Heat kernels for reflected diffusions with jumps on inner uniform domains
Authors: Chen Zhen-Qing , Kim Panki , Kumagai Takashi , Wang Jian 
Published: March 2022

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain

D
D

in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When

D
D

is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on

D
D

, whose infinitesimal generators are non-local (pseudo-differential) operators

L

\mathcal {L}

on

D
D

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L u left-parenthesis x right-parenthesis equals one half sigma-summation Underscript i comma j equals 1 Overscript d Endscripts StartFraction partial-differential Over partial-differential x Subscript i Baseline EndFraction left-parenthesis a Subscript i j Baseline left-parenthesis x right-parenthesis StartFraction partial-differential u left-parenthesis x right-parenthesis Over partial-differential x Subscript j Baseline EndFraction right-parenthesis plus limit Underscript epsilon down-arrow 0 Endscripts integral Underscript StartSet y element-of upper D colon rho Subscript upper D Baseline left-parenthesis y comma x right-parenthesis greater-than epsilon EndSet Endscripts left-parenthesis u left-parenthesis y right-parenthesis minus u left-parenthesis x right-parenthesis right-parenthesis upper J left-parenthesis x comma y right-parenthesis d y">

L

u
(
x
)

=

1
2

∑

i
,
j
=
1

d

∂

∂

x
i

(

a

i
j

(
x
)

∂
u
(
x
)

∂

x
j

)

+

lim

ε
↓
0

∫

{
y
∈
D
:

ρ
D

(
y
,
x
)
>
ε
}

(
u
(
y
)
−
u
(
x
)
)
J
(
x
,
y
)

d
y

\mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy

\]
satisfying “Neumann boundary condition”. Here,

ρ
D

(
x
,
y
)

\rho _D(x,y)

is the length metric on

D
D

,

A
(
x
)
=
(

a

i
j

(
x
)

)

1
≤
i
,
j
≤
d

A(x)=(a_{ij}(x))_{1\leq i,j\leq d}

is a measurable

d
×
d

d\times d

matrix-valued function on

D
D

that is uniformly elliptic and bounded, and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis x comma y right-parenthesis colon-equal StartFraction 1 Over normal upper Phi left-parenthesis rho Subscript upper D Baseline left-parenthesis x comma y right-parenthesis right-parenthesis EndFraction integral Underscript left-bracket alpha 1 comma alpha 2 right-bracket Endscripts StartFraction c left-parenthesis alpha comma x comma y right-parenthesis Over rho Subscript upper D Baseline left-parenthesis x comma y right-parenthesis Superscript d plus alpha Baseline EndFraction nu left-parenthesis d alpha right-parenthesis comma">

J
(
x
,
y
)
≔

1

Φ
(

ρ
D

(
x
,
y
)
)

∫

[

α
1

,

α
2

]

c
(
α
,
x
,
y
)

ρ
D

(
x
,
y

)

d
+
α

ν
(
d
α
)
,

J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ),

\]
where

ν
\nu

is a finite measure on

[

α
1

,

α
2

]
⊂
(
0
,
2
)

[\alpha _1, \alpha _2] \subset (0, 2)

,

Φ
\Phi

is an increasing function on

[
0
,
∞
)

[ 0, \infty )

with

c
1

e

c
2

r

β

≤
Φ
(
r
)
≤

c
3

e

c
4

r

β

c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }}

for some

β
∈
[
0
,
∞
]

\beta \in [0,\infty ]

, and

c
(
α
,
x
,
y
)

c(\alpha , x, y)

is a jointly measurable function that is bounded between two positive constants and is symmetric in

(
x
,
y
)

(x, y)

.