mathbf{e}}_{d}=(\widetilde{0},1)$. Define for $a\in(0,1]$,

$S(a)=\\{(\widetilde{z},z_{d})\in{\mathbb{R}}^{d}_{+}:|\widetilde{z}|<2/a,\,a/(2a+2)<z_{d}<(2a+2)/a\\}.$

Note that $B({\mathbf{e}}_{d},1/8)\subset S(a)$ for all $a\in(0,1]$. Moreover,
for any $a\in(0,1]$, $x\in B({\mathbf{e}}_{d},1/8)$ and
$y=(\widetilde{y},y_{d})\in{\mathbb{R}}^{d}_{+}$ with $x_{d}\wedge
y_{d}>a|x-y|$, we have
$|\widetilde{y}|\leq|\widetilde{x}|+|y-x|<1/8+x_{d}/a<2/a$, $y_{d}+y_{d}/a\geq
x_{d}-|x-y|+|x-y|\geq 7/8$ and $y_{d}\leq x_{d}+|x-y|\leq
x_{d}+x_{d}/a<(2a+2)/a$. Thus,

$\displaystyle\big{\\{}(t,(\widetilde{x},x_{d}),(\widetilde{y},y_{d}))\in{\mathbb{R}}^{1}_{+}\times
B({\mathbf{e}}_{d},1/8)\times{\mathbb{R}}^{d}_{+}:x_{d}\wedge
y_{d}>a(t^{1/\alpha}\vee|x-y|)\big{\\}}$
$\displaystyle\subset(0,2/a^{\alpha})\times S(a)\times S(a)\qquad\text{for
every}\;\,a\in(0,1].$ (3.10)

######  Proposition 3.2.

For any $a\in(0,1]$, there exists a constant $C=C(a)>0$ such that the
following estimates hold: For any $t>0$ and
$x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$, there is a measurable set
$N_{t,x}\subset{\mathbb{R}}^{d}_{+}$ of zero Lebesgue measure such that for
all $y\in{\mathbb{R}}^{d}_{+}\setminus N_{t,x}$ with $x_{d}\wedge
y_{d}>a(t^{1/\alpha}\vee|x-y|)$,

$\overline{p}(t,x,y)\geq p^{\kappa}(t,x,y)\geq
C\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$ (3.11)

Proof. By (2.5), it suffices to prove the second inequality in (3.11). Since
${\cal N}$ has zero Lebesgue measure, there exist $r>0$ and
$\widetilde{z}\in{\mathbb{R}}^{d-1}$ such that $r^{-1}(x-(\widetilde{z},0))\in
B({\mathbf{e}}_{d},1/8)\setminus{\cal N}$. By Lemmas 2.1 and 2.2, we have

$\displaystyle
p^{\kappa}(t,x,y)=r^{-d}p^{\kappa}(r^{-\alpha}t,r^{-1}(x-(\widetilde{z},0)),r^{-1}(y-(\widetilde{z},0)))\qquad\text{for
a.e.}\;\,y\in{\mathbb{R}}^{d}_{+}.$ (3.12)

Therefore, by (3), it is enough to prove that there exists a constant
$c=c(a)>0$ such that for all $0<t<2/a^{\alpha}$, $x\in
B({\mathbf{e}}_{d},1/8)\setminus\mathcal{N}$ and a.e. $y\in S(a)\setminus{\cal
N}$,

$\displaystyle p^{\,\kappa,S(a/2)}(t,x,y)\geq
c\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),$ (3.13)

which implies the second inequality in (3.11) by (3.3) and (3.12).

Note that $\delta_{S(a/2)}(y)\geq a/(a^{2}+3a+2)$ for every $y\in S(a)$. Hence
for all $0<t<2/a^{\alpha}$ and $z,y\in S(a/2)$ it holds that
$\delta_{S(a/2)}(z)\wedge\delta_{S(a/2)}(y)\geq(a^{2}/(2a^{2}+6a+4))t^{1/\alpha}$.
Now it follows from Proposition 3.1 (with $b=(a^{2}/(2a^{2}+6a+4))$) that
(3.13) holds. $\Box$

As a direct consequence of Proposition 3.2, for any $a\in(0,1]$, there exists
a constant $C>0$ such that

$\overline{p}(t,x,y)\geq p^{\kappa}(t,x,y)\geq Ct^{-d/\alpha}$ (3.14)

for all $t>0$, $x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$ and a.e.
$y\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$ with $x_{d}\wedge
y_{d}>at^{1/\alpha}$ and $|x-y|\leq t^{1/\alpha}$.

By repeating the proofs of [21, Lemmas 6.1 and 6.3], we obtain the following
two results from (3.3), Proposition 2.7 and (3.14).

######  Lemma 3.3.

There exist constants $C>0$ and $\eta\in(0,1/4)$ such that for all
$x\in{\mathbb{R}}^{d}_{+}$, $r\in(0,x_{d})$, $t\in(0,(\eta r)^{\alpha}]$ and
$z\in B(x,\eta t^{1/\alpha})\setminus{\cal N}$,

$\overline{p}^{B(x,r)}(t,z,y)\geq p^{\kappa,B(x,r)}(t,z,y)\geq
Ct^{-d/\alpha}\quad\text{for a.e. }y\in B(x,\eta t^{1/\alpha}).$

######  Lemma 3.4.

There exists a constant $C>1$ such that for all
$x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$ and $r\in(0,x_{d})$,

$C^{-1}r^{\alpha}\leq{\mathbb{E}}_{x}[\tau^{\kappa}_{B(x,r)}]\leq\sup_{z\in
B(x,r)\setminus{\cal N}}{\mathbb{E}}_{z}[\overline{\tau}_{B(x,r)}]\leq
Cr^{\alpha}.$ (3.15)

The Lévy system formula (see [23, Theorem 5.3.1] and the arguments in [15,
p.40]) state that for any non-negative Borel function $F$ on
${\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$ vanishing on the diagonal and
any stopping time $T$ for $Y^{\kappa}$, it holds that

$\displaystyle{\mathbb{E}}_{x}\sum_{s\leq
T}F(Y^{\kappa}_{s-},Y^{\kappa}_{s})={\mathbb{E}}_{x}\int^{T}_{0}\int_{{\mathbb{R}}^{d}_{+}}F(Y^{\kappa}_{s},y)J(Y^{\kappa}_{s},y)dyds,\quad
x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}.$ (3.16)

Here $Y^{\kappa}_{s-}=\lim_{t\uparrow s}Y^{\kappa}_{t}$ denotes the left limit
of the process $Y$ at time $s>0$. Similarly, for any non-negative Borel
function $F$ on
${\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$
vanishing on the diagonal and any stopping time $T$ for $\overline{Y}$, it
holds that

$\displaystyle{\mathbb{E}}_{x}\sum_{s\leq
T}F(\overline{Y}_{s-},\overline{Y}_{s})={\mathbb{E}}_{x}\int^{T}_{0}\int_{{\mathbb{R}}^{d}_{+}}F(\overline{Y}_{s},y)J(\overline{Y}_{s},y)dyds,\quad
x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}.$ (3.17)

See [31, (3.3) and (3.4)] for a simper form following from (3.16), which will
be used in this paper too.

For $x=(\widetilde{x},x_{d})\in{\overline{\mathbb{R}}}^{d}_{+}$ and $t>0$, we
define

$\displaystyle V_{x}(t)$
$\displaystyle=\\{z=(\widetilde{z},z_{d})\in{\mathbb{R}}^{d}:\,|\widetilde{z}-\widetilde{x}|<2t^{1/\alpha},\,z_{d}\in[0,2t^{1/\alpha})\\},$
$\displaystyle W_{x}(t)$
$\displaystyle=\\{z=(\widetilde{z},z_{d})\in{\mathbb{R}}^{d}:\,|\widetilde{z}-\widetilde{x}|<2t^{1/\alpha},\,z_{d}\in[x_{d}+5t^{1/\alpha},x_{d}+8t^{1/\alpha})\\}.$
(3.18)

In dimension 1, we abuse notation and use $V_{x}(t)=[0,2t^{1/\alpha})$ and
$W_{x}(t)=[x+5t^{1/\alpha},x+8t^{1/\alpha})$.

######  Lemma 3.5.

(i) There exists $C>0$ such that for all $t>0$ and
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$,

${\mathbb{P}}_{x}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\big{)}\geq C.$

(ii) There exists a constant $C>0$ such that for all $n\geq 1$, $t>0$ and
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$,

${\mathbb{P}}_{x}(\overline{\tau}_{V_{x}(t)}>nt)\leq 2e^{-Cn}.$

Proof. (i) Define $V_{x}(t,r)=\\{z\in
V_{x}(t):\delta_{V_{x}(t)}(z)>rt^{1/\alpha}\\}$ for $r>0$. For any
$r\in(0,1)$, $z\in V_{x}(t,r)$, $u\in B(z,rt^{1/\alpha}/2)$ and $w\in
W_{x}(t)$, we have

$\displaystyle rt^{1/\alpha}/2\leq u_{d}\leq 5t^{1/\alpha}\leq
w_{d}\quad\text{and}\quad|u-w|\leq|u-z|+|z-x|+|x-w|<16t^{1/\alpha}.$ (3.19)

Using the Lévy system formula in (3.17), (A3)(I), (3.19) and Lemma 3.4, we get
that for any $r\in(0,1)$ and $z\in V_{x}(t,r)\setminus{\cal N}$,

$\displaystyle{\mathbb{P}}_{z}(\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t))$
$\displaystyle\geq{\mathbb{P}}_{z}(\overline{Y}_{\overline{\tau}_{B(z,rt^{1/\alpha}/2)}}\in
W_{x}(t))={\mathbb{E}}_{z}\bigg{[}\int_{0}^{\overline{\tau}_{B(z,rt^{1/\alpha}/2)}}\int_{W_{x}(t)}J(\overline{Y}_{s},w)dwds\bigg{]}$
$\displaystyle\geq
c_{1}{\mathbb{E}}_{z}\bigg{[}\int_{0}^{\overline{\tau}_{B(z,rt^{1/\alpha}/2)}}\int_{W_{x}(t)}\left(\frac{r/2}{16}\right)^{\beta_{1}}\left(\frac{5}{16}\right)^{\beta_{2}}(16t^{1/\alpha})^{-d-\alpha}dwds\bigg{]}$
$\displaystyle\geq
c_{2}r^{\beta_{1}}t^{-1-d/\alpha}{\mathbb{E}}_{z}\big{[}\overline{\tau}_{B(z,rt^{1/\alpha}/2)}\big{]}\int_{W_{x}(t)}dw\geq
c_{3}r^{\beta_{1}+\alpha}.$ (3.20)

By Proposition 2.7 and [25, Remark 3.3], the condition (i) in [25, Theorem
3.1] holds true with $\rho(s)=s^{1/\alpha}$. Then, since $\overline{Y}$ is
conservative, by the implication (i) $\Rightarrow$ (ii) of [25, Theorem 3.1],
there exists a constant ${\varepsilon}_{0}>0$ independent of $t$ and $x$ such
that

$\displaystyle{\mathbb{P}}_{x}(\overline{\tau}_{V_{x}}(t)>{\varepsilon}_{0}t)\geq
1/2.$ (3.21)

On the other hand, for all $r>0$, we get from Proposition 2.7 that

$\displaystyle{\mathbb{P}}_{x}(\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r)}>{\varepsilon}_{0}t)\leq{\mathbb{P}}_{x}(\overline{Y}_{{\varepsilon}_{0}t}\in
V_{x}(t)\setminus V_{x}(t,r))$ $\displaystyle\leq\int_{V_{x}(t)\setminus
V_{x}(t,r)}p({\varepsilon}_{0}t,x,y)dy\leq
c_{4}t^{-d/\alpha}\int_{V_{x}(t)\setminus V_{x}(t,r)}dy\leq c_{5}r.$ (3.22)

Set $r_{0}:=1/(4c_{5}+1)$. Using the strong Markov property and (3)–(3), we
obtain

$\displaystyle{\mathbb{P}}_{x}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\big{)}\geq{\mathbb{P}}_{x}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t),\,\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r_{0})}\leq{\varepsilon}_{0}t<\overline{\tau}_{V_{x}(t)}\big{)}$
$\displaystyle\geq{\mathbb{P}}_{x}\Big{(}{\mathbb{P}}_{\overline{Y}_{\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r_{0})}}}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\big{)}\,:\,\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r_{0})}\leq{\varepsilon}_{0}t<\overline{\tau}_{V_{x}(t)}\Big{)}$
$\displaystyle\geq
c_{3}r_{0}^{\beta_{1}+\alpha}{\mathbb{P}}_{x}\left(\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r_{0})}\leq{\varepsilon}_{0}t<\overline{\tau}_{V_{x}(t)}\right)$
$\displaystyle\geq
c_{3}r_{0}^{\beta_{1}+\alpha}\left({\mathbb{P}}_{x}\left(\overline{\tau}_{V_{x}(t)}>{\varepsilon}_{0}t\right)-{\mathbb{P}}_{x}\left(\overline{\tau}_{V_{x}(t)\setminus
V_{x}(t,r_{0})}>{\varepsilon}_{0}t\right)\right)\geq
c_{3}r_{0}^{\beta_{1}+\alpha}/4.$

(ii) By Proposition 2.7, there exists a constant $k_{0}>0$ independent of $t$
and $x$ such that for any $z\in V_{x}(t)\setminus{\cal N}$,

$\displaystyle{\mathbb{P}}_{z}\big{(}\overline{\tau}_{V_{x}(t)}>k_{0}t\big{)}\leq{\mathbb{P}}_{z}\big{(}\overline{Y}_{k_{0}t}\in
V_{x}(t)\big{)}\leq\int_{V_{x}(t)}\overline{p}(k_{0}t,z,y)dy\leq
c_{1}t^{d/\alpha}(k_{0}t)^{-d/\alpha}\leq\frac{1}{2}.$ (3.23)

For $r>0$, let $\lfloor r\rfloor:=\sup\\{m\in{\mathbb{N}}:m\leq r\\}$. Now for
any $n\geq 1$, using the Markov property and (3.23), we get that

$\displaystyle{\mathbb{P}}_{x}(\overline{\tau}_{V_{x}(t)}>nt)\leq{\mathbb{P}}_{x}(\overline{\tau}_{V_{x}(t)}>\lfloor
n/k_{0}\rfloor k_{0}t)$
$\displaystyle\leq{\mathbb{P}}_{x}\big{(}{\mathbb{P}}_{X_{(\lfloor
n/k_{0}\rfloor-1)k_{0}t}}(\overline{\tau}_{V_{x}(t)}>k_{0}t):\overline{\tau}_{V_{x}(t)}>(\lfloor
n/k_{0}\rfloor-1)k_{0}t\big{)}$ $\displaystyle\leq
2^{-1}{\mathbb{P}}_{x}\big{(}\overline{\tau}_{V_{x}(t)}>(\lfloor
n/k_{0}\rfloor-1)k_{0}t\big{)}\leq\cdots\leq 2^{-\lfloor n/k_{0}\rfloor}\leq
2e^{-(\log 2)n/k_{0}}.$

$\Box$

######  Lemma 3.6.

There exist constants $M>1$ and $C>0$ such that for all $t>0$ and
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$,

$\operatorname*{ess\,inf}_{z\in W_{x}(t)}\,\overline{p}(Mt,x,z)\geq
Ct^{-d/\alpha}.$

Proof. Suppose $t>0$ and $x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal
N}$. For all $w,z\in W_{x}(t)$, we have $w_{d}\wedge z_{d}\geq 5t^{1/\alpha}$
and $|w-z|<7t^{1/\alpha}$. Therefore, by Proposition 3.2, for any $M>1$, there
is $c_{1}=c_{1}(M)>0$ independent of $t$ and $x$ such that for all $w\in
W_{x}(t)\setminus{\cal N}$, $z\in W_{x}(t)$ and
${\varepsilon}\in(0,t^{1/\alpha})$,

$\displaystyle\inf_{t\leq s\leq
Mt}\,{\mathbb{P}}_{w}\big{(}\overline{Y}_{s}\in
B(z,{\varepsilon})\big{)}=\inf_{t\leq s\leq
Mt}\int_{B(z,{\varepsilon})}\overline{p}(s,w,y)dy\geq
c_{1}t^{-d/\alpha}m_{d}(B(z,{\varepsilon})).$ (3.24)

By the strong Markov property and (3.24), for all $M>1$, $z\in W_{x}(t)$ and
${\varepsilon}\in(0,t^{1/\alpha})$,

$\displaystyle{\mathbb{P}}_{x}\left(\overline{Y}_{Mt}\in
B(z,{\varepsilon})\right)\geq{\mathbb{E}}_{x}\left[{\mathbb{P}}_{\overline{Y}_{\overline{\tau}_{V_{x}(t)}}}\big{(}\overline{Y}_{Mt-\overline{\tau}_{V_{x}(t)}}\in
B(z,{\varepsilon})\big{)}:\overline{\tau}_{V_{x}(t)}\leq(M-1)t,\,\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\right]$ $\displaystyle\geq\Big{(}\inf_{t\leq s\leq Mt}\,\inf_{w\in
W_{x}(t)\setminus{\cal N}}{\mathbb{P}}_{w}\big{(}\overline{Y}_{s}\in
B(z,{\varepsilon})\big{)}\Big{)}\,{\mathbb{P}}_{x}\left(\overline{\tau}_{V_{x}(t)}\leq(M-1)t,\,\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\right)$ $\displaystyle\geq
c_{1}t^{-d/\alpha}m_{d}(B(z,{\varepsilon}))\left({\mathbb{P}}_{x}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\big{)}-{\mathbb{P}}_{x}\big{(}\overline{\tau}_{V_{x}(t)}>(M-1)t\big{)}\right).$
(3.25)

By Lemma 3.5(i)-(ii), there are constants $c_{2},c_{3},c_{4}>0$ such that for
all $M>1$,

$\displaystyle{\mathbb{P}}_{x}\big{(}\overline{Y}_{\overline{\tau}_{V_{x}(t)}}\in
W_{x}(t)\big{)}-{\mathbb{P}}_{x}\big{(}\overline{\tau}_{V_{x}(t)}>(M-1)t\big{)}\geq
c_{2}-c_{3}e^{-c_{4}M}.$

Choosing $M=c_{4}^{-1}\log(2c_{3}/c_{2})+1$, we arrive at the result by (3)
and the Lebesgue differentiation theorem. $\Box$

######  Lemma 3.7.

There exists $C>0$ such that for all $t>0$ and
$x,y\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$ with $|x-y|\leq
t^{1/\alpha}$,

$\displaystyle\overline{p}(t,x,y)\geq Ct^{-d/\alpha}.$

Proof. Let $M>1$ be the constant in Lemma 3.6. Note that for all $(z,w)\in
W_{x}(t/M)\times W_{y}(t/M)$, we have $z_{d}\wedge w_{d}\geq
5(t/M)^{1/\alpha}$ and $|z-w|\leq|z-x|+|x-y|+|y-w|\leq 21(t/M)^{1/\alpha}$.
Therefore, by Proposition 3.2, there is $c_{1}>0$ independent of $t,x,y$ such
that

$\displaystyle\operatorname*{ess\,inf}_{z\in
W_{x}(t/(3M))}\operatorname*{ess\,inf}_{w\in
W_{y}(t/(3M))}\overline{p}(t/3,z,w)\geq c_{1}t^{-d/\alpha}.$ (3.26)

By the semigroup property, (3.26) and Lemma 3.6,

$\displaystyle\overline{p}(t,x,y)$
$\displaystyle\geq\int_{W_{x}(t/(3M))}\int_{W_{y}(t/(3M))}\overline{p}(t/3,x,z)\overline{p}(t/3,z,w)\overline{p}(t/3,w,y)dzdw$
$\displaystyle\geq\left(\operatorname*{ess\,inf}_{z\in
W_{x}(t/(3M))}\overline{p}(t/3,x,z)\right)\left(\operatorname*{ess\,inf}_{w\in
W_{y}(t/(3M))}\overline{p}(t/3,y,w)\right)$
$\displaystyle\qquad\times\left(\operatorname*{ess\,inf}_{z\in
W_{x}(t/(3M))}\operatorname*{ess\,inf}_{w\in
W_{y}(t/(3M))}\overline{p}(t/3,z,w)\right)\int_{W_{x}(t/(3M))}\int_{W_{y}(t/(3M))}dzdw$
$\displaystyle\geq c_{2}t^{-3d/\alpha+2d/\alpha}=c_{2}t^{-d/\alpha}.$

The proof is complete. $\Box$

For $x\in{\overline{\mathbb{R}}}^{d}_{+}$ and $r>0$, we denote
${B_{+}}(x,r):=B(x,r)\cap{\overline{\mathbb{R}}}^{d}_{+}$. We observe that

$\displaystyle m_{d}({B_{+}}(x,r))\asymp r^{d}\qquad\text{for
all}\;\,x\in{\overline{\mathbb{R}}}^{d}_{+},\,r>0.$ (3.27)

Now, using (2.6) and Lemma 3.7 (instead of (3.14)), we extend the results in
Lemmas 3.3 and 3.4 to $\overline{Y}$ removing the restrictions on $x$ and $r$.

######  Lemma 3.8.

There exist constants $C>0$ and $\eta\in(0,1/4)$ such that for all
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$, $r>0$, $t\in(0,(\eta
r)^{\alpha}]$ and $z\in{B_{+}}(x,\eta t^{1/\alpha})\setminus{\cal N}$,

$\overline{p}^{{B_{+}}(x,r)}(t,z,y)\geq Ct^{-d/\alpha}\quad\text{for a.e.
}y\in{B_{+}}(x,\eta t^{1/\alpha})\setminus{\cal N}.$

######  Lemma 3.9.

There exists a constant $C>1$ such that for all
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$ and $r>0$,

$C^{-1}r^{\alpha}\leq{\mathbb{E}}_{x}\left[\overline{\tau}_{{B_{+}}(x,r)}\right]\leq\sup_{z\in{B_{+}}(x,r)\setminus{\cal
N}}{\mathbb{E}}_{z}\left[\overline{\tau}_{{B_{+}}(x,r)}\right]\leq
Cr^{\alpha}.$ (3.28)

Let $\overline{X}:=(T_{s},\overline{Y}_{s})_{s\geq 0}$ and
$X^{\kappa}:=(T_{s},Y^{\kappa}_{s})_{s\geq 0}$ be time-space processes where
$T_{s}=T_{0}-s$. The law of the time-space process $s\mapsto\overline{X}_{s}$
or $s\mapsto X^{\kappa}_{s}$ starting from $(t,x)$ will be denoted by
${\mathbb{P}}_{(t,x)}$. For every open subset $U$ of
$[0,\infty)\times{\mathbb{R}}^{d}$, define
$\overline{\tau}_{U}=\inf\\{s>0:\,\overline{X}_{s}\notin U\\}$ and
$\tau^{\kappa}_{U}=\inf\\{s>0:\,X_{s}^{\kappa}\notin U\\}$. We also define
$\overline{\sigma}_{U}=\inf\\{s>0:\,\overline{X}_{s}\in U\\}$ and
$\sigma^{\kappa}_{U}=\inf\\{s>0:\,X_{s}^{\kappa}\in U\\}$.

A Borel function
$u:[0,\infty)\times\overline{\mathbb{R}}^{d}_{+}\to{\mathbb{R}}$ is said to be
parabolic in
$(a,b]\times{B_{+}}(x,r)\subset(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}$
with respect to $\overline{Y}$ if for every relatively compact open set
$U\subset(a,b]\times{B_{+}}(x,r)$ with respect to the topology on
$[0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}$, it holds that
$u(t,z)={\mathbb{E}}_{(t,z)}u(\overline{X}_{\overline{\tau}_{U}})$ for all
$(t,z)\in U$ with $z\notin{\cal N}$. Similarly, a Borel function
$u:[0,\infty)\times{\mathbb{R}}^{d}_{+}\to{\mathbb{R}}$ is said to be
parabolic in $(a,b]\times B(x,r)\subset(0,\infty)\times{\mathbb{R}}^{d}_{+}$
with respect to $Y^{\kappa}$ if for every relatively compact open set
$U\subset(a,b]\times B(x,r)$, it holds that
$u(t,z)={\mathbb{E}}_{(t,z)}u(X^{\kappa}_{\tau^{\kappa}_{U}})$ for all
$(t,z)\in U$ with $z\notin{\cal N}$.

######  Lemma 3.10.

(i) Let $\eta\in(0,1/4)$ be the constant from Lemma 3.3. For every
$\delta\in(0,\eta]$, there exists a constant $C>0$ such that for all
$x\in{\mathbb{R}}^{d}_{+}\setminus{\cal N}$, $r\in(0,x_{d})$, $t\geq\delta
r^{\alpha}$, and any compact set $A\subset[t-\delta r^{\alpha},t-\delta
r^{\alpha}/2]\times B(x,(\eta\delta/2)^{1/\alpha}r)$,

${\mathbb{P}}_{(t,x)}\big{(}\sigma^{\kappa}_{A}<\tau^{\kappa}_{[t-\delta
r^{\alpha},t]\times B(x,r)}\big{)}\geq C\frac{m_{d+1}(A)}{r^{d+\alpha}}.$

(ii) Let $\eta\in(0,1/4)$ be the constant from Lemma 3.8. For every
$\delta\in(0,\eta]$, there exists a constant $C>0$ such that for all
$x\in{\overline{\mathbb{R}}}^{d}_{+}\setminus{\cal N}$, $r>0$, $t\geq\delta
r^{\alpha}$, and any compact set $A\subset[t-\delta r^{\alpha},t-\delta
r^{\alpha}/2]\times{B_{+}}(x,(\eta\delta/2)^{1/\alpha}r)$,

${\mathbb{P}}_{(t,x)}\left(\overline{\sigma}_{A}<\overline{\tau}_{[t-\delta
r^{\alpha},t]\times{B_{+}}(x,r)}\right)\geq C\frac{m_{d+1}(A)}{r^{d+\alpha}}.$

Proof. By repeating the proofs of [21, Lemma 6.5] (using the Lévy system
formulas in (3.16) and (3.17)), we deduce the results from Lemmas 3.3 and 3.8
respectively. $\Box$

######  Theorem 3.11.

(i) For any $\delta\in(0,1)$, there exist $\lambda\in(0,1]$ and $C>0$ such
that for all $x\in{\mathbb{R}}^{d}_{+}$, $r\in(0,x_{d})$, $t_{0}\geq 0$, and
any function $u$ on $(0,\infty)\times{\mathbb{R}}^{d}_{+}$ which is parabolic
in $(t_{0},t_{0}+r^{\alpha}]\times B(x,r)$ with respect to $Y^{\kappa}$ and
bounded in $(t_{0},t_{0}+r^{\alpha}]\times{\mathbb{R}}^{d}_{+}$, we have

$|u(s,y)-u(t,z)|\leq
C\bigg{(}\frac{|s-t|^{1/\alpha}+|y-z|}{r}\bigg{)}^{\lambda}\operatorname*{ess\,sup}_{[t_{0},t_{0}+r^{\alpha}]\times{\mathbb{R}}^{d}_{+}}|u|,$
(3.29)

for every $s,t\in(t_{0}+(1-\delta^{\alpha})r^{\alpha},t_{0}+r^{\alpha}]$ and
$y,z\in B(x,\delta r)\setminus{\cal N}$.

(ii) For any $\delta\in(0,1)$, there exist $\lambda\in(0,1]$ and $C>0$ such
that for all $x\in{\overline{\mathbb{R}}}^{d}_{+}$, $r>0$, $t_{0}\geq 0$, and
any function $u$ on $(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}$ which is
parabolic in $(t_{0},t_{0}+r^{\alpha}]\times{B_{+}}(x,r)$ with respect to
$\overline{Y}$ and bounded in
$(t_{0},t_{0}+r^{\alpha}]\times\overline{\mathbb{R}}^{d}_{+}$, (3.29) holds
true for every $s,t\in(t_{0}+(1-\delta^{\alpha})r^{\alpha},t_{0}+r^{\alpha}]$
and $y,z\in{B_{+}}(x,\delta r)\setminus{\cal N}$.

Proof. Using Lemmas 3.3, 3.4 and 3.10(i) for (i), and (3.27) and Lemmas 3.8,
3.9 and 3.10(ii) for (ii), we get the desired results using the same argument
as in the proof of [15, Theorem 4.14] (see also the proof of [18, Proposition
3.8]). We omit details here. $\Box$

######  Remark 3.12.

By Theorem 3.11, since the heat kernels $\overline{p}(t,x,y)$ and
$p^{\kappa}(t,x,y)$ are parabolic with respect to $\overline{Y}$ and
$Y^{\kappa}$ respectively, they can be extended continuously to
$(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$
and $(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$
respectively. As consequences, by Proposition 2.7, $\overline{Y}$ and
$Y^{\kappa}$ can be refined to be a strongly Feller processes starting from
every point in ${\overline{\mathbb{R}}}^{d}_{+}$ and ${\mathbb{R}}^{d}_{+}$
respectively, and the exceptional set ${\cal N}$ in Proposition 2.7 can be
taken to be the empty set.

In the remainder of this paper, we take the jointly continuous version of
$\overline{p}:(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}\to[0,\infty)$
and
$p^{\kappa}:(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}\to[0,\infty)$,
take the exceptional set ${\cal N}$ in Proposition 2.7 to be empty set, and
replace the ${\rm essinf}$ in Lemma 3.6 by $\inf$. Again, for notational
convenience, we extend the domain of $p^{\kappa}(t,x,y)$ to
$(0,\infty)\times(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$
by letting $p^{\kappa}(t,x,y)=0$ if $x\in\partial{\mathbb{R}}^{d}_{+}$ or
$y\in\partial{\mathbb{R}}^{d}_{+}$.

The following scaling and horizontal translation invariance properties of the
heat kernels (the latter for $d\geq 2$), which come from Lemmas 2.1 and 2.2,
will be used throughout the paper: For any
$(t,x,y)\in(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}\times{\overline{\mathbb{R}}}^{d}_{+}$,
$r>0$ and $\widetilde{z}\in{\mathbb{R}}^{d-1}$,

$\overline{p}(t,x,y)=r^{-d}\overline{p}(t/r^{\alpha},x/r,y/r)=\overline{p}(t,x+(\widetilde{z},0),y+(\widetilde{z},0)),$
$p^{\kappa}(t,x,y)=r^{-d}p^{\kappa}(t/r^{\alpha},x/r,y/r)=p^{\kappa}(t,x+(\widetilde{z},0),y+(\widetilde{z},0)).$
(3.30)

From Proposition 2.7, since the exceptional set ${\cal N}$ is removed, we
obtain

######  Corollary 3.13.

If $d>\alpha$,

$G^{\kappa}(x,y)\leq\overline{G}(x,y)\leq\frac{c}{|x-y|^{d-\alpha}},\quad
x,y\in{\overline{\mathbb{R}}}_{+}^{d}.$

######  Remark 3.14.

The assumption $d>(\alpha+\beta_{1}+\beta_{2})\wedge 2$ in [31, 32] is only
used to show $G^{\kappa}(x,y)\leq c|x-y|^{-d+\alpha}$. Thus, by Corollary
3.13, all results in [31, 32] with the assumption
$d>(\alpha+\beta_{1}+\beta_{2})\wedge 2$ hold under the weaker assumption
$d>\alpha$.

Since the heat kernels $\overline{p}(t,x,y)$ and $p^{\kappa}(t,x,y)$ are
jointly continuous, we get the next lemma from the strong Markov properties of
$\overline{Y}$ and $Y^{\kappa}$. This lemma is a refined version of [13, Lemma
4.2] which was inspired by [35]. In this paper, this lemma will play an
important role in the bootstrap method to prove sharp upper estimates on the
heat kernels. Although the proof of next lemma is standard, we give it for the
reader’s convenience.

######  Lemma 3.15.

(i) Let $V_{1}$ and $V_{3}$ be open subsets of
${\overline{\mathbb{R}}}^{d}_{+}$ with ${\rm dist}(V_{1},V_{3})>0$. Set
$V_{2}:={\overline{\mathbb{R}}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. For any
$x\in V_{1}$, $y\in V_{3}$ and $t>0$, it holds that

$\displaystyle\overline{p}(t,x,y)$
$\displaystyle\leq{\mathbb{P}}_{x}(\overline{\tau}_{V_{1}}<t)\sup_{s\leq
t,\,z\in V_{2}}\overline{p}(s,z,y)$ $\displaystyle\quad+{\rm
dist}(V_{1},V_{3})^{-d-\alpha}\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}\overline{p}^{V_{1}}(t-s,x,u){\cal
B}(u,w)\overline{p}(s,y,w)dudwds.$

(ii) Let $V_{1}$ and $V_{3}$ be open subsets of ${\mathbb{R}}^{d}_{+}$ with
${\rm dist}(V_{1},V_{3})>0$. Set
$V_{2}:={\mathbb{R}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. For any $x\in V_{1}$,
$y\in V_{3}$ and $t>0$, it holds that

$\displaystyle p^{\kappa}(t,x,y)$
$\displaystyle\leq{\mathbb{P}}_{x}(\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa})\sup_{s\leq
t,\,z\in V_{2}}p^{\kappa}(s,z,y)$ $\displaystyle\quad+{\rm
dist}(V_{1},V_{3})^{-d-\alpha}\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{\kappa,V_{1}}(t-s,x,u){\cal
B}(u,w)p^{\kappa}(s,y,w)dudwds.$

Proof. Since the proofs are the same, we only give the proof of (ii).

By the strong Markov property, the Lévy system formula in (3.16) and symmetry,
we get that for any $x\in V_{1}$ and $y\in V_{3}$,

$\displaystyle p^{\kappa}(t,x,y)$
$\displaystyle={\mathbb{E}}_{x}\left[p^{\kappa}(t-\tau^{\kappa}_{V_{1}},Y^{\kappa}_{\tau^{\kappa}_{V_{1}}},y):\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa}\right]$
$\displaystyle={\mathbb{E}}_{x}\left[p^{\kappa}(t-\tau^{\kappa}_{V_{1}},Y^{\kappa}_{\tau^{\kappa}_{V_{1}}},y):\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa},Y^{\kappa}_{\tau^{\kappa}_{V_{1}}}\in
V_{2}\right]$
$\displaystyle\quad+{\mathbb{E}}_{x}\left[p^{\kappa}(t-\tau^{\kappa}_{V_{1}},Y^{\kappa}_{\tau^{\kappa}_{V_{1}}},y):\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa},Y^{\kappa}_{\tau^{\kappa}_{V_{1}}}\in
V_{3}\right]$
$\displaystyle\leq{\mathbb{P}}_{x}(\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa})\sup_{s\leq
t,\,z\in V_{2}}p^{\kappa}(s,z,y)$
$\displaystyle\quad+\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{\kappa,V_{1}}(t-s,x,u)\frac{{\cal
B}(u,w)}{|u-w|^{d+\alpha}}p^{\kappa}(s,w,y)dudwds$
$\displaystyle\leq{\mathbb{P}}_{x}(\tau^{\kappa}_{V_{1}}<t<\zeta^{\kappa})\sup_{s\leq
t,\,z\in V_{2}}p^{\kappa}(s,z,y)$ $\displaystyle\quad+{\rm
dist}(V_{1},V_{3})^{-d-\alpha}\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{\kappa,V_{1}}(t-s,x,u){\cal
B}(u,w)p^{\kappa}(s,y,w)dudwds.$

$\Box$

##  4\. Parabolic Harnack inequality and preliminary lower bound of heat
kernels

In this section, we prove that the parabolic Harnack inequality holds for
$\overline{Y}$ and $Y^{\kappa}$, and get some preliminary lower bounds for the
heat kernels $\overline{p}(t,x,y)$ and $p^{\kappa}(t,x,y)$.

Recall that ${B_{+}}(x,r)=B(x,r)\cap{\overline{\mathbb{R}}}^{d}_{+}$.

######  Lemma 4.1.

(i) Let $\eta\in(0,1/4)$ be the constant from Lemma 3.3 and let
$\delta\in(0,2^{-\alpha-2}\eta)$. There exists a constant $C>0$ such that for
all $y\in{\mathbb{R}}^{d}_{+}$, $R\in(0,y_{d})$,
$r\in(0,(\eta\delta/2)^{1/\alpha}R/2]$, $\delta R^{\alpha}/2\leq t-s\leq
4\delta(2R)^{\alpha}$, $x\in B(y,(\eta\delta/2)^{1/\alpha}R/2)$ and $z\in
B(x_{0},(\eta\delta/2)^{1/\alpha}R)$,

${\mathbb{P}}_{(t,z)}\big{(}\sigma^{\kappa}_{\\{s\\}\times
B(x,r)}\leq\tau^{\kappa}_{[s,t]\times B(y,R)}\big{)}\geq C(r/R)^{d}.$

(ii) Let $\eta\in(0,1/4)$ be the constant from Lemma 3.8 and let
$\delta\in(0,2^{-\alpha-2}\eta)$. There exists a constant $C>0$ such that for
all $y\in{\overline{\mathbb{R}}}^{d}_{+}$, $R>0$,
$r\in(0,(\eta\delta/2)^{1/\alpha}R/2]$, $\delta R^{\alpha}/2\leq t-s\leq
4\delta(2R)^{\alpha}$, $x\in{B_{+}}(y,(\eta\delta/2)^{1/\alpha}R/2)$ and
$z\in{B_{+}}(x_{0},(\eta\delta/2)^{1/\alpha}R)$,

${\mathbb{P}}_{(t,z)}\big{(}\overline{\sigma}_{\\{s\\}\times{B_{+}}(x,r)}\leq\overline{\tau}_{[s,t]\times{B_{+}}(y,R)}\big{)}\geq
C(r/R)^{d}.$

Proof. Using Lemmas 3.3 and 3.8, and (3.27), the result can be proved by the
same argument as that of [21, Lemma 6.7]. We omit details here. $\Box$

In order to obtain the parabolic Harnack inequality for $\overline{Y}$ and
$Y^{\kappa}$, we introduce two conditions:

(UBS) There exists a constant $C>0$ such that for all
$x,y\in{\overline{\mathbb{R}}}_{+}^{d}$ and $0<r\leq|x-y|/2$,

$\displaystyle{\cal B}(x,y)\leq\frac{C}{r^{d}}\int_{{B_{+}}(x,r)}{\cal
B}(z,y)dz.$ (4.1)

(IUBS) There exists a constant $C>0$ such that (4.1) holds for all
$x,y\in{\mathbb{R}}_{+}^{d}$ and $0<r\leq(|x-y|\wedge x_{d})/2$. (Note that,
${B_{+}}(x,r)=B(x,r)$ for this range of $r$.)

######  Lemma 4.2.

If (A3)(II) also holds, then (UBS) is satisfied. In particular, (IUBS) is
satisfied.

Proof. Let $x,y\in{\overline{\mathbb{R}}}_{+}^{d}$ and $0<r\leq|x-y|/2$. Note
that for all $z\in{B_{+}}(x,r)$, $|x-y|/2\leq|z-y|\leq 2|x-y|$ by the triangle
inequality. Thus, by (A3), there is $c_{1}>0$ independent of $x,y$ and $r$
such that for all $z\in{B_{+}}(x,r)$ with $z_{d}\geq x_{d}$, ${\cal
B}(x,y)\leq c_{1}{\cal B}(z,y)$. Using this, we get

$\displaystyle\frac{1}{r^{d}}\int_{{B_{+}}(x,r)}{\cal B}(z,y)dz\geq\frac{{\cal
B}(x,y)}{c_{1}r^{d}}\int_{B(x,r):z_{d}\geq x_{d}}dz\geq c_{2}{\cal B}(x,y).$

$\Box$

We now show that the following parabolic Harnack inequalities hold.

######  Theorem 4.3.

(i) Suppose that ${\cal B}(x,y)$ satisfies (IUBS). Then there exist constants
$\delta>0$ and $C,M\geq 1$ such that for all $t_{0}\geq 0$,
$x\in{\mathbb{R}}^{d}_{+}$ and $R\in(0,x_{d})$, and any non-negative function
$u$ on $(0,\infty)\times{\mathbb{R}}^{d}_{+}$ which is parabolic on
$Q:=(t_{0},t_{0}+4\delta R^{\alpha}]\times B(x,R)$ with respect to
$\overline{Y}$ or $Y^{\kappa}$, we have

$\sup_{(t_{1},y_{1})\in Q_{-}}u(t_{1},y_{1})\leq C\inf_{(t_{2},y_{2})\in
Q_{+}}u(t_{2},y_{2}),$ (4.2)

where $Q_{-}=[t_{0}+\delta R^{\alpha},t_{0}+2\delta R^{\alpha}]\times
B(x,R/M)$ and $Q_{+}=[t_{0}+3\delta R^{\alpha},t_{0}+4\delta R^{\alpha}]\times
B(x,R/M)$.

(ii) Suppose that ${\cal B}(x,y)$ satisfies (UBS). Then there exist constants
$\delta>0$ and $C,M\geq 1$ such that for all $t_{0}\geq 0$,
$x\in{\overline{\mathbb{R}}}^{d}_{+}$ and $R>0$, and any non-negative function
$u$ on $(0,\infty)\times{\overline{\mathbb{R}}}^{d}_{+}$ which is parabolic on
$Q^{0}:=(t_{0},t_{0}+4\delta R^{\alpha}]\times{B_{+}}(x,R)$ with respect to
$\overline{Y}$, we have

$\sup_{(t_{1},y_{1})\in Q^{0}_{-}}u(t_{1},y_{1})\leq C\inf_{(t_{2},y_{2})\in
Q^{0}_{+}}u(t_{2},y_{2}),$ (4.3)

where $Q^{0}_{-}=[t_{0}+\delta R^{\alpha},t_{0}+2\delta
R^{\alpha}]\times{B_{+}}(x,R/M)$ and $Q^{0}_{+}=[t_{0}+3\delta
R^{\alpha},t_{0}+4\delta R^{\alpha}]\times{B_{+}}(x,R/M)$.

Proof. (i) By (IUBS), there is a constant $C>0$ such that for all
$x,y\in{\mathbb{R}}^{d}_{+}$ and $0<r\leq(|x-y|\wedge x_{d})/2$,

$\displaystyle J(x,y)=\frac{{\cal
B}(x,y)}{|x-y|^{d+\alpha}}\leq\frac{C}{r^{d}}\int_{{B_{+}}(x,r)}\frac{{\cal
B}(z,y)}{|z-y|^{d+\alpha}}dz=\frac{C}{r^{d}}\int_{{B_{+}}(x,r)}J(z,y)dz.$
(4.4)

Using Proposition 2.7 and (4.4), one can follow the proof of [21, Lemma 6.10]
and see that [21, Lemma 6.10] is also valid for our case. (Note that, in the
proof of [21, Lemma 6.10], a pointwise comparison for the jump kernel from
[21, Proposition 6.8] was used to bound the term $I_{2}$ therein which can be
replaced by (4.4).) Using this and Lemmas 3.3, 3.4, 3.10, 4.1, the result can
be proved using the same argument as in the proof of [9, Lemma 5.3] (see also
the proof of [18, Lemma 4.1]). We omit details here.

(ii) Since (UBS) implies that (4.4) is satisfied for all
$x,y\in{\mathbb{R}}^{d}_{+}$ and $0<r\leq|x-y|/2$, using Proposition 2.7 and
(3.27), one can also deduce that [21, Lemma 6.10] is valid for this case with
${B_{+}}(x_{0},\cdot)$ instead of $B(x_{0},\cdot)$ in the definitions of
$Q_{i}$, $1\leq i\leq 4$ therein. Then using Lemmas 3.8, 3.9, 3.10(ii),
4.1(ii), one can follow the arguments in the proof of [9, Lemma 5.3] and
conclude the result. $\Box$

Using Lemma 3.3, and Theorem 4.3(i), we obtain the following lemma.

######  Lemma 4.4.

Suppose that ${\cal B}(x,y)$ satisfies (IUBS). For any positive constants
$a,b$, there exists a constant $C=C(a,b,\kappa)>0$ such that for all
$z\in{\mathbb{R}}_{+}^{d}$ and $r>0$ with
$B(z,2br)\subset{\mathbb{R}}^{d}_{+}$,

$\inf_{y\in
B(z,br/2)}{\mathbb{P}}_{y}\big{(}\tau^{\kappa}_{B(z,br)}>ar^{\alpha}\big{)}\,\geq\,C.$

Proof. By Lemma 3.3, there exist constants $c_{1},c_{2},{\varepsilon}_{1}>0$
such that for all $z\in{\mathbb{R}}^{d}_{+}$ and $r>0$ with
$B(z,2br)\subset{\mathbb{R}}^{d}_{+}$,

$\displaystyle{\mathbb{P}}_{z}\big{(}\tau^{\kappa}_{B(z,br/2)}>{\varepsilon}_{1}r^{\alpha}\big{)}=\int_{B(z,br/2)}p^{\kappa,B(z,br/2)}({\varepsilon}_{1}r^{\alpha},z,w)dw$
$\displaystyle\geq\int_{B(z,c_{1}r)}p^{\kappa,B(z,br/2)}({\varepsilon}_{1}r^{\alpha},z,w)dw\geq
c_{2}.$ (4.5)

Thus it suffices to prove the lemma for $a>{\varepsilon}_{1}$. Applying the
parabolic Harnack inequality (Theorem 4.3) repeatedly, we conclude that there
exists $c_{3}>0$ such that for any $w,y\in B(z,br/2)$,

$p^{\kappa,B(z,br)}(ar^{\alpha},y,w)\geq
c_{3}\,p^{\kappa,B(z,br)}({\varepsilon}_{1}r^{\alpha},z,w).$

Thus, using (4), we deduce that for any $y\in B(z,br/2)$,

$\displaystyle{\mathbb{P}}_{y}\big{(}\tau^{\kappa}_{B(z,br)}>ar^{\alpha}\big{)}=\int_{B(z,br)}p^{\kappa,B(z,br)}(ar^{\alpha},y,w)dw$
$\displaystyle\geq
c_{3}\int_{B(z,br/2)}p^{\kappa,B(z,br)}({\varepsilon}_{1}r^{\alpha},z,w)dw\geq
c_{3}{\mathbb{P}}_{z}\big{(}\tau^{\kappa}_{B(z,br/2)}>{\varepsilon}_{1}r^{\alpha}\big{)}\geq
c_{2}c_{3}.$

This proves the lemma. $\Box$

Now, we follow the proof of [8, Proposition 3.5] to get the following
preliminary lower bound.

######  Proposition 4.5.

Suppose that ${\cal B}(x,y)$ satisfies (IUBS). For any $a>0$, there exists a
constant $C=C(a,\kappa)>0$ such that for any
$(t,x,y)\in(0,\infty)\times{\mathbb{R}}_{+}^{d}\times{\mathbb{R}}_{+}^{d}$
with $at^{1/\alpha}\leq(4|x-y|)\wedge x_{d}\wedge y_{d}$,

$\overline{p}(t,x,y)\geq p^{\kappa}(t,x,y)\geq CtJ(x,y).$

Proof. The first inequality holds true by (2.5) and Remark 3.12.

By Lemma 4.4, starting at $z\in B(y,\,(12)^{-1}at^{1/\alpha})$, with
probability at least $c_{1}=c_{1}(a)>0$, the process $Y^{\kappa}$ does not
move more than $(18)^{-1}at^{1/\alpha}$ by time $t$. Thus, using the strong
Markov property and the Lévy system formula in (3.16), we obtain that for
$a(t/2)^{1/\alpha}\leq(4|x-y|)\wedge x_{d}\wedge y_{d}$,

$\displaystyle{\mathbb{P}}_{x}\left(Y^{\kappa}_{t}\in
B\big{(}y,\,6^{-1}at^{1/\alpha}\big{)}\right)$ $\displaystyle\geq
c_{1}{\mathbb{P}}_{x}\big{(}\,Y^{\kappa}_{t\wedge\tau^{\kappa}_{B(x,(18)^{-1}at^{1/\alpha})}}\in
B(y,\,(12)^{-1}at^{1/\alpha})\hbox{ and
}t\wedge\tau^{\kappa}_{B(x,(18)^{-1}at^{1/\alpha})}\hbox{ is a jumping
time\,}\big{)}$
$\displaystyle=c_{1}{\mathbb{E}}_{x}\bigg{[}\int_{0}^{t\wedge\tau^{\kappa}_{B(x,(18)^{-1}at^{1/\alpha})}}\int_{B(y,\,(12)^{-1}at^{1/\alpha})}J(Y^{\kappa}_{s},u)duds\bigg{]}.$
(4.6)

By (4.4), we obtain that for $a(t/2)^{1/\alpha}\leq(4|x-y|)\wedge x_{d}\wedge
y_{d}$,

$\displaystyle{\mathbb{E}}_{x}\bigg{[}\int_{0}^{t\wedge\tau^{\kappa}_{B(x,(18)^{-1}at^{1/\alpha})}}\int_{B(y,\,(12)^{-1}at^{1/\alpha})}J(Y^{\kappa}_{s},u)duds\bigg{]}$
$\displaystyle={\mathbb{E}}_{x}\bigg{[}\int_{0}^{t}\int_{B(y,\,(12)^{-1}at^{1/\alpha})}J(Y^{\kappa,B(x,(18)^{-1}at^{1/\alpha})}_{s},u)duds\bigg{]}$
$\displaystyle\geq
c_{2}t^{d/\alpha}\int_{0}^{t}{\mathbb{E}}_{x}\left[J(Y^{\kappa,B(x,(18)^{-1}at^{1/\alpha})}_{s},y)\right]ds$
$\displaystyle\geq
c_{2}t^{d/\alpha}\int_{t/2}^{t}\int_{B(x,(72)^{-1}a(t/2)^{1/\alpha})}J(w,y)p^{\kappa,B(x,(18)^{-1}at^{1/\alpha})}(s,x,w)dwds.$
(4.7)

Note that, for $t/2<s<t$ and $w\in B(x,(72)^{-1}a(t/2)^{1/\alpha})$, we have

$\delta_{B(x,(18)^{-1}at^{1/\alpha})}(w)\geq(18)^{-1}at^{1/\alpha}-(72)^{-1}a(t/2)^{1/\alpha}\geq(36)^{-1}as^{-1/\alpha}$

and

$|x-w|<(72)^{-1}a(t/2)^{1/\alpha}\leq 4^{-1}(18)^{-1}as^{1/\alpha}.$

Thus by Lemma 3.3 and the parabolic Harnack inequality (Theorem 4.3) we see
that for $t/2<s<t$ and $w\in B(x,(72)^{-1}a(t/2)^{1/\alpha})$,

$\displaystyle p^{\kappa,B(x,(18)^{-1}at^{1/\alpha})}(s,x,w)\geq
c_{3}\,t^{-d/\alpha}.$ (4.8)

Combining (4.6), (4.7) with (4.8) and applying (4.4) again, we get that for
$a(t/2)^{1/\alpha}\leq(4|x-y|)\wedge x_{d}\wedge y_{d}$,

$\displaystyle{\mathbb{P}}_{x}\left(Y^{\kappa}_{t}\in
B\big{(}y,\,6^{-1}at^{1/\alpha}\big{)}\right)\geq
c_{4}t\int_{B(x,(72)^{-1}a(t/2)^{1/\alpha})}J(w,y)dw\geq
c_{5}t^{1+d/\alpha}J(x,y).$ (4.9)

Note that for $y_{d}\geq at^{1/\alpha}$ and $z\in B(y,a(t/2)^{1/\alpha}/6)$,

$\displaystyle z_{d}\geq y_{d}-|z-y|\geq
a(1-(1/2)^{1/\alpha}/6)t^{1/\alpha}\geq(5a/6)t^{1/\alpha}$ (4.10)

The proposition now follows from the Chapman-Kolmogorov equation along with
(4.9) and Proposition 3.2 (using (4.10)): for $at^{1/\alpha}\leq(4|x-y|)\wedge
x_{d}\wedge y_{d}$,

$\displaystyle p^{\kappa}(t,x,y)$
$\displaystyle=\int_{D}p^{\kappa}(t/2,x,z)p^{\kappa}(t/2,z,y)dz\geq\int_{B(y,\,a(t/2)^{1/\alpha}/6)}p^{\kappa}(t/2,x,z)p^{\kappa}(t/2,z,y)dz$
$\displaystyle\geq
c_{6}t^{-d/\alpha}\,{\mathbb{P}}_{x}\left(Y^{\kappa}_{t/2}\in
B(y,a(t/2)^{1/\alpha}/6)\right)\geq c_{7}\,{t}J(x,y).$

$\Box$

##  5\. Preliminary upper bounds of heat kernels

The goal of this section is to prove the following proposition.

######  Proposition 5.1.

Suppose that (A1)–(A4) and (1.8) hold true. Then there exists a constant $C>0$
such that

$p^{\kappa}(t,x,y)\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),\quad
t>0,\,x,y\in{\mathbb{R}}^{d}_{+},$ (5.1)

where $q_{\kappa}\in[(\alpha-1)_{+},\alpha+\beta_{1})$ is the constant from
(1.8).

Note that, since $t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}$ is comparable
to the transition density of the isotropic $\alpha$-stable process in
${\mathbb{R}}^{d}$, there exists a constant $C>0$ such that for all $t,s>0$
and $x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle\int_{{\mathbb{R}}^{d}_{+}}\left(t^{-d/\alpha}\wedge\frac{t}{|x-z|^{d+\alpha}}\right)dz\leq
C,$ (5.2)

and

$\displaystyle\int_{{\mathbb{R}}^{d}_{+}}\left(t^{-d/\alpha}\wedge\frac{t}{|x-z|^{d+\alpha}}\right)\left(s^{-d/\alpha}\wedge\frac{s}{|y-z|^{d+\alpha}}\right)dz\leq
C\left((t+s)^{-d/\alpha}\wedge\frac{t+s}{|x-y|^{d+\alpha}}\right),$ (5.3)

where in (5.3) we used the semigroup property.

Before giving the proof of Proposition 5.1, we record its simple consequence.

######  Corollary 5.2.

There exists a constant $C>0$ such that

${\mathbb{P}}_{x}(\zeta^{\kappa}>t)\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}},\quad
t>0,\;x\in{\mathbb{R}}^{d}_{+}.$

Proof. By Proposition 5.1 and (5.2),

$\displaystyle{\mathbb{P}}_{x}(\zeta^{\kappa}>t)=\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t,x,y)dy$
$\displaystyle\leq
c_{1}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}\int_{{\mathbb{R}}^{d}_{+}}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)dy\leq
c_{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q_{\kappa}}.$

$\Box$

When $q_{\kappa}=0$, Proposition 5.1 follows from Proposition 2.7. Hence, in
the remainder of this section, we assume that (A1)–(A4) hold, fix
$\kappa\in[0,\infty)$ such that $q_{\kappa}>0$ and denote by $q$ the constant
$q_{\kappa}$ in (1.8). For the proof we will need several results from [30,
31, 32] that we now recall for the convenience of the reader.

For $r>0$, we define a subset $U(r)$ of ${\overline{\mathbb{R}}}^{d}_{+}$,
$d\geq 2$, by

$U(r):=\\{x=(\widetilde{x},x_{d})\in{\mathbb{R}}^{d}:\,|\widetilde{x}|<r/2,\,0\leq
x_{d}<r/2\\}.$

When $d=1$, we abuse notation and use $U(r)=[0,r/2)$. For $t>0$ and an open
set $V\subset{\mathbb{R}}^{d}_{+}$, denote by $Y_{t}^{\kappa,d}$ and
$Y_{t}^{\kappa,V,d}$ the last coordinates of $Y_{t}^{\kappa}$ and
$Y_{t}^{\kappa,V}$ respectively.

######  Lemma 5.3.

There exists a constant $C>0$ such that for all $x,y\in{\mathbb{R}}^{d}_{+}$
satisfying $|x-y|\geq x_{d}$, it holds that

$\displaystyle{\cal B}(x,y)\leq Cx_{d}^{\beta_{1}}(|\log
x_{d}|^{\beta_{3}}\vee 1)\big{(}1+{\bf 1}_{|y|\geq
1}(\log|y|)^{\beta_{3}}\big{)}|x-y|^{-\beta_{1}}.$

Proof. See [30, Lemma 5.2(a)]. $\Box$

######  Lemma 5.4.

For any $R>0$, there exists a constant $C>0$ such that for all $r\in(0,R]$ and
$x\in U(2^{-4}r)\cap{\mathbb{R}}^{d}_{+}$,

$\displaystyle{\mathbb{E}}_{x}\int_{0}^{\tau^{\kappa}_{U(r)}}(Y^{\kappa,d}_{t})^{\beta_{1}}|\log
Y^{\kappa,d}_{t}|^{\beta_{3}}dt\leq Cx_{d}^{q}.$

Proof. The result follows from scaling (Lemma 2.1), and [30, Lemma 5.7(a)] if
$\kappa>0$ and [32, Lemma 5.3] if $\kappa=0$. $\Box$

######  Lemma 5.5.

There exists a constant $C>0$ such that for all $r>0$ and $x\in
U(2^{-4}r)\cap{\mathbb{R}}^{d}_{+}$,

${\mathbb{P}}_{x}\big{(}\tau^{\kappa}_{U(r)}<\zeta^{\kappa}\big{)}={\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(r)}}\in{\mathbb{R}}^{d}_{+}\big{)}\leq
C\left(\frac{x_{d}}{r}\right)^{q}.$

Proof. When $\kappa>0$, the result follows from [31, Lemma 3.4]. When
$\kappa=0$, using [32, Lemma 5.5] and Lemma 5.4, one can follow the proof of
[31, Lemma 3.4] and deduce the result. $\Box$

######  Lemma 5.6.

There exists a constant $C>0$ such that for all $r>0$ and $x\in
U(2^{-4}r)\cap{\mathbb{R}}^{d}_{+}$,

${\mathbb{E}}_{x}\big{[}\tau^{\kappa}_{U(r)}\big{]}\leq
C\left(\frac{x_{d}}{r}\right)^{q}.$

Proof. The result follows from scaling (Lemma 2.1), and [30, Lemma 5.13] if
$\kappa>0$ and [32, Lemma 4.5] if $\kappa=0$. $\Box$

######  Proposition 5.7.

There exists a constant $C>0$ such that for any
$w\in\partial{\mathbb{R}}^{d}_{+}$, $r>0$ and any non-negative function $f$ in
${\mathbb{R}}^{d}_{+}$ that is harmonic in ${\mathbb{R}}^{d}_{+}\cap B(w,r)$
with respect to $Y^{\kappa}$ and vanishes continuously on
$\partial{\mathbb{R}}^{d}_{+}\cap B(w,r)$, we have

$\displaystyle f(x)\leq Cf(\widehat{x})\quad\text{for
all}\;\,x\in{\mathbb{R}}^{d}_{+}\cap B(w,r/2),$

where $\widehat{x}\in{\mathbb{R}}^{d}_{+}\cap B(w,r)$ with
$\widehat{x}_{d}\geq r/4$.

Proof. The result follows from [30, Theorem 1.2] if $\kappa>0$ and [32,
Theorem 5.6] if $\kappa=0$. $\Box$

After recalling the known results above, we now continue with several
auxiliary lemmas leading to the proof of Proposition 5.1.

######  Lemma 5.8.

For all $\gamma\geq 0$, $t>0$ and $x\in U(1)\cap{\mathbb{R}}^{d}_{+}$, it
holds that

$\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t,x,z)(1\wedge
z_{d})^{\gamma}dz\leq{\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,U(1),d}_{t})^{\gamma}:\tau^{\kappa}_{U(1)}>t\right]+{\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+}\big{)}.$
(5.4)

In particular, it holds that

${\mathbb{P}}_{x}(\zeta^{\kappa}>t)\leq
t^{-1}{\mathbb{E}}_{x}\big{[}\tau^{\kappa}_{U(1)}\big{]}+{\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+}\big{)}.$
(5.5)

Proof. Since $Y^{\kappa,U(1)}_{t}=Y^{\kappa}_{t}$ for
$t<\tau^{\kappa}_{U(1)}$, we have

$\displaystyle\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t,x,z)(1\wedge
z_{d})^{\gamma}dz={\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,d}_{t})^{\gamma}:t<\zeta^{\kappa}\right]$
$\displaystyle={\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,d}_{t})^{\gamma}:\tau^{\kappa}_{U(1)}>t\right]+{\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,d}_{t})^{\gamma}:\tau^{\kappa}_{U(1)}\leq t<\zeta^{\kappa}\right]$
$\displaystyle\leq{\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,U(1),d}_{t})^{\gamma}:\tau^{\kappa}_{U(1)}>t\right]+{\mathbb{E}}_{x}\big{[}1:\tau^{\kappa}_{U(1)}<\zeta^{\kappa}\big{]}$
$\displaystyle={\mathbb{E}}_{x}\left[(1\wedge
Y^{\kappa,U(1),d}_{t})^{\gamma}:\tau^{\kappa}_{U(1)}>t\right]+{\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+}\big{)}.$

By taking $\gamma=0$ in (5.4) and using Markov’s inequality, we get

${\mathbb{P}}_{x}(\zeta^{\kappa}>t)\leq{\mathbb{P}}_{x}(\tau^{\kappa}_{U(1)}>t)+{\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+}\big{)}\leq
t^{-1}{\mathbb{E}}_{x}\big{[}\tau^{\kappa}_{U(1)}\big{]}+{\mathbb{P}}_{x}\big{(}Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+}\big{)}.$

$\Box$

######  Lemma 5.9.

There exists $C>0$ such that

$p^{\kappa}(t,x,y)\leq
Ct^{-d/\alpha}{\mathbb{P}}_{x}(\zeta^{\kappa}>t/3){\mathbb{P}}_{y}(\zeta^{\kappa}>t/3),\quad
t>0,\,x,y\in{\mathbb{R}}^{d}_{+}.$

Proof. By the semigroup property, the symmetry of $p^{\kappa}(t,\cdot,\cdot)$
and Proposition 2.7, we obtain

$\displaystyle
p^{\kappa}(t,x,y)=\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t/3,x,z)p^{\kappa}(t/3,z,w)p^{\kappa}(t/3,y,w)dzdw$
$\displaystyle\leq
c_{1}t^{-d/\alpha}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t/3,x,z)dz\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t/3,y,w)dw=c_{1}t^{-d/\alpha}{\mathbb{P}}_{x}(\zeta^{\kappa}>t/3){\mathbb{P}}_{y}(\zeta^{\kappa}>t/3).$

$\Box$

The next lemma shows that (5.1) (hence Proposition 5.1) is a consequence of
the following, seemingly weaker, inequality: There exists $C>0$ such that

$p^{\kappa}(t,x,y)\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}t^{-d/\alpha},\quad
t>0,\,x,y\in{\mathbb{R}}^{d}_{+}.$ (5.6)

######  Lemma 5.10.

If (5.6) holds true, then (5.1) also holds.

Proof. We claim that there exists a constant $c_{1}>0$ such that

$p^{\kappa}(t,x,y)\leq
c_{1}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$
(5.7)

By (3.30), we can assume $\widetilde{x}=\widetilde{0}$ and
$t=t_{0}=(1/2)^{\alpha}$. If $x_{d}\geq 2^{-4}t_{0}^{1/\alpha}$ or $|x-y|\leq
4t_{0}^{1/\alpha}$, then (5.7) follows from Proposition 2.7 or the assumption
(5.6) respectively. Hence, we assume $x_{d}<2^{-4}t_{0}^{1/\alpha}$ and
$|x-y|>4t_{0}^{1/\alpha}$, and will show that

$p^{\kappa}(t_{0},x,y)\leq c_{1}(t_{0})\frac{x_{d}^{q}}{|x-y|^{d+\alpha}}.$
(5.8)

Let $V_{1}=U(t_{0}^{1/\alpha})$,
$V_{3}=\\{w\in{\mathbb{R}}^{d}_{+}:|w-y|<|x-y|/2\\}$ and
$V_{2}={\mathbb{R}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. By Lemma 5.5, we have

$\displaystyle{\mathbb{P}}_{x}(\tau^{\kappa}_{V_{1}}<t_{0}<\zeta^{\kappa})\leq{\mathbb{P}}_{x}(Y^{\kappa}_{\tau^{\kappa}_{V_{1}}}\in{\mathbb{R}}^{d}_{+})\leq
c_{2}(t_{0}^{-1/\alpha}x_{d})^{q}.$ (5.9)

Also, we get from Proposition 2.7 that

$\sup_{s\leq t_{0},\,z\in V_{2}}p^{\kappa}(s,z,y)\leq c_{3}\sup_{s\leq
t_{0},\,z\in{\mathbb{R}}^{d}_{+},|z-y|>|x-y|/2}\frac{s}{|z-y|^{d+\alpha}}=2^{d+\alpha}c_{3}\frac{t_{0}}{|x-y|^{d+\alpha}}.$
(5.10)

Next, we note that by the triangle inequality, for any $u\in V_{1}$ and $w\in
V_{3}$,

$|u-w|\geq|x-y|-|x-u|-|y-w|\geq|x-y|-t_{0}^{1/\alpha}-\frac{|x-y|}{2}\geq\frac{|x-y|}{4}\geq
t_{0}^{1/\alpha}\geq u_{d}.$ (5.11)

In particular, recalling that $\beta_{1}>0$ if $\beta_{3}>0$, we see that
$\big{(}1+{\bf 1}_{|w|\geq
1}(\log|w|)^{\beta_{3}}\big{)}|u-w|^{-\beta_{1}}\leq c_{4}$ for $u\in V_{1}$
and $w\in V_{3}$, so by Lemma 5.3, we have that for any $u\in V_{1}$ and $w\in
V_{3}$,

${\cal B}(u,w)\leq c_{5}u_{d}^{\beta_{1}}(|\log u_{d}|^{\beta_{3}}\vee
1)\big{(}1+{\bf 1}_{|w|\geq
1}(\log|w|)^{\beta_{3}}\big{)}|u-w|^{-\beta_{1}}\leq
c_{6}u_{d}^{\beta_{1}}|\log u_{d}|^{\beta_{3}}.$ (5.12)

Thus, by (A3)(II), Lemma 5.4, (5.11) and (5.12) we get that

$\displaystyle\int_{0}^{t_{0}}\int_{V_{3}}\int_{V_{1}}p^{\kappa,V_{1}}(t-s,x,u){\cal
B}(u,w)p^{\kappa}(s,y,w)dudwds$ $\displaystyle\leq
c_{7}\int_{0}^{t_{0}}\left(\int_{V_{1}}p^{\kappa,V_{1}}(t-s,x,u)u_{d}^{\beta_{1}}|\log
u_{d}|^{\beta_{3}}du\right)\left(\int_{V_{3}}p^{\kappa}(s,y,w)dw\right)ds$
$\displaystyle\leq
c_{7}\int_{0}^{\infty}\left(\int_{V_{1}}p^{\kappa,V_{1}}(s,x,u)u_{d}^{\beta_{1}}|\log
u_{d}|^{\beta_{3}}du\right)ds$
$\displaystyle=c_{7}{\mathbb{E}}_{x}\int_{0}^{\tau^{\kappa}_{V_{1}}}(Y_{s}^{\kappa,d})^{\beta_{1}}|\log
Y_{s}^{\kappa,d}|^{\beta_{3}}ds\leq c_{8}x_{d}^{q}.$ (5.13)

Now (5.8) (and so (5.7)) follows from (5.9)–(5.11), (5.13) and Lemma 3.15.

Finally, by the semigroup property, symmetry, (5.3) and (5.7),

$\displaystyle
p^{\kappa}(t,x,y)=\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(t/2,x,z)p^{\kappa}(t/2,y,z)dz$
$\displaystyle\leq
c_{1}^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\int_{{\mathbb{R}}^{d}_{+}}\left((t/2)^{-d/\alpha}\wedge\frac{t/2}{|x-z|^{d+\alpha}}\right)\left((t/2)^{-d/\alpha}\wedge\frac{t/2}{|y-z|^{d+\alpha}}\right)dz$
$\displaystyle\leq
c_{9}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$

$\Box$

Now we prove (5.6) holds. We first consider the case $q<\alpha$.

######  Lemma 5.11.

If $q<\alpha$, then (5.6) holds true.

Proof. By (3.30), it suffices to prove (5.6) when $t=1$. By Lemma 5.9, it
suffices to prove that there exists a constant $c_{1}>0$ such that
${\mathbb{P}}_{x}(\zeta^{\kappa}>1/3)\leq c_{1}(1\wedge x_{d})^{q}$ for all
$x\in{\mathbb{R}}^{d}_{+}$. By Lemma 2.2 and the fact that
${\mathbb{P}}_{x}(\zeta>1/3)\leq 1$, without loss of generality, we can assume
$\widetilde{x}=0$ and $x_{d}<2^{-5}$. Then, since $q<\alpha$, by (5.5) and
Lemmas 5.5–5.6 (with $r=1$), we get ${\mathbb{P}}_{x}(\zeta^{\kappa}>1/3)\leq
c_{2}x_{d}^{q}$. $\Box$

To remove the assumption $q<\alpha$ in Lemma 5.11, we will make use of a
result from [31, 32]. Recall from Remark 3.14 that all results in [31, 32] are
valid when $d>\alpha$. Now we state, and will prove later, that the desired
result still holds true for $d=1\leq\alpha$ (thus for all $d$).

######  Lemma 5.12.

Let $\gamma>q-\alpha$. There exists $C>0$ such that for any $R>0$,
$U(R)\subset D\subset U(2R)$ and any
$x=(\widetilde{0},x_{d})\in{\mathbb{R}}^{d}_{+}$ with $x_{d}\leq R/10$, it
holds that

${\mathbb{E}}_{x}\int_{0}^{\tau^{\kappa}_{D}}(Y^{\kappa,D,d}_{s})^{\gamma}ds=\int_{0}^{\infty}\int_{D}p^{\kappa,D}(t,x,z)z_{d}^{\gamma}dzdt\leq
CR^{\gamma+\alpha-q}x_{d}^{q}.$

Proof. When $d>\alpha$, by Remark 3.14, the result follows from [31,
Proposition 6.10] if $\kappa>0$, and from [32, Proposition 6.8] if $\kappa=0$.
The proof of the case $d=1\leq\alpha$ is postponed to the end of this section.
$\Box$

######  Lemma 5.13.

Inequality (5.6) holds true.

Proof. Again using (3.30), it suffices to prove (5.6) when $t=1$.

Inequality (5.6) holds when $q<\alpha$ by Lemma 5.11. Now, assume that (5.6)
holds for all $q<k\alpha$ for some $k\in\mathbb{N}$. We now show (5.6) also
holds for $q\in[k\alpha,(k+1)\alpha)$ and hence (5.6) always holds by
induction.

Fix ${\varepsilon}\in(0,\alpha)$ such that $q-\alpha+{\varepsilon}<k\alpha$.
Then $q-\alpha+{\varepsilon}<q$ and $C(\alpha,q-\alpha+{\varepsilon},{\cal
B})<C(\alpha,q,{\cal B})=\kappa$. By (3.30) and the induction hypothesis, it
holds that for any $s,u\in[0,1/4]$ and $z,w\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle p^{\kappa}(1-s-u,z,w)$
$\displaystyle=(1-s-u)^{-d/\alpha}p^{\kappa}(1,(1-s-u)^{-1/\alpha}z,(1-s-u)^{-1/\alpha}w)$
$\displaystyle\leq 2^{d/\alpha}p^{C(\alpha,q-\alpha+{\varepsilon},{\cal
B})}(1,(1-s-u)^{-1/\alpha}z,(1-s-u)^{-1/\alpha}w)$ $\displaystyle\leq
c_{3}(1\wedge z_{d})^{q-\alpha+{\varepsilon}}(1\wedge
w_{d})^{q-\alpha+{\varepsilon}}.$

Here in the first inequality above we used the fact that
$p^{\kappa}(1,x,y)\leq p^{C(\alpha,q-\alpha+{\varepsilon},{\cal B})}(1,x,y)$,
which is a consequence of $C(\alpha,q,{\cal
B})>C(\alpha,q-\alpha+{\varepsilon},{\cal B})$. Therefore, by the semigroup
property and symmetry, we get

$\displaystyle
p^{\kappa}(1,x,y)=16\int_{0}^{1/4}\int_{0}^{1/4}p^{\kappa}(1,x,y)dsdu$
$\displaystyle=16\int_{0}^{1/4}\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,x,z)p^{\kappa}(1-s-u,z,w)p^{\kappa}(u,y,w)dzdwdsdu$
$\displaystyle\leq
16c_{3}\bigg{(}\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,x,z)(1\wedge
z_{d})^{q-\alpha+{\varepsilon}}dzds\bigg{)}\bigg{(}\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(u,y,w)(1\wedge
w_{d})^{q-\alpha+{\varepsilon}}dwdu\bigg{)}.$

Thus, to conclude (5.6) by induction, it suffices to show that there exists a
constant $c_{4}>0$ such that

$\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,v,z)(1\wedge
z_{d})^{q-\alpha+{\varepsilon}}dzds\leq c_{4}(1\wedge v_{d})^{q},\quad
v\in{\mathbb{R}}^{d}_{+}.$ (5.14)

By Lemma 2.2, we can assume $\widetilde{v}=0$. If $v\notin U(2^{-4})$, then we
get

$(1\wedge v_{d})^{q}\geq 2^{-5q}\geq
2^{-5q}\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,v,z)dzds\geq
2^{-5q}\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,v,z)(1\wedge
z_{d})^{q-\alpha+{\varepsilon}}dzds.$

Otherwise, if $v\in U(2^{-4})$, then by (5.4), Fubini’s theorem, Lemmas 5.5
and 5.12,

$\displaystyle\int_{0}^{1/4}\int_{{\mathbb{R}}^{d}_{+}}p^{\kappa}(s,v,z)(1\wedge
z_{d})^{q-\alpha+{\varepsilon}}dzds$
$\displaystyle\leq\int_{0}^{1/4}{\mathbb{E}}_{v}\left[(1\wedge
Y^{\kappa,U(1),d}_{s})^{q-\alpha+{\varepsilon}}:\tau^{\kappa}_{U(1)}>s\right]ds+\int_{0}^{1/4}{\mathbb{P}}_{v}(Y^{\kappa}_{\tau^{\kappa}_{U(1)}}\in{\mathbb{R}}^{d}_{+})ds$
$\displaystyle\leq{\mathbb{E}}_{v}\int_{0}^{\tau^{\kappa}_{U(1)}}(1\wedge
Y^{\kappa,U(1),d}_{s})^{q-\alpha+{\varepsilon}}ds+\frac{1}{4}{\mathbb{P}}_{v}(Y^{\kappa}_{\tau_{U(1)}}\in{\mathbb{R}}^{d}_{+})\leq
c_{5}v_{d}^{q}.$

This completes the proof. $\Box$

Proof of Proposition 5.1: The assertion is a direct consequence of Lemmas 5.13
and 5.10. $\Box$

In the remainder of this section we complete the proof of Lemma 5.12.

######  Lemma 5.14.

If $d=1\leq\alpha$, then there exists $C>0$ such that for all
$x,y\in(0,\infty)$,

$G^{\kappa}(x,y)\leq C\left(1\wedge\frac{x\wedge
y}{|x-y|}\right)^{q\wedge(\alpha-\frac{1}{2})}(x\vee
y)^{\alpha-1}\log\left(e+\frac{x\vee y}{|x-y|}\right).$ (5.15)

Proof. Let
$\widetilde{q}:=q\wedge(\alpha-\frac{1}{2})\in(0,q]\cap[\alpha-1,\alpha)$. By
symmetry and scaling (3.30), we can assume $|x-y|=1$ and $x\leq y$ without
loss of generality. Then $y=x+1>x\vee 1$. It suffices to show that

$\displaystyle G^{\kappa}(x,y)\leq Cx^{\widetilde{q}}y^{\alpha-1}\log(e+y).$

Note that $1+2\widetilde{q}>1+\widetilde{q}\geq\alpha$. Since
$C(\alpha,\widetilde{q},{\cal B})\leq C(\alpha,q,{\cal B})=\kappa$ implies
that $p^{\kappa}(t,x,y)\leq p^{C(\alpha,\widetilde{q},{\cal B})}(t,x,y)$,
using the fact $y>x\vee 1$ and Lemmas 5.11 and 5.10, we get

$\displaystyle
G^{\kappa}(x,y)\leq\int_{0}^{\infty}p^{C(\alpha,\widetilde{q},{\cal
B})}(t,x,y)dt$ $\displaystyle\leq
c_{1}\left(x^{\widetilde{q}}\int_{0}^{1}t^{(\alpha-\widetilde{q})/\alpha}dt+x^{\widetilde{q}}\int_{1}^{y^{\alpha}}t^{-(1+\widetilde{q})/\alpha}dt+x^{\widetilde{q}}y^{\widetilde{q}}\int_{y^{\alpha}}^{\infty}t^{-(1+2\widetilde{q})/\alpha}dt\right)$
$\displaystyle\leq
c_{1}x^{\widetilde{q}}\left(1+\log(y^{\alpha})+\frac{\alpha}{1+2\widetilde{q}-\alpha}y^{\alpha-\widetilde{q}-1}\right)\leq
c_{2}x^{\widetilde{q}}y^{\alpha-1}\log(e+y).$

The proof is complete. $\Box$

We now improve (5.15) by removing the term $\alpha-\frac{1}{2}$ from the power
of the first factor.

######  Lemma 5.15.

If $d=1\leq\alpha$, then there exists $C>0$ such that for all
$x,y\in(0,\infty)$,

$\displaystyle G^{\kappa}(x,y)\leq C\left(1\wedge\frac{x\wedge
y}{|x-y|}\right)^{q}(x\vee y)^{\alpha-1}\log\left(e+\frac{x\vee
y}{|x-y|}\right).$

Proof. Let $r=2^{-6}$. By symmetry, (3.30) and Lemma 5.14, without loss of
generality, we can assume $|x-y|=1$ and $x<y\wedge(r/2)$. Note that
$y=x+1\in(1,1+r/2)$. It suffices to show that

$\displaystyle G^{\kappa}(x,y)\leq Cx^{q}.$

For $z\in(0,r/2)$ and $w\in(r,\infty)$, we have $z<w/2\leq w-z<w$ and $w\vee
y\asymp w$. Thus, for any $z\in(0,r/2)$, using Lemmas 5.3 and 5.14, we obtain

$\displaystyle\int_{r}^{\infty}G^{\kappa}(w,y){\cal
B}(z,w)(w-z)^{-1-\alpha}dw$ $\displaystyle\leq c_{1}z^{\beta_{1}}(|\log
z|^{\beta_{3}}\vee
1)\int_{r}^{\infty}\frac{G^{\kappa}(w,y)}{w^{1+\alpha+\beta_{1}}}\big{(}1+{\bf
1}_{\\{w\geq 1\\}}(\log w)^{\beta_{3}}\big{)}dw$ $\displaystyle\leq
c_{2}z^{\beta_{1}}|\log
z|^{\beta_{3}}\int_{r}^{\infty}\log\bigg{(}e+\frac{w}{|w-y|}\bigg{)}\frac{\big{(}1+{\bf
1}_{\\{w\geq 1\\}}(\log w)^{\beta_{3}}\big{)}}{w^{2+\beta_{1}}}dw.$

Hence, by the Lévy system formula and Lemma 5.4, we get

$\displaystyle{\mathbb{E}}_{x}\left[G^{\kappa}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}},y);Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\notin(0,r)\right]$
$\displaystyle\leq
c_{2}{\mathbb{E}}_{x}\int^{\tau^{\kappa}_{(0,r/2)}}_{0}(Y^{\kappa,d}_{t})^{\beta_{1}}|\log(Y^{\kappa,d}_{t})|^{\beta_{3}}dt\int_{r}^{\infty}\log\bigg{(}e+\frac{w}{|w-y|}\bigg{)}\frac{\big{(}1+{\bf
1}_{\\{w\geq 1\\}}(\log w)^{\beta_{3}}\big{)}}{w^{2+\beta_{1}}}dw$
$\displaystyle\leq
c_{3}x^{q}\int_{r}^{\infty}\log\bigg{(}e+\frac{w}{|w-y|}\bigg{)}\frac{\big{(}1+{\bf
1}_{\\{w\geq 1\\}}(\log w)^{\beta_{3}}\big{)}}{w^{2+\beta_{1}}}dw.$ (5.16)

Since $w-y\geq w/2$ for $w\in(2y,\infty)$ and $y\in(1,1+r/2)$, using a change
of the variables, we obtain

$\displaystyle\int_{r}^{\infty}\log\bigg{(}e+\frac{w}{|w-y|}\bigg{)}\frac{\big{(}1+{\bf
1}_{\\{w\geq 1\\}}(\log w)^{\beta_{3}}\big{)}}{w^{2+\beta_{1}}}dw$
$\displaystyle\leq\frac{1+(\log(2+r))^{\beta_{3}}}{r^{2+\beta_{1}}}\int_{r}^{2y}\log\bigg{(}e+\frac{2+r}{|w-y|}\bigg{)}dw+\log(e+2)\int_{2y}^{\infty}\frac{1+(\log
w)^{\beta_{3}}}{w^{2+\beta_{1}}}dw$
$\displaystyle\leq\frac{2(1+(\log(2+r))^{\beta_{3}})}{r^{2+\beta_{1}}}\int_{0}^{1+r/2}\log\bigg{(}e+\frac{2+r}{v}\bigg{)}dv+c_{4}\leq
c_{5}.$

Thus, by (5), it holds that

$\displaystyle{\mathbb{E}}_{x}\left[G^{\kappa}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}},y);Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\notin(0,r)\right]\leq
c_{3}c_{5}x^{q}.$ (5.17)

Since $z\mapsto G^{\kappa}(z,y)$ is harmonic in $(0,2r)$ with respect to
$Y^{\kappa}$ and vanishes continuously as $z\to 0$ by Lemma 5.14, we get from
Proposition 5.7 that $G^{\kappa}(z,y)\leq c_{6}G^{\kappa}(r,y)$ for all
$z\in(0,r)$. Therefore, using Lemmas 5.5 and 5.14, since $y\in(1,1+r/2)$, we
obtain

$\displaystyle{\mathbb{E}}_{x}\left[G^{\kappa}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}},y);Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\in(0,r)\right]\leq
c_{6}G^{\kappa}(r,y){\mathbb{P}}_{x}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\in(0,r))$
$\displaystyle\leq
c_{7}\bigg{(}\frac{x}{r}\bigg{)}^{q}\bigg{(}\frac{r}{y-r}\bigg{)}^{q\wedge(\alpha-\frac{1}{2})}y^{\alpha-1}\log\bigg{(}e+\frac{y}{y-r}\bigg{)}\leq
c_{8}x^{q}.$ (5.18)

Combining (5.17) and (5) and using the harmonicity of $z\mapsto
G^{\kappa}(z,y)$ in $(0,2r)$, we arrive at

$\displaystyle
G^{\kappa}(x,y)={\mathbb{E}}_{x}\left[G^{\kappa}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}},y);Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\notin(0,r)\right]+{\mathbb{E}}_{x}\left[G^{\kappa}(Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}},y);Y^{\kappa}_{\tau^{\kappa}_{(0,r/2)}}\in(0,r)\right]\leq
c_{9}x^{q}.$

The proof is complete. $\Box$

Proof of Lemma 5.12 for $d=1\leq\alpha$: Assume that $d=1\leq\alpha$. Since
$D\subset U(2R)=[0,R)$, using Fubini’s theorem and Lemma 5.15, we have

$\displaystyle\int_{0}^{\infty}\int_{D}p^{\kappa,D}(t,x,z)z^{\gamma}dzdt\leq\int_{0}^{R}\int_{0}^{\infty}p^{\kappa,D}(t,x,z)dt\,z^{\gamma}dz\leq\int_{0}^{R}G^{\kappa}(x,z)z^{\gamma}dz$
$\displaystyle\leq
c_{1}\int_{0}^{x/2}\frac{z^{q+\gamma}}{|x-z|^{q}}x^{\alpha-1}\log\bigg{(}e+\frac{x}{|x-z|}\bigg{)}dz+c_{1}\int_{x/2}^{2x}z^{\gamma}(2x)^{\alpha-1}\log\bigg{(}e+\frac{2x}{|x-z|}\bigg{)}dz$
$\displaystyle\quad+c_{1}\int_{2x}^{R}\frac{x^{q}}{|x-z|^{q}}z^{\gamma+\alpha-1}\log\bigg{(}e+\frac{z}{|x-z|}\bigg{)}dz$
$\displaystyle=:c_{1}(I+II+III).$

Since $\gamma+\alpha-q>0$ and $x\leq R/10$, we see that

$\displaystyle I\leq
x^{q+\gamma+\alpha-1}\int_{0}^{x/2}\frac{1}{(x/2)^{q}}\log\bigg{(}e+\frac{x}{(x/2)}\bigg{)}dz\leq
2^{q-1}\log(e+2)x^{\gamma+\alpha}\leq c_{2}R^{\gamma+\alpha-q}x^{q}.$

Next, using the change of the variables $z=xy$, we also get

$\displaystyle II$
$\displaystyle\leq(2x)^{\gamma+\alpha-1}\int_{x/2}^{2x}\log\bigg{(}e+\frac{2x}{|x-z|}\bigg{)}dz=2^{\gamma+\alpha-1}x^{\gamma+\alpha}\int_{1/2}^{2}\log\bigg{(}e+\frac{2}{|1-y|}\bigg{)}dy$
$\displaystyle\leq c_{3}x^{\gamma+\alpha}\leq c_{3}R^{\gamma+\alpha-q}x^{q}.$

Lastly, since $|x-z|\geq z/2$ for all $z\geq 2x$, we obtain

$\displaystyle III\leq
2^{q}\log(e+2)x^{q}\int_{2x}^{R}z^{\gamma+\alpha-q-1}dz\leq
c_{4}R^{\gamma+\alpha-q}x^{q}.$

The proof is complete. $\Box$

##  6\. Sharp lower bounds of heat kernels

In this and the next section, whenever we consider $\overline{Y}$, we assume
that (A1), (A3) and (A4) hold, and whenever we consider $Y^{\kappa}$, we
assume that all (A1)–(A4) hold. Note that (IUBS) holds under this setting by
Lemma 4.2.

We fix $\kappa\in[0,\infty)$. The following notational convenience will be
used throughout this and the next section. When we consider $Y^{\kappa}$, we
assume that $\kappa>0$ if $\alpha\in(0,1]$ (see Lemma 2.3), and denote by $q$
the strictly positive constant $q_{k}\in((\alpha-1)_{+},\alpha+\beta_{1})$
from (1.8). Additionally, we write $Y$, $p(t,x,y)$, $p^{D}(t,x,y)$, $\tau_{D}$
and $\zeta$ instead of $Y^{\kappa}$, $p^{\kappa}(t,x,y)$,
$p^{\kappa,D}(t,x,y)$, $\tau_{D}^{\kappa}$ and $\zeta^{\kappa}$. When we
consider $\overline{Y}$, the letter $q$ denotes 0, and we write $Y$,
$p(t,x,y)$, $p^{D}(t,x,y)$, $\tau_{D}$ and $\zeta$ instead of $\overline{Y}$,
$\overline{p}(t,x.y)$, $\overline{p}^{D}(t,x,y)$, $\overline{\tau}_{D}$ and
$\infty$.

Recall the definitions of $V_{x}(t)$ and $W_{x}(t)$ from (3). We let
$V_{x}:=V_{x}(1)$ and $W_{x}:=W_{x}(1)$.

######  Lemma 6.1.

There exist constants $M>1$ and $C>0$ such that for all $t>0$ and
$x\in{\mathbb{R}}^{d}_{+}$,

$\inf_{z\in W_{x}(t)}p(Mt,x,z)\geq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}t^{-d/\alpha}.$

Proof. When $q=0$, the result is given in Lemma 3.6.

Suppose $q>0$. By the scaling property (3.30), it suffices to prove the lemma
for $t=1$. If $x_{d}\geq 1$, then the result follows from Proposition 3.2.
Assume $x_{d}<1$. By Proposition 3.2, for any $M>1$, there is
$c_{1}=c_{1}(M)>0$ such that

$\displaystyle\inf_{w,z\in W_{x},\,1\leq s\leq M}p(s,w,z)\geq
c_{1}\quad\text{for all $x\in{\mathbb{R}}^{d}_{+}$}.$ (6.1)

Using the strong Markov property and (6.1), we see that for all $M>1$ and
$z\in W_{x}$,

$\displaystyle p(M,x,z)$
$\displaystyle\geq{\mathbb{E}}_{x}\left[\,p(M-\tau_{V_{x}},Y_{\tau_{V_{x}}},z):\tau_{V_{x}}\leq
M-1,\,Y_{\tau_{V_{x}}}\in W_{x}\right]$ $\displaystyle\geq\bigg{(}\inf_{w\in
W_{x},\,1\leq s\leq M}p(s,w,z)\bigg{)}{\mathbb{P}}_{x}\left(\tau_{V_{x}}\leq
M-1,\,Y_{\tau_{V_{x}}}\in W_{x}\right)$ $\displaystyle\geq
c_{1}\left({\mathbb{P}}_{x}(Y_{\tau_{V_{x}}}\in
W_{x})-{\mathbb{P}}_{x}(\tau_{V_{x}(t)}>M-1)\right).$

Note that, by [30, Lemma 5.10] for $\kappa>0$ and [32, Theorem 1.1] for
$\kappa=0$, we have

${\mathbb{P}}_{x}(Y_{\tau_{V_{x}}}\in W_{x})\geq 2c_{2}x_{d}^{q}$

and, by Corollary 5.2, we also have

${\mathbb{P}}_{x}(\tau_{V_{x}}>M-1)\leq{\mathbb{P}}_{x}(\zeta>M-1)\leq
c_{3}(x_{d}/(M-1)^{1/\alpha})^{q}.$

Thus, we can choose $M=1+(c_{3}/c_{2})^{\alpha/q}$ so that
$2c_{2}-c_{3}(M-1)^{-q/\alpha}=c_{2}$, which implies $p(M,x,z)\geq
c_{2}x_{d}^{q}$. $\Box$

For any $b_{1},b_{2},b_{3},b_{4}\geq 0$, we define for $t\geq 0$ and
$x,y\in{\overline{\mathbb{R}}}^{d}_{+}$,

$\displaystyle A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)$
$\displaystyle:=\bigg{(}\frac{(x_{d}\wedge y_{d})\vee
t^{1/\alpha}}{|x-y|}\wedge 1\bigg{)}^{b_{1}}\bigg{(}\frac{(x_{d}\vee
y_{d})\vee t^{1/\alpha}}{|x-y|}\wedge 1\bigg{)}^{b_{2}}$
$\displaystyle\times\log^{b_{3}}\bigg{(}e+\frac{((x_{d}\vee y_{d})\vee
t^{1/\alpha})\wedge|x-y|}{((x_{d}\wedge y_{d})\vee
t^{1/\alpha})\wedge|x-y|}\bigg{)}\,\log^{b_{4}}\bigg{(}e+\frac{|x-y|}{((x_{d}\vee
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$ (6.2)

We first note that
$A_{b_{1},b_{2},b_{3},b_{4}}(0,x,y)=B_{b_{1},b_{2},b_{3},b_{4}}(x,y)$ for
$x,y\in{\mathbb{R}}^{d}_{+}$ and

$\displaystyle A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)\asymp
B_{b_{1},b_{2},b_{3},b_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\quad\text{
for }t\geq 0,x,y\in{\mathbb{R}}^{d}_{+},$ (6.3)

since $a\vee b\asymp a+b$ for all $a,b\geq 0$. Note also that for any $a>0$,
there exists $c>0$ such that

$A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)\geq c(a\wedge 1)^{b_{1}+b_{2}},$ (6.4)

for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$ with $(x_{d}\wedge
y_{d})+t^{1/\alpha}\geq a|x-y|$.

######  Proposition 6.2.

There exists a constant $C>0$ such that for all $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle p(t,x,y)\geq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$

Proof. Without loss of generality, we assume $x_{d}\leq y_{d}$. Let $M>1$ be
the constant in Lemma 6.1. By the semigroup property and Lemma 6.1,

$\displaystyle p(t,x,y)$
$\displaystyle\geq\int_{W_{x}(t/(3M))}\int_{W_{y}(t/(3M))}p(t/3,x,z)\,p(t/3,z,w)\,p(t/3,w,y)dzdw$
$\displaystyle\geq c_{1}t^{2d/\alpha}\left(\inf_{z\in
W_{x}(t/(3M))}p(t/3,x,z)\right)\left(\inf_{w\in
W_{y}(t/(3M))}p(t/3,y,w)\right)$ $\displaystyle\quad\times\left(\inf_{(z,w)\in
W_{x}(t/(3M))\times W_{y}(t/(3M))}p(t/3,z,w)\right)$ $\displaystyle\geq
c_{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\inf_{(z,w)\in
W_{x}(t/(3M))\times W_{y}(t/(3M))}p(t/3,z,w).$ (6.5)

For all $(z,w)\in W_{x}(t/(3M))\times W_{y}(t/(3M))$, we have $z_{d}\geq
x_{d}+5(t/(3M))^{1/\alpha}$, $w_{d}\geq y_{d}+5(t/(3M))^{1/\alpha}$,
$|z-w|\leq|z-x|+|x-y|+|y-w|\leq|x-y|+20(t/(3M))^{1/\alpha}$ and
$|z-w|\geq|x-y|-|z-x|-|y-w|\geq|x-y|-20(t/(3M))^{1/\alpha}$. Hence, if
$|x-y|\leq 21(t/(3M))^{1/\alpha}$, then

$\displaystyle z_{d}\wedge w_{d}\geq
5(\frac{t}{3M})^{1/\alpha}\geq\frac{5}{41}\big{(}(\frac{t}{3M})^{1/\alpha}\vee|z-w|\big{)},\quad(z,w)\in
W_{x}(\frac{t}{3M})\times W_{y}(\frac{t}{3M}).$ (6.6)

If $|x-y|>21(t/(3M))^{1/\alpha}$, then using (A3)(I) in the first inequality
below, $|z-w|\leq 2|x-y|$ and Lemma 10.9 in the second and (6.3) in the third,
we get

$\displaystyle\inf_{(z,w)\in W_{x}(t/(3M))\times W_{y}(t/(3M))}J(z,w)\geq
C_{2}^{-1}\inf_{(z,w)\in W_{x}(t/(3M))\times
W_{y}(t/(3M))}\frac{B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(z,w)}{|z-w|^{d+\alpha}}$
$\displaystyle\geq
c_{3}\frac{B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+5(t/(3M))^{1/\alpha}{\mathbf{e}}_{d},y+5(t/(3M))^{1/\alpha}{\mathbf{e}}_{d})}{|x-y|^{d+\alpha}}\geq
c_{4}\frac{A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)}{|x-y|^{d+\alpha}}.$
(6.7)

Now, by Propositions 3.2 and 4.5, (6.6), (6) and (6.4), we have

$\displaystyle\inf_{(z,w)\in W_{x}(t/(3M))\times W_{y}(t/(3M))}p(t/3,z,w)$
$\displaystyle\geq c_{5}\begin{cases}t^{-d/\alpha}&\text{ if }|x-y|\leq
21(t/(3M))^{1/\alpha},\\\\[6.0pt]
tA_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)|x-y|^{-d-\alpha}&\text{ if
}|x-y|>21(t/(3M))^{1/\alpha}\end{cases}$ $\displaystyle\geq
c_{6}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$
(6.8)

Combining (6.5) and (6), we get the desired result. $\Box$

######  Corollary 6.3.

It holds that for any $t>0$ and $x\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle{\mathbb{P}}_{x}(\zeta>t)\asymp\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}.$

Proof. Using Proposition 6.2 and (6.4), we get

$\displaystyle{\mathbb{P}}_{x}(\zeta>t)$
$\displaystyle\geq\int_{B(x,2t^{1/\alpha}):y_{d}\geq
t^{1/\alpha}}p(t,x,y)dy\geq
c_{1}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}t^{-d/\alpha}\int_{B(x,2t^{1/\alpha}):y_{d}\geq
t^{1/\alpha}}dy$ $\displaystyle\geq
c_{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}.$

Combining the above with Corollary 5.2, we arrive at the result. $\Box$

######  Lemma 6.4.

There exists a constant $C>0$ such that for all $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,

$\displaystyle p(t,x,y)\geq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)$
$\displaystyle\quad\times\big{(}|x-y|^{\alpha}\wedge t\big{)}\int_{(y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}.$

Proof. Case (i): $(x_{d}\wedge y_{d})\vee t^{1/\alpha}\geq|x-y|/4$. In this
case, we see from (6.4) that
$A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$ is bounded below by a
positive constant. Moreover, since
$\sup_{s>0,\,z,w\in{\mathbb{R}}^{d}_{+}}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(s,z,w)\leq
c_{1}<\infty$, we get

$\displaystyle|x-y|^{\alpha}\int_{(y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\leq
c_{1}^{2}|x-y|^{\alpha}\int_{|x-y|/4}^{\infty}\frac{dr}{r^{\alpha+1}}=4^{\alpha}c_{1}^{2}/\alpha.$

Therefore, we get the result from Proposition 6.2 in this case.

Case (ii): $(x_{d}\wedge y_{d})\vee t^{1/\alpha}<|x-y|/4$. For $r>0$, set
$x(r):=x+r{\mathbf{e}}_{d}$ and

$K(r):=\left\\{z=(\widetilde{z},z_{d})\in{\mathbb{R}}^{d}_{+}:|\widetilde{z}-\widetilde{x}|<\frac{r}{2},\,z_{d}=x_{d}+r\right\\}.$

Let

$K:=\left\\{z\in{\mathbb{R}}^{d}_{+}:z\in K(r)\text{ for some }(y_{d}\vee
t^{1/\alpha})\wedge\frac{|x-y|}{4}<r<\frac{|x-y|}{2}\right\\}.$

For any $z=(\widetilde{z},x_{d}+r)\in K$, since