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이학�‹ 학위|8

Dirichlet heat kernel estimatesfor subordinate Brownian

motion and its perturbation:stable and beyond

(종� �|운 운Ù의 �‹클� 열 커� 근‹치와그것의 -Ù)

2019D 2월

�울�학교 �학원

수‹과학�

0 주 학



(종� �|운 운Ù의 �‹클� 열 커� 근‹치와그것의 -Ù)

지˜교수 김 판 기

이 |8을 이학�‹ 학위|8으\ 제출함

2018D 10월

�울�학교 �학원

수‹과학�

0 주 학0 주 학의 이학�‹ 학위|8을 인준함

2018D 12월

위 원 장 (인)

� 위 원 장 (인)

위 원 (인)

위 원 (인)

위 원 (인)



A dissertation

submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

to the faculty of the Graduate School ofSeoul National University

by

Joohak Bae

Dissertation Director : Professor Panki Kim

Department of Mathematical SciencesSeoul National University

February 2019

Abstract

In this thesis, we first consider a subordinate Brownian motion X with Gaus-

sian components when the scaling order of purely discontinuous part is be-

tween 0 and 2 including 2. We establish sharp two-sided bounds for tran-

sition density of X in Rd and C1,1 open sets. As a corollary, we obtain a

sharp Green function estimates. Second, we show that, when potentials are

in appropriate Kato classes, Dirichlet heat kernel estimates for a large class

of non-local operators are stable under (non-local) Feynman-Kac perturba-

tions. Especially, our operators include infinitesimal generators for killed

subordinate Brownian motions whose scaling order is 2.

Key words: Dirichlet heat kernel, transition density, Laplace exponent,

Levy measure, subordinator, subordinate Brownian motion, Brownian mo-

tion, Green function, non-local operator, Feynman-Kac perturbation, Feynman-

Kac transform

Student Number: 2013-20234

i

Contents

Abstract i

1 Introduction 1

2 Preliminaries 4

3 SBM with Gaussian components 7

3.1 Setting and main results . . . . . . . . . . . . . . . . . . . . . 7

3.2 Heat kernel estimates in Rd . . . . . . . . . . . . . . . . . . . 12

3.2.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Dirichlet heat kernel estimates in C1,1 open sets . . . . . . . . 19

3.3.1 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.2 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Green function estimates . . . . . . . . . . . . . . . . . . . . . 40

4 Feynman-Kac perturbation of DHK of SBM 44

4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Local and Non-local 3P inequalities . . . . . . . . . . . . . . . 49

4.2.1 3P inequality for Rd . . . . . . . . . . . . . . . . . . . 49

4.2.2 Integral 3P inequality for local perturbation . . . . . . 53

4.2.3 Integral 3P inequality for nonlocal perturbation . . . . 59

4.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Large time heat kernel estimates and Green function estimates 78

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

ii

CONTENTS

Abstract (in Korean) 101

Acknowledgement (in Korean) 102

iii

Chapter 1

Introduction

One of the most important notions in probability theory and analysis is the

heat kernel. The transition density p(t, x, y) of a Markov process X is the

heat kernel of infinitesimal generator L of X(also called the fundamental

solution of ∂tu = Lu), whose explicit form does not exists usually. Thus

obtaining sharp estimates of heat kernel p(t, x, y) is a fundamental problem

in both fields. Recently, for a large class of purely discontinuous Markov

processes, the sharp heat kernel estimates were obtained in [3, 6, 7, 15, 16].

A common property of all purely discontinuous Markov processes considered

so far in the estimates of the heat kernel was that the scaling order was always

strictly between 0 and 2. In [29], Ante Mimica succeeded in obtaining sharp

heat kernel estimates for purely discontinuous subordinate Brownian motions

when the scaling order is between 0 and 2 including 2. For heat kernel

estimates of processes with diffusion parts, mixture of Brownian motion and

stable process was considered in [35] and diffusion process with jumps was

considered in [17].

For any open subset D ⊂ Rd, let XD be a subprocess of X killed upon

leaving D and pD(t, x, y) be a transition density of XD. An infinitesimal

generator L|D of XD is the infinitesimal generator L with zero exterior con-

dition. pD(t, x, y) is also called the Dirichlet heat kernel for L|D since it is

the fundamental solution to exterior Dirichlet problem with respect to L|D.

There are many results for Dirichlet heat kernel estimates in open subsets of

Rd (see [4, 8, 9, 10, 11, 12, 14]). In [14], Zhen-Qing Chen, Panki Kim and

1

CHAPTER 1. INTRODUCTION

Renming Song, obtains sharp two-sided estimates for the Dirichlet heat ker-

nels in C1,1 open sets of a large class of subordinate Brownian motions with

Gaussian components. Very recently, In [24], Panki Kim and Ante Mimica

establish sharp two-sided estimates for the Dirichlet heat kernels in C1,1 open

sets of subordinate Brownian motions without Gaussian components whose

scaling order is not necessarily strictly below 2.

In this thesis, we first continue the journey on investigating the sharp

two-sided estimates of heat kernels both in the whole space and C1,1 open

sets. Here, we consider subordinate Brownian motions with Gaussian com-

ponents when the scaling order of purely discontinuous part is between 0 and

2 including 2. Such processes were not considered in [35, 17, 14, 24]. Espe-

cially, due to an additional exponential term and a new function H (which

also appear in the heat kernel estimates in [29]), the heat kernel estimates

of our processes are different from estimates in [14]. (See Theorems 3.1.1,

3.1.3 and 3.2.1.) Specifically, in our setting, [14, (1.8)] does not hold and [14,

Theorem 1.1(i)] is not sharp. We will handle new exponential term and H

carefully. See Figure 3.1 below.

Second, we show that the Dirichlet heat kernel estimates in [24] are sta-

ble under non-local Feynman-Kac perturbations. In [13], Zhen-Qing Chen,

Panki Kim and Renming Song studied Dirichlet heat kernel estimates for

non-local Feynman-Kac semigroups of stable-type discontinuous processes,

which include symmetric stable-like processes on closed d-sets in Rd, killed

symmetric stable processes, censored stable processes in C1,1 open sets and

stable processes with drifts in bounded C1,1 open sets. In fact, our result

in this thesis provides the sharp Dirichlet heat kernel estimates under non-

local Feynman-Kac perturbations in more general discontinuous processes

than processes considered in [13]. See (4.1.2). As an application, by using

Feynman-Kac transform of the killed subordinate Brownian motion in [24],

we obtain Dirichlet heat kernel estimates of jump processes (including non-

Levy processes) whose upper weak scaling index is not necessarily strictly

less than 2. To cover the case that the upper scaling order is 2, we consider

the general form of the Dirichlet heat kernel estimates in [24] which has ex-

ponential terms. The main difference and difficulty, compared with [13], is

to control the exponential terms in the Dirichlet heat kernel estimates. To

2

CHAPTER 1. INTRODUCTION

overcome the difficulty, we use ideas in the proof of [29, Proposition 3.4] and

show in Propositions 4.2.1 and 4.2.2 that one can control constants in the

exponent when t is small enough. Our result also covers heat kernel estimates

in whole space and the large times.

Throughout this thesis the constants r0, R0,Λ0, a1, a2, and Ci, i = 0, 1, 2, ...

will be fixed. While, we use c1, c2, ... to denote generic constants, whose ex-

act values are not important and the labeling of the constants c1, c2, ... starts

anew in the statement of each result and its proof. For a, b ∈ R we denote

a ∧ b := min{a, b} and a ∨ b := max{a, b}. For D ⊂ Rd, we also denote the

distance from x ∈ D and Dc by δD(x). Notation f(x) � g(x) means that

there exists constants c1, c2 > 0 such that c1f(x) ≤ g(x) ≤ c2g(x) in the

common domain of the definition of f and g, f+ = f ∨ 0 and f− = −(f ∧ 0).

3

Chapter 2

Preliminaries

Let S = (St)t≥0 be a subordinator (increasing 1-dimensional Levy process)

whose Laplace transform of St is of the form

Ee−λSt = e−tψ(λ), λ > 0.

where ψ is called the Laplace exponent of S. It is well known that ψ is a

Bernstein function with ψ(0+) = 0, that is (−1)nψ(n) ≤ 0, for all n ≥ 1.

Thus it has a representation

ψ(λ) = bλ+ φ(λ) with φ(λ) :=

ˆ(0,∞)

(1− e−λt)µ(dt). (2.0.1)

Here, µ is a Levy measure of S satisfying´(0,∞)

(1 ∧ t)µ(dt) <∞.

Let X = (Xt)t≥0 = (WSt)t≥0 be a subordinate Brownian motion in Rd

with subordinator S = (St)t≥0, where W = (Wt)t≥0 is a Brownian motion

in Rd independent of S. Then X is rotationally invariant Levy process in

Rd whose characteristic function is ψ(|ξ|2) = b|ξ|2 + φ(|ξ|2). We denote

H(λ) := φ(λ) − λφ′(λ). The Levy density (jumping kernel) J of X is given

by

J(x) = j(|x|) =

ˆ ∞0

(4πt)−d/2e−|x|2/4tµ(dt).

The function J(x) determines a Levy system for X: for any non-negative

measurable function f on R+ × Rd × Rd with f(s, y, y) = 0 for all y ∈ Rd,

4

CHAPTER 2. PRELIMINARIES

any stopping time T (with respect to the filtration of X) and any x ∈ Rd,

Ex

[∑s≤T

f(s,Xs−, Xs)

]= Ex

[ˆ T

0

(ˆRdf(s,Xs, y)J(Xs − y)dy

)ds

].

(2.0.2)

We first introduce the following scaling conditions for a function f : (0,∞)→(0,∞).

Definition 2.0.1. Suppose that f is a positive Borel function on (0,∞).

(1) We say that f satisfies La(γ, CL) if there exist a ≥ 0, γ > 0, and CL ∈(0, 1] such that

f(λx)

f(λ)≥ CLx

γ for all λ > a and x ≥ 1,

(2) We say that f satisfies Ua(δ, CU) if there exist a ≥ 0, δ > 0, and CU ∈[1,∞) such that

f(λx)

f(λ)≤ CUx

δ for all λ > a and x ≥ 1.

Remark 2.0.2. According to [24, Remark 2.2], if we assume in addition f

is increasing, then the following holds.

(1) If f satisfies Lb(γ, CL) with b > 0 then f satisfies La(γ, (ab)γCL) for all

a ∈ (0, b]:f(λx)

f(λ)≥(ab

)γCLx

γ, x ≥ 1, λ ≥ a.

(2) If f satisfies Ub(δ, CU) with b > 0 then f satisfies Ua(δ,f(b)f(a)

CU) for all

a ∈ (0, b]:f(λx)

f(λ)≤ f(b)

f(a)CUx

δ, x ≥ 1, λ ≥ a.

The following are properties of φ and H. We will be used several times

in this thesis.

5

CHAPTER 2. PRELIMINARIES

Lemma 2.0.3 ([29, Lemma 2.1(a)]). For any λ > 0 and x ≥ 1,

φ(λx) ≤ xφ(λ) and H(λx) ≤ x2H(λ) .

Lemma 2.0.4 ([29, Lemma 2.1(b)]). For a ≥ 0 if H satisfies La(γ, CL)

(resp. Ua(δ, CU)), then φ satisfies La(γ, CL)(resp. Ua(δ ∧ 1, CU)).

Lemma 2.0.5 ([29, Lemma 3.1]). (i) Let l := limλ→0+ φ(λ) and u :=

limλ→∞ φ(λ). Then for any l < λ < u and x ≥ 1 such that λx < u we

have φ−1(λx)φ−1(λ)

≥ x.

(ii) If φ satisfies La(γ, CL) for some a ≥ 0, then

φ−1(λx)

φ−1(λ)≤ C

−1/γL x1/γ for all λ > φ(a), x ≥ CL.

In this thesis, we call D ⊂ Rd (when d ≥ 2) is a C1,1 open set with

C1,1 characteristics (R0,Λ0), if there exists a localization radius R0 > 0 and

a constant Λ0 > 0 such that for every z ∈ ∂D there exist a C1,1-function

ϕ = ϕz : Rd−1 → R satisfying ϕ(0) = 0, ∇ϕ(0) = (0, ..., 0), ||∇ϕ||∞ ≤ Λ0,

|∇ϕ(x) − ∇ϕ(w)| ≤ Λ0|x − w| and an orthonormal coordinate system CSzof z = (z1, · · · , zd−1, zd) := (z, zd) with origin at z such that D ∩ B(z, R0) =

{y = (y, yd) ∈ B(0, R0) in CSz : yd > ϕ(y)}. The pair (R0,Λ0) will be called

the C1,1 characteristics of the open set D. By a C1,1 open set in R with a

characteristic R0 > 0, we mean an open set that can be written as the union

of disjoint intervals so that the infimum of the lengths of all these intervals

is at least R0 and the infimum of the distances between these intervals is at

least R0.

6

Chapter 3

Subordinate Brownian motions

with Gaussian components

3.1 Setting and main results

In this chapter, we assume that S = (St)t≥0 is a subordinator whose Laplace

exponent ψ is λ + φ(λ), where φ is defined in (2.0.1). Let X be a sub-

ordinate Brownian motion in Rd with subordinator S = (St)t≥0. Then X

is rotationally invariant Levy process in Rd whose characteristic function

is ψ(|ξ|2) = |ξ|2 + φ(|ξ|2). One can view X as an independent sum of a

Brownian motion and purely discontinuous subordinate Brownian motion

i.e., Xt = Bt + Yt where B is a Brownian motion and Y is a subordinate

Brownian motion, independent of B, with subordinator T whose Laplace ex-

ponent of T is φ. If the scaling order of φ is 2, one can say that the process

X is very close to Brownian motion. (See Corollary 3.1.2.)

Throughout this chapter we denote p(2)(t, x) the transition density of B

(and W ). i.e.,

p(2)(t, x) = (4πt)−d/2 exp(−|x|2

4t).

We assume that µ(0,∞) =∞ and denote q(t, x) the transition density of Y

7

CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS

and p(t, x) the transition density of X. p(t, x) and q(t, x) are of the forms

p(t, x) =

ˆ(0,∞)

(4πs)−d/2e−|x|24s P(St ∈ ds),

q(t, x) =

ˆ(0,∞)

(4πs)−d/2e−|x|24s P(Tt ∈ ds) (3.1.1)

for x, y ∈ Rd and t > 0 . These imply that for all t > 0, p(t, x) ≤ p(t, y) and

q(t, x) ≤ q(t, y) if |x| ≥ |y|.The following is the first main result of this chapter.

Theorem 3.1.1. Let X = (Xt)t≥0 be a subordinate Brownian motion whose

characteristic exponent is ψ(|ξ|2) = |ξ|2 + φ(|ξ|2).

(1) For any t > 0 and x ∈ Rd,

p(t, x) ≤ q(t, 0) ∧ p(2)(t, 0) ∧ (p(2)(t, x/2) + q(t, x/2)).

(2) Suppose H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2 for some a > 0.

For every T > 0, there exists positive constant c1 such that for all 0 < t ≤ T

and x ∈ Rd,

p(t, x) ≥ c1

(p(2)(t, 0) ∧

(p(2)(t, 2x) + q(t, 2x)

)).

(3) Suppose H satisfies L0(γ, CL) and U0(δ, CU) with δ < 2. Then there

exists positive constant c2 such that for all t > 0 and x ∈ Rd,

p(t, x) ≥ c2

(q(t, 0) ∧ p(2)(t, 0) ∧

(p(2)(t, 2x) + q(t, 2x)

)).

As an application, we obtain sharp two-sided estimate for Green function

of transient subordinate Brownian motion X = (Xt)t≥0 (d ≥ 3). If X is

transient, then the following Green function is well-defined and finite.

G(x, y) = G(x− y) =

ˆ ∞0

p(t, x− y)dt, x, y ∈ Rd, x 6= y.

Corollary 3.1.2. Let d ≥ 3. Suppose H satisfies L0(γ, CL) and U0(δ, CU)

8


with δ < 2. Then

G(x) � |x|−d(|x|2 ∧ φ(|x|−2)−1

), x ∈ Rd.

Consider φ is given in Example 3.1.6 (2) below. Then

|x|−dφ(|x|−2)−1 �

{|x|−d+2 log 1

|x| |x| < 12

|x|−d+2 |x| ≥ 12.

Thus G(x) � G(2)(x), x ∈ Rd where G(2)(x) = c|x|−d+2 is the Green function

of the Brownian motion. This shows that how close this process is to the

Brownian motion and Green function estimates may not detect the difference

between our X and the Brownian motion.

Let D ⊂ Rd (when d ≥ 2) be a C1,1 open set with C1,1 characteristics

(R0,Λ0). Throughout this chapter we denote pD(t, x, y) the transition density

of XD. The following are the second main results in this chapter.


characteristic exponent is ψ(|ξ|2) = |ξ|2+φ(|ξ|2). Suppose H satisfies La(γ, CL)

and Ua(δ, CU) with δ < 2 for some a > 0 and D is a bounded C1,1 open set

in Rd with characteristics (R0,Λ0). Then for every T > 0 there exist positive

constants c1, c2, c3, aU , aL such that

(1) For any (t, x, y) ∈ (0, T ]×D ×D, we have

pD(t, x, y) ≤ c1

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧ (p(2)(t, c2(x− y)) +

tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1))

).

(3.1.2)

9



pD(t, x, y) ≥ c−11

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧ (p(2)(t, c3(x− y)) +

tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aL|x−y|2φ−1(t−1))

).

(3.1.3)

(3) For any (t, x, y) ∈ [T,∞)×D ×D, we have

pD(t, x, y) � e−λ1tδD(x)δD(y),

where −λ1 < 0 is the largest eigenvalue of the generator of XD.

We say that the path distance in a domain (connected open set) U is

comparable to the Euclidean distance with characteristic λ0 if for every x and

y in U there is a rectifiable curve l in U which connects x to y such that the

length of l is less than or equal to λ0|x − y|. Clearly, such a property holds

for all bounded C1,1 domains, C1,1 domains with compact complements, and

domain consisting of all the points above the graph of a bounded globally

C1,1 function.


characteristic exponent is ψ(|ξ|2) = |ξ|2+φ(|ξ|2). Suppose H satisfies L0(γ, CL)

and U0(δ, CU) with δ < 2 and D is an unbounded C1,1 open set in Rd with

characteristics (R0,Λ0). Then for every T > 0 there exists c1, c2, c3, aU , aLsuch that


pD(t, x, y) ≤ c1

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧ (p(2)(t, c2(x− y)) +

tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1))

).

(2) If the path distance in each connected component of D is comparable to

the Euclidean distance with characteristic λ0, then for any (t, x, y) ∈ (0, T ]×

10


D ×D, we have

pD(t, x, y) ≥ c−11

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧ (p(2)(t, c3(x− y)) +

tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aL|x−y|2φ−1(t−1))

).

Define GD(x, y) =´∞0pD(t, x, y)dt, Green function of XD. The following

is Green function estimate of XD.

Corollary 3.1.5. Suppose H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2

for some a > 0 and D is a bounded C1,1 open set in Rd with characteristics

(R0,Λ0). Then

GD(x, y) � gD(x, y), x, y ∈ D,

where

gD(x, y) :=

1

|x−y|d−2

(1 ∧ δD(x)δD(y)

|x−y|2

)when d ≥ 3,

log

(1 + δD(x)δD(y)

|x−y|2

)when d = 2,(

δD(x)δD(y))1/2 ∧ δD(x)δD(y)

|x−y| when d = 1.

(3.1.4)

Denote by G(2)D (x, y) the Green function of Brownian motion in D. It is

known (see [20]) that G(2)D � gD(x, y) when x and y are in the same compo-

nent of D, and G(2)D (x, y) = 0 otherwise. Thus when D is a bounded C1,1

connected open subset of Rd, GD(x, y) � G(2)D (x, y), while our heat kernel

estimates (Theorem 3.1.3) are different from heat kernel estimates of Brow-

nian motion in D.

These are examples where the scaling order of φ is not strictly between 0

and 2.

Example 3.1.6. (1) Let φ(λ) = λlog(1+λβ/2)

, where β ∈ (0, 2). Then

φ−1(λ) �

{λ

22−β 0 < λ < 2

λ log λ λ ≥ 2H(λ) �

{λ1−β/2 0 < λ < 2

λ(log λ)2

λ ≥ 2

11


Hence, H satisfies L0(γ, CL) and U0(δ, CU) with some γ, CL, CU and δ < 2.

(2) Let φ(λ) = λlog(1+λ)

− 1. Then

φ−1(λ) �{λ 0 < λ < 2

λ log λ λ ≥ 2H(λ) �

{λ2 0 < λ < 2

λ(log λ)2

λ ≥ 2

Hence, H satisfies L0(γ, CL) and U2(δ, CU) with some γ, CL, CU and δ < 2.

Suppose D is a bounded C1,1 open set with diam(D) < 1/2 and ψ(λ) =

λ + φ(λ), where φ is the one in above two cases. Then for t < 1/2, there

exist positive constants c1, c2, c3, aU , and aL such that

pD(t, x, y) ≤ c1

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧

(p(2)(t, c2(x− y))

+t

|x− y|d+2(log 1|x−y|)

2+ t−d/2

(log

1

t

)d/2e−aU

|x−y|2t

log 1t

)),

and

pD(t, x, y) ≥ c−11

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧

(p(2)(t, c3(x− y))

+t

|x− y|d+2(log 1|x−y|)

2+ t−d/2

(log

1

t

)d/2e−aL

|x−y|2t

log 1t

)).

3.2 Heat kernel estimates in Rd

In this section we obtain estimates of transition density of the subordinate

Brownian motion X. The following are heat kernel estimates for q(t, x),

which is transition density of Y . Recall that Y is a subordinate Brownian

motion with subordinator T whose Laplace exponent of T is φ and H(λ) =

φ(λ)− λφ′(λ).

Theorem 3.2.1 ([29, 24]). (i) If φ satisfies La(γ, CL) for some a > 0, then

for every T > 0 there exist C = C(T ) > 1 and aU > 0 such that for all t ≤ T

12


and x ∈ Rd,

q(t, x) ≤ C(φ−1(t−1)d/2 ∧

(t|x|−dH(|x|−2) + φ−1(t−1)d/2e−aU |x|

2φ−1(t−1))),

(3.2.1)

and

q(t, x) ≥ C−1φ−1(t−1)d/2, if tφ(|x|−2) ≥ 1. (3.2.2)

Consequently, the Levy density (jumping kernel) J satisfies

J(x) = limt→0

q(t, x)/t ≤ C|x|−dH(|x|−2).

Furthermore, if a = 0, then (3.2.1) and (3.2.2) hold for every t > 0 and

x ∈ Rd.

(ii) If H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2 for some a > 0, then

for every T,M > 0 there exist C = C(a, γ, CL, δ, CU , T,M) > 0 and aL > 0

such that for all t ≤ T and |x| < M ,

q(t, x) ≥ C−1(φ−1(t−1)d/2 ∧

(t|x|−dH(|x|−2) + φ−1(t−1)d/2e−aL|x|

2φ−1(t−1))).

(3.2.3)

Consequently, the Levy density J satisfies

J(x) = limt→0

q(t, x)/t � |x|−dH(|x|−2), |x| < M. (3.2.4)

Furthermore, if a = 0, then (3.2.3) and (3.2.4) hold for all t > 0 and x ∈ Rd.

We will use following formula of transition density p(t, x) of X, which is

given by

p(t, x) =

ˆRdp(2)(t, x− y)q(t, y)dy.

3.2.1 Upper bounds

In this subsection we will prove the upper bounds for the transition density.

13


Lemma 3.2.2. For any t > 0 and x ∈ Rd,

p(t, x) ≤ q(t, 0) ∧ p(2)(t, 0) ∧ (p(2)(t, x/2) + q(t, x/2)).

Proof. By the radial monotonicity,

p(t, x) =

ˆRdp(2)(t, x− y)q(t, y)dy ≤ p(2)(t, 0) = c1t

−d/2

Similarly, p(t, x) ≤ q(t, 0). Since

p(t, x) =

ˆRdp(2)(t, x− y)q(t, y)dy

=

ˆ|y−x|>|x|/2

p(2)(t, x− y)q(t, y)dy +

ˆ|y−x|≤|x|/2

p(2)(t, x− y)q(t, y)dy

≤ˆ|y−x|>|x|/2

p(2)(t, x− y)q(t, y)dy +

ˆ|y|≥|x|/2


≤ p(2)(t, x/2) + q(t, x/2),

we obtain this lemma. 2

Remark 3.2.3. By Lemma 3.2.2 and (3.2.1), if φ satisfies La(γ, CL) for

some a > 0, then for 0 < t ≤ T and x ∈ Rd

p(t, x) ≤ c1

(t−d/2∧(t−d/2e−|x|

2/(c2t)+tH(|x|−2)|x|d

+φ−1(t−1)d/2e−c3|x|2φ−1(t−1))

),

and if φ satisfies L0(γ, CL), then for t > 0 and x ∈ Rd

p(t, x) ≤c4(

(t−d/2 ∧ φ−1(t−1)d/2)

∧ (t−d/2e−|x|2/(c5t) +

tH(|x|−2)|x|d

+ φ−1(t−1)d/2e−c6|x|2φ−1(t−1))

).

3.2.2 Lower bounds

In this subsection we will prove the lower bounds for the transition density.

We provide the proof for the case a > 0 only.

14


Lemma 3.2.4. Suppose φ satisfies La(γ, CL) for some a > 0 (L0(γ, CL),

respectively). For T > 0 there exists a positive constant c such that for

0 < t ≤ T (t > 0, respectively) and x ∈ Rd satisfying tφ(|x|−2) ≤ 1,

p(t, x) ≥ cp(2)(t, 2x).

Proof. If |y| ≤ φ−1(t−1)−1/2, then |y| ≤ φ−1(t−1)−1/2 ≤ |x|. Therefore

|y − x| ≤ 2|x| and hence exp(−|x − y|2/(4t)) ≥ exp(−|x|2/t). By (3.2.2),

q(t, y) ≥ C−1φ−1(t−1)d/2 for 0 < t ≤ T . Thus,

p(t, x) ≥ˆB(0,φ−1(t−1)−1/2)


≥ c1(4πt)−d/2 exp(−|x|2/t)φ−1(t−1)d/2(φ−1(t−1)−1/2)d = c2p

(2)(t, 2x).

2

Lemma 3.2.5. There exists a positive constant c such that for any t > 0

and x ∈ Rd satisfying t ≤ |x|2,

p(t, x) ≥ cq(t, 3x/2).

Proof. Note that if |x − y| ≤ |x|/2, then |y| ≤ 3|x|/2 hence q(t, y) ≥q(t, 3x/2). Using this and change of variable, we have

p(t, x) = q(t, 3x/2)

ˆRd

q(t, y)

q(t, 3x/2)p(2)(t, x− y)dy

≥ q(t, 3x/2)

ˆB(x,|x|/2)

p(2)(t, x− y)dy

= q(t, 3x/2)

ˆB(0,t−1/2 |x|

2)

p(2)(1, u)du

≥(ˆ

B(0,1/2)

p(2)(1, u)du

)q(t, 3x/2) = c1q(t, 3x/2).

2

15


Lemma 3.2.6. Suppose H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2 for

some a > 0 (L0(γ, CL) and U0(δ, CU), respectively). For T,M > 0 there exists

a constant c such that for all 1 ≤ t ≤ T and x ∈ Rd satisfying |x| < M/2

(t ≥ 1 and x ∈ Rd, respectively) and tφ(|x|−2) ≥ 1,

p(t, x) ≥ cq(t, 0).

Proof. Assume tφ(|x|−2) ≥ 1 and let b = Mφ−1(T−1)1/2/2. Note that we

have q(t, 0) ≤ Cφ−1(t−1)d/2 by (3.2.1).

If |y − x| ≤ bφ−1(t−1)−1/2, then |y| ≤ |x − y| + |x| ≤ (b + 1)φ−1(t−1)−1/2

and |y| ≤ |x− y|+ |x| ≤ bφ−1(t−1)−1/2 + |x| ≤M . Thus by (3.2.3), we have

that for |y − x| ≤ bφ−1(t−1)−1/2,

q(t, y) ≥ C−1φ−1(t−1)d/2e−aL|y|2φ−1(t−1) ≥ c1φ

−1(t−1)d/2.

Therefore, using the above inequality and change of variables, we have

p(t, x) ≥ˆ|x−y|≤bφ−1(t−1)−1/2


≥ c2φ−1(t−1)d/2

ˆ|x−y|≤bφ−1(t−1)−1/2

p(2)(t, x− y)dy

= c2φ−1(t−1)d/2

ˆ|u|≤bt−1/2φ−1(t−1)−1/2

p(2)(1, u)du

≥ c2φ−1(t−1)d/2

ˆ|u|≤bφ−1(1)−1/2

p(2)(1, u)du

≥ c3φ−1(t−1)d/2

≥ c4q(t, 0).

In the third inequality, we use φ−1(1)φ−1(t−1)

≥ t which follows from Lemma 2.0.5(i).

2

Lemma 3.2.7. Suppose φ satisfies La(γ, CL) for some a ≥ 0. For T ≥ 1

there exists a positive constant c such that for all t ≤ T and tφ(|x|−2) ≥ 1,

p(t, x) ≥ cp(2)(t, 0).

16


Proof. We may assume that a < φ−1(t−1) by Remark 2.0.2. By Lemma

2.0.5(ii) and Lemma 2.0.5(i), we have for t ≤ T ,

C−1/γL T 1/γφ−1(t−1) ≥ φ−1(Tt−1) ≥ Tt−1φ−1(1). (3.2.5)

If |y| ≤ φ−1(t−1)−1/2, then q(t, y) ≥ C−1φ−1(t−1)d/2 by (3.2.2). Also |x−y| ≤|x|+ |y| ≤ 2φ−1(t−1)−1/2 and (3.2.5) imply

exp(−|x− y|2

4t) ≥ exp(−4φ−1(t−1)−1

4t) ≥ exp (−C−1/γL φ−1(1)−1T 1/γ−1).

Therefore for t ≤ T and tφ(|x|−2) ≥ 1,

p(t, x) ≥ˆ|y|≤φ−1(t−1)−1/2


≥ c1φ−1(t−1)d/2

ˆ|y|≤φ−1(t−1)−1/2

p(2)(t, x− y)dy

≥ c1φ−1(t−1)d/2(4πt)−d/2e−C

−1/γL φ−1(1)−1T 1/γ−1

ˆ|y|≤φ−1(t−1)−1/2

dy

≥ c2(4πt)−d/2 = c2p

(2)(t, 0).

2

Proof of Theorem 3.1.1. By Lemma 3.2.2, we get the upper bound of

p(t, x). Combining Lemmas 2.0.4, 3.2.4, 3.2.5, 3.2.6 and 3.2.7, we get the

lower bound of p(t, x). See Figure 3.1. 2

Note that heat kernel estimate is different to [17, Theorem 1.4], when

tφ(|x|−2) ≤ 1 and |x| > 1. In Figure 3.1, p(2)(t, c1x) additionally appear in

our case.

17


t

|x|

q(t, 0) q(t, 0)

p(2)(t, c1x) + q(t, c2x)

p(2)(t, 0)

p(2)(t, 0)

p(2)(t, c1x)+q(t, c2x)

Figure 3.1: Regions of heat kernel estimates for p(t, x). Dotted line corre-

sponds to t = |x|2 and full line corresponds to tφ(|x|−2) = 1.

18


3.3 Dirichlet heat kernel estimates in C1,1 open

sets

Recall that pD(t, x, y) is the transition density of XD. In this section we

obtain the sharp estimates of pD(t, x, y) in C1,1 open sets.

3.3.1 Lower bounds

In this subsection we derive the lower bound estimate on pD(t, x, y) when D

is a C1,1 open set. When D is unbounded, we assume that the path distance

in each connected component of D is comparable to the Euclidean distance.

Since the proofs are almost identical, we will provide a proof when D is a

bounded C1,1 open set. We will use some relation between killed subordinate

Brownian motions and subordinate killed Brownian motions.

Let Tt be a subordinator whose Laplace exponent φ is given by (2.0.1).

Then t + Tt is a subordinator which has the same law as St. So {Xt; t ≥ 0}starting from x has the same distribution as {Bt+Tt ; t ≥ 0} starting from x.

Suppose that U is an open subset of Rd. We denote by BU the part process

of B killed upon leaving U . The process {ZUt ; t ≥ 0} defined by ZU

t = BUt+Tt

is called a subordinate killed Brownian motion in U . Let qU(t, x, y) be the

transition density of ZU . Denote by ζZ,U the lifetime of ZU . Clearly, ZUt =

Bt+Tt for every t ∈ [0, ζZ,U). Therefore we have

pU(t, z, w) ≥ qU(t, z, w) for (t, z, w) ∈ (0,∞)× U × U.

In the next proposition we will use [29, Proposition 2.4]. Note that there

is a typo in [29, Proposition 2.4]. αφ−1(β−1) in the display there should be

αφ−1(βt−1).

Proposition 3.3.1. Suppose that D is a C1,1 open set in Rd with characteris-

tics (R0,Λ0). If D is bounded, we assume that φ satisfies La(γ, CL) for some

a > 0. If D is unbounded, we assume that φ satisfies L0(γ, CL) and the path

distance in each connected component of D is comparable to the Euclidean

distance with characteristic λ0. For any T > 0 there exist positive constants

c1 = c1(R0,Λ0, λ0, T, φ) and c2 = (R0,Λ0, λ0) such that for all t ∈ (0, T ] and

19


x, y in the same connected component of D,

pD(t, x, y) ≥ c1

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)φ−1(t−1)d/2e−c2|x−y|

2φ−1(t−1).

Proof. Suppose that x and y are in the same component of D and ρ ∈ (0, 1)

is the constant in [29, Proposition 2.4] for α = 2 and β = 1. Without loss

of generality, let T ≥ 1 and a satisfies T < ρφ(a)−1 by Remark 2.0.2. Let

pD(t, z, w) be the transition density of BD (killed Brownian motion) and

qD(t, x, y) be the transition density of BDSt

(subordinate killed Brownian mo-

tion). By [19, Theorem 3.3] (see also [36, Theorem 1.2]) (where the com-

parability condition on the path distance in each component of D with the

Euclidean distance is used if D is unbounded), there exists positive con-

stants c3 = c3(R0,Λ0, λ0, T, φ) and c4 = c4(R0,Λ0, λ0) such that for any

(s, z, w) ∈ (0, φ−1(ρT−1)−1]×D ×D,

pD(s, z, w) ≥ c3

(1 ∧ δD(z)√

s

)(1 ∧ δD(w)√

s

)s−d/2e−c4|x−w|

2/s.

(Although not explicitly mentioned in [19], a careful examination of the

proofs in [19] reveals that the constants c3 and c4 in the above lower bound

estimate can be chosen to depend only on (R0,Λ0, λ0, T, φ) and (R0,Λ0, λ0),

respectively.)

We have that for 0 < t ≤ T ,

pD(t, x, y) ≥ qD(t, x, y)

=

ˆ(0,∞)

pD(s, x, y)P(St ∈ ds)

≥ c3

ˆ[φ−1(t−1)−1/2,φ−1(ρt−1)−1]

(1 ∧ δD(x)√

s

)(1 ∧ δD(y)√

s

)s−d/2e−c4

|x−y|2s P(St ∈ ds)

≥ c3

(1 ∧ δD(x)√

φ−1(ρt−1)−1

)(1 ∧ δD(y)√

φ−1(ρt−1)−1

)φ−1(ρt−1)d/2e−2c4|x−y|

2φ−1(t−1)

× P(2−1φ−1(t−1)−1 ≤ St ≤ φ−1(ρt−1)−1).

20


Since 0 < t < ρφ(a)−1, using the Lemma 2.0.5(ii), we have

φ−1(ρt−1) = φ−1(t−1)φ−1(ρt−1)

φ−1(t−1)≥ C

1/γL ρ1/γφ−1(t−1).

Using this and (3.2.5), we have

φ−1(ρt−1) ≥ C2/γL ρ1/γT 1−1/γφ−1(1)t−1.

Using the last two displays and [29, Proposition 2.4] we get

pD(t, x, y) ≥ c5

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)φ−1(ρt−1)d/2e−2c4|x−y|

2φ−1(t−1)

× P(2−1φ−1(t−1)−1 ≤ St ≤ φ−1(ρt−1)−1)

≥ c6

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)Cd/2γL ρd/2γφ−1(t−1)d/2e−2c4|x−y|

2φ−1(t−1)τ,

where τ is the constant in [29, Proposition 2.4] for α = 2 and β = 1. 2

Lemma 3.3.2. For any positive constants a, b and T , there exists c > 0 such

that for all z ∈ Rd and 0 < t ≤ T ,

infy∈B(z,at1/2/2)

Py(τB(z,at1/2) > bt) ≥ c

Proof. See [14, Lemma 2.3]. 2

Although the proof of the following Lemma is similar to that of Lemma

2.4 of [14], we give the proof again to make the thesis self-contained.

Lemma 3.3.3. Suppose H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2

for some a > 0 (L0(γ, CL) and U0(δ, CU), respectively). Then for every

T > 0,M > 0 and b > 0 there exists c > 0 such that we have that for all

t ∈ (0, T ], and u, v ∈ Rd satisfying |u− v| ≤M/2 (u, v ∈ Rd, respectively)

pE(t, u, v) ≥ c(t−d/2 ∧ t|u− v|−dH(|u− v|−2))

21


where E := B(u, bt1/2) ∪B(v, bt1/2).

Proof. We fix b > 0 and u, v ∈ Rd satisfying |u − v| ≤ M/2, and let

rt := bt1/2. If |u − v| ≤ rt/2, by [14, Lemma 2.1] (with√λ = rt and

D = B(0, 1)),

pE(t, u, v) ≥ inf|z|<rt/2

pB(0,rt)(t, 0, z) = inf|z|<rt/2

pB(0,rt)(r2t (t/r

2t ), 0, z)

≥ c1t−d/2

(1 ∧ rt√

t

)(1 ∧ rt

2√t

)e−c2r

2t /t ≥ c3t

−d/2.

If |u − v| ≥ rt/2, since the distance between B(u, rt/8) and B(v, rt/8) is at

least rt/4, we have by the strong Markov property and the Levy system of

X in (2.0.2) that

pE(t, u, v)

≥ Eu[pE(t− τB(u,rt/8), XτB(u,rt/8), v) : τB(u,rt/8) < t,XτB(u,rt/8)

∈ B(v, rt/8)]

=

ˆ t

0

( ˆB(u,rt/8)

pB(u,rt/8)(s, u, w)(ˆ

B(v,rt/8)

J(w, z)pE(t− s, z, v)dz)dw)ds

≥(

infw∈B(u,rt/8)z∈B(v,rt/8)

J(w, z)) ˆ t

0

Pu(τB(u,rt/8) > s)(ˆ

B(v,rt/8)

pE(t− s, z, v)dz)ds

≥ Pu(τB(u,rt/8) > t)(


J(w, z))ˆ t

0

ˆB(v,rt/8)

pB(v,rt/8)(t− s, z, v)dzds

= P0(τB(0,rt/8) > t)(


j(|w − z|))ˆ t

0

P0(τB(0,rt/8) > s)

≥ t(P0(τB(0,rt/8) > t))2(


j(|w − z|))

≥ c5t(


j(|w − z|)).

In the last inequality we have used Lemma 3.3.2. Note that if w ∈ B(u, rt/8)

22


and z ∈ B(v, rt/8), then

|w − z| ≤ |u− w|+ |u− v|+ |v − z| ≤ |u− v|+ rt4≤ 2|u− v|.

Thus using (3.2.4) and Lemma 2.0.3 we have

pE(t, u, v) ≥ c5tj(2|u− v|) ≥ c6t2−d|u− v|−dH(|u− v|−2/4)

≥ 2−d−4c6t|u− v|−dH(|u− v|−2).

2

The next lemma say that if x and y are far away, the jumping kernel

component dominates the Gaussian component and another off-diagonal es-

timate component.

Lemma 3.3.4. Suppose φ satisfies La(γ, CL) for some a ≥ 0. For any

given positive constants c1, c2, R and T , there is a positive constant c3 =

c3(R, T, c1, c2) so that

t−d/2e−r2/(c1t) + φ−1(t−1)d/2e−c2r

2φ−1(t−1) ≤ c3tr−dH(r−2)

for every r ≥ R and t ∈ (0, T ].

Proof. By Lemma 2.0.3 there exist c4 > 0 and c5 > 0 such that

r−dH(r−2) ≥ H(1)r−d−4 ≥ c4e−c5r for every r > 1.

For r > 1 ∨ (2c1c5T ) ∨ 2c5c2φ−1(T−1)−1 and t ∈ (0, T ], we have following

inequalities

r2/(2c1t) > c5r, c2r2φ−1(t−1)/2 > c5r,

t−d/2−1e−r2/(2c1t) ≤ t−d/2−1e−1/(2c1t) ≤ sup

0<s≤Ts−d/2−1e−1/(2c1s) =: c6 <∞,

23


and

φ−1(t−1)d/2t−1e−c2r2φ−1(t−1)/2

≤ sup0<s≤T

φ−1(s−1)d/2s−1e−c2φ−1(s−1)/2

≤ T 1/γ−1C−1/γL φ−1(1)−1 sup

0<s≤Tφ−1(s−1)d/2+1e−c2φ

−1(s−1)/2 =: c7 <∞.

In the last inequality we have used (3.2.5). (Without loss of generality, we can

assume that a ≤ φ−1(T−1).) Therefore when r > 1∨(2c1c5T )∨ 2c5c2φ−1(T−1)−1

and t ∈ (0, T ], we have

t−d/2e−r2/(c1t) ≤ c6te

−r2/(2c1t) ≤ c6te−c5r ≤ (c6/c4)tr

−dH(r−2)

and

φ−1(t−1)d/2e−c2r2φ−1(t−1) ≤ c7te

−c2r2φ−1(t−1)/2 ≤ c7te−c5r ≤ (c7/c4)tr

−dH(r−2).

When R ≤ r ≤ 1 ∨ (2c1c5T ) ∨ 2c5c2φ−1(T−1)−1 and t ∈ (0, T ], clearly

t−d/2e−r2/(c1t) ≤ t

(sups≤T

s−d/2−1e−R2/(c1s)

)≤ c8tr

−dH(r−2)

and

φ−1(t−1)d/2e−c2r2φ−1(t−1) ≤ t

(sups≤T

φ−1(s−1)d/2s−1e−c2R2φ−1(s−1)

)≤ c9tr

−dH(r−2).

2

Proof of Theorem 3.1.3 (2) and Theorem 3.1.4 (2). Since two proofs

are almost identical, we just prove Theorem 3.1.3 (2). First note that the

distance between two distinct connected components of D is at least R0.

Since D is a C1,1 open set, it satisfies the uniform interior ball condition with

radius r0 = r0(R0,Λ0) ∈ (0, R0]: there exists r0 = r0(R0,Λ0) ∈ (0, R0] such

that for any x ∈ D with δD(x) < r0, there are zx ∈ ∂D so that |x−zx| = δD(x)

24


and that B(x0, r0) ⊂ D for x0 = zx + r0(x − zx)/|x − zx|. Set T0 = (r0/4)2.

Using such uniform interior ball condition, by considering the cases δD(x) <

r0 and δD(x) > r0, there exists L = L(r0) > 1 such that, for all t ∈ (0, T0]

and x, y ∈ D, we can choose ξtx ∈ D ∩B(x, L√t) and ξty ∈ D ∩B(y, L

√t) so

that B(ξtx, 2√t) and B(ξty, 2

√t) are subsets of the connected components of

D that contains x and y, respectively.

We first consider the case t ∈ (0, T0]. Note that by the semigroup prop-

erty,

pD(t, x, y) ≥ˆB(ξtx,

√t)

ˆB(ξty ,

√t)

pD(t/3, x, u)pD(t/3, u, v)pD(t/3, v, y)dudv.

(3.3.1)

For u ∈ B(ξtx,√t), we have

δD(u) ≥√t and |x− u| ≤ |x− ξtx|+ |ξtx − u| ≤ L

√t+√t = (L+ 1)

√t.

Thus by [14, Lemma 2.1], for t ∈ (0, T0],

ˆB(ξtx,

√t)

pD(t/3, x, u)du

≥ c3

(1 ∧ δD(x)√

t

) ˆB(ξtx,

√t)

(1 ∧ δD(u)√

t

)t−d/2e−c4|x−u|

2/tdu

≥ c3

(1 ∧ δD(x)√

t

)t−d/2e−c4(L+1)2|B(ξtx,

√t)| ≥ c5

(1 ∧ δD(x)√

t

). (3.3.2)

Similarly, for t ∈ (0, T0] and v ∈ B(ξty,√t),

ˆB(ξty ,

√t)

pD(t/3, y, v)dv ≥ c5

(1 ∧ δD(y)√

t

). (3.3.3)

25


Using (3.3.1), Lemma 3.3.3, symmetry and (3.3.2)–(3.3.3), we have

pD(t, x, y)

≥ˆB(ξty ,

√t)

ˆB(ξtx,

√t)

pD(t/3, x, u)pB(u,√t/2)∪B(v,

√t/2)(t/3, u, v)pD(t/3, v, y)dudv

≥ c6

ˆB(ξty ,

√t)

ˆB(ξtx,

√t)

pD(t/3, x, u)

(t−d/2 ∧ tH(|u− v|−2)

|u− v|d

)pD(t/3, v, y)dudv

≥ c6

(inf

u∈B(ξtx,√t)

v∈B(ξty ,√t)

(t−d/2 ∧ tH(|u− v|−2)

|u− v|d

))

×ˆB(ξty ,

√t)

ˆB(ξtx,

√t)

pD(t/3, x, u)pD(t/3, v, y)dudv

≥ c6c25

(inf

u∈B(ξtx,√t)

v∈B(ξty ,√t)

(t−d/2 ∧ tH(|u− v|−2)

|u− v|d

))(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

).

(3.3.4)

Suppose that |x− y| ≥√t/8 and t ∈ (0, T0]. Then we have that for (u, v) ∈

B(ξtx,√t)×B(ξty,

√t),

|u− v| ≤ |u− ξtx|+ |ξtx − x|+ |x− y|+ |y − ξty|+ |ξty − v|≤ 2(1 + L)

√t+ |x− y| ≤ (16(1 + L)|x− y|).

Thus using Lemma 2.0.3 we have

inf(u,v)∈B(ξtx,

√t)×B(ξty ,

√t)

(t−d/2 ∧ tH(|u− v|−2)

|u− v|d

)≥ c7

(t−d/2 ∧ tH(|x− y|−2)

|x− y|d

).

Therefore, for |x− y| ≥√t/8 and t ∈ (0, T0]

pD(t, x, y) ≥ c8

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧ tH(|x− y|−2)

|x− y|d

). (3.3.5)

Using the inequality (3.3.5), we will obtain the sharp lower bound esti-

mates by considering the following three cases.

26


Case (1): Suppose that |x − y| ≥√t/8, t ∈ (0, T0], and x and y are con-

tained in same connected component of D. Combining with (3.3.5), Propo-

sition 3.3.1, and [14, Lemma 2.1], we conclude that

pD(t, x, y) ≥ c9

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×((

t−d/2 ∧ tH(|x− y|−2)|x− y|d

)+ φ−1(t−1)d/2e−c10|x−y|2φ−1(t−1) + t−d/2e

− |x−y|2

c11t

)≥ c9

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧ (t−d/2e

− |x−y|2

c11t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c10|x−y|2φ−1(t−1))

).

(3.3.6)

Case (2): Suppose that |x−y| ≥√t/8, t ∈ (0, T0], and x and y are contained

in two distinct connected components of D. By (3.3.5) and Lemma 3.3.4, we

have the same conclusion in (3.3.6).

Case (3): Suppose that |x− y| <√t/8 and t ∈ (0, T0]. In this case x and y

are in the same connected component. For (u, v) ∈ B(ξtx,√t)×B(ξty,

√t),

|u− v| ≤ 2(1 + L)√t+ |x− y| ≤ (2(1 + L) + 8−1)

√t.

Thus by [14, Lemma 2.1], we have that for every (u, v) ∈ B(ξtx,√t) ×

B(ξty,√t),

pD(t/3, u, v) ≥ c12

(1 ∧ δD(u)√

t

)(1 ∧ δD(v)√

t

)t−d/2e−c13|u−v|2/t ≥ c14t

−d/2.

27


Therefore by (3.3.1)–(3.3.3), for t ≤ T0,

pD(t, x, y) ≥ c14c25

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)t−d/2

≥ c14c25

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧

(t−d/2e

− |x−y|2

c15t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c16|x−y|2φ−1(t−1)))

.

(3.3.7)

Combining the above three cases, we get (3.1.3) for t ∈ (0, T0]. When T > T0and t ∈ (T0, T ], observe that T0/3 ≤ t−2T0/3 ≤ T−2T0/3 ≤ (T/T0−2/3)T0,

that is, t− 2T0/3 is comparable to T0/3 with some universal constants that

depend only on T and T0. Using the inequality

pD(t, x, y)

≥ˆB(ξ

T0x ,√T0)

ˆB(ξ

T0y ,√T0)

pD(T0/3, x, u)pD(t− 2T0/3, u, v)pD(T0/3, v, y)dudv

≥ˆB(ξ

T0x ,√T0)

ˆB(ξ

T0y ,√T0)

pD(T0/3, x, u)pB(u,√T0/2)∪B(v,

√T0/2)(t− 2T0/3, u, v)

× pD(T0/3, v, y)dudv

instead of (3.3.1) and following the argument in (3.3.4) and (3.3.5) we have

pD(t, x, y) ≥ c17

(1 ∧ δD(x)√

T0

)(1 ∧ δD(y)√

T0

)(T−d/20 ∧ T0H(|x− y|−2)

|x− y|d

).

Consider the cases |x − y| ≥√T0/8 and |x − y| <

√T0/8 separately and

follow the above three cases. Then since tT0

T≤ T0 < t for t ∈ [T0, T ], we can

obtain (3.1.3) for t ∈ [T0, T ] and hence for t ∈ (0, T ]. 2

3.3.2 Upper bounds

In this subsection we derive the upper bound estimate on pD(t, x, y) when D

is a C1,1 open set(not necessarily bounded). We use the following lemma in

28


[14, Lemma 3.1].

Lemma 3.3.5 ([14, Lemma 3.1]). Suppose that U1, U3, E are open subsets of

Rd with U1, U3 ⊂ E and dist(U1, U3) > 0. Let U2 := E \ (U1 ∪U3). If x ∈ U1

and y ∈ U3, then for every t > 0,

pE(t, x, y) ≤ Px(XτU1

∈ U2

)(sup

s<t,z∈U2

pE(s, z, y)

)+

ˆ t

0

Px(τU1 > s)Py(τE > t− s)ds(

supu∈U1,z∈U3

J(u, z)

)(3.3.8)

≤ Px(XτU1

∈ U2

)(sup

s<t,z∈U2

p(s, z, y)

)+ (t ∧ Ex[τU1 ])

(sup

u∈U1,z∈U3

J(u, z)

).

(3.3.9)

Note that by Remark 3.2.3 or Theorem 3.1.1, there exist positive con-

stants c, aU , and cU such that

p(t, x, y)

≤ c(t−d/2 ∧

(t−d/2e

− |x−y|2

cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1)

)).

(3.3.10)

The boundary Harnack principle for subordinate Brownian motions with

Gaussian components was proved in [27] for any C1,1 open set, see [27, The-

orem 1.2]. In [27, Theorem 1.2], it is assumed that φ is a complete Bernstein

function and that the Levy density µ of S satisfies growth condition near

zero, i.e., for any K > 0, there exists c = c(K) > 1 such that µ(r) ≤ cµ(2r).

Note that in the proof of [27, Theorem 1.2], as a consequence of the

growth condition of Levy density of S and assumption that φ is a complete

Bernstein function, in fact, the following conditions of Levy density j of X

are actually used (see [27, (2.7), (2.8)]):

for any K > 0, there exists c1 = c1(K) > 1 such that

j(r) ≤ c1j(2r), for r ∈ (0, K), (3.3.11)

29


and there exists c2 > 1 such that

j(r) ≤ c2j(r + 1), for r > 1. (3.3.12)

If, instead of the assumption that φ is a complete Bernstein function and Levy

density µ satisfies growth condition near zero, we assume that H satisfies

L0(γ, CL) and U0(δ, CU) with δ < 2, then by (3.2.4), (3.3.11) and (3.3.12)

hold. Thus the boundary Harnack principle still hold. But if we assume that

H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2 for some a > 0, then (3.3.12)

may not holds. Nonetheless, if we only consider harmonic functions not only

vanishing continuously on Dc ∩ B(Q, r), Q ∈ ∂D, but also zero on Dc, then

we don’t need the condition (3.3.12). Thus we have the following modified

theorem.

Theorem 3.3.6. Let D is a C1,1 open set in Rd with characteristics (R0,Λ0).

If D is bounded, then we assume that H satisfies La(γ, CL) and Ua(δ, CU) with

δ < 2 for some a > 0. If D is unbounded, then we assume that H satisfies

L0(γ, CL) and U0(δ, CU) with δ < 2. Then there exists a positive constant

c = c(d,Λ0, R0) such that for r ∈ (0, R0], Q ∈ ∂D and any nonnegative

function f in Rd which is harmonic in D ∩ B(Q, r) with respect to X, zero

on Dc and vanishes continuously on Dc ∩B(Q, r), we have

f(x)

δD(x)≤ c

f(y)

δD(y)for every x, y ∈ D ∩B(Q, r/2). (3.3.13)

Proof. Since we have explained before the statement of the theorem why

theorem holds for the unbounded case, we will just prove the theorem when

D is bounded and H satisfies La(γ, CL) and Ua(δ, CU) with δ < 2 for some

a > 0.

Let Rd+ = {x = (x1, ..., xd−1, xd) := (x, xd) ∈ Rd : xd > 0}, V is the poten-

tial measure of the ladder height process of Xdt , where Xd

t is d-th component

of Xt, and w(x) = V ((xd)+). We first show that [27, Proposition 3.3] holds

under our assumptions, i.e., we claim that for any positive constants r0 and

30


M , we have

supx∈Rd:0<xd<M

ˆB(x,r0)c∩Rd+

w(y)j(|x− y|)dy <∞, (3.3.14)

Once we have (3.3.14), then for f which is harmonic in D ∩B(Q, r) with

respect to X, zero on Dc and vanishes continuously on Dc ∩ B(Q, r) where

r ∈ (0, R0] and Q ∈ ∂D, we can follow the proofs of [27, Theorem 5.3 and

Theorem 1.2] line by line without using (3.3.12).

By [2, Theorem 5, page 79] and [26, Lemma 2.1], V is absolutely continu-

ous and has a continuous and strictly positive density v such that v(0+) = 1.

It is also well known that V is subadditive, i.e., V (s + t) ≤ V (s) + V (t),

s, t ∈ R (See [2, page 74].) and V (∞) = ∞. Without loss of generality we

assume that x = 0. Note that for 0 < xd < M and y ∈ B(x, r0)c,

w(y) = V ((yd)+) ≤ V (|y|) ≤ V (M + |x− y|) ≤ V (M) + V (|x− y|).

(3.3.15)

Let L(r) =´∞rrd−1j(r)dr, then by [5, (2.23)], L(r) ≤ c1/V (r)2. Using

(3.3.15), the integration by parts and [5, (2.23)] twice, we have

supx∈Rd:0<xd<M

ˆB(x,r0)c∩Rd+

w(y)j(|x− y|)dy

≤ supx∈Rd:0<xd<M

ˆB(x,r0)c

(V (M) + V (|x− y|)j(|x− y|)dy

≤ c2

ˆ ∞r0

(V (M) + V (r))rd−1j(r)dr

= c2L(r0)(V (M) + V (r0)) + c2

ˆ ∞r0

V ′(r)L(r)dr

≤ c2L(r0)(V (M) + V (r0)) + c3

ˆ ∞r0

V ′(r)

V (r)2dr

= c2L(r0)(V (M) + V (r0)) +c3

V (r0)<∞.

We have proved (3.3.14). 2

31


For the remainder of this section, we follow proofs of [14, Proposition

3.2 and Theorem 1.1(i)]. First note that for C1,1 open set D ⊂ Rd with

characteristics (R0,Λ0), there exists r0 = r0(R0,Λ0) ∈ (0, R0] such that D

satisfies the uniform interior and uniform exterior ball conditions with radius

r0. We will use such r0 > 0 in the proof of the next proposition and Theorem

3.1.3 (1).

Proposition 3.3.7. Let D is a C1,1 open set in Rd with characteristics

(R0,Λ0). If D is bounded, then we assume that H satisfies La(γ, CL) and

Ua(δ, CU) with δ < 2 for some a > 0. If D is unbounded, then we assume

that H satisfies L0(γ, CL) and U0(δ, CU) with δ < 2. For every T > 0, there

exists c > 0 such that for all (t, x, y) ∈ (0, T ]×D ×D,

pD(t, x, y) ≤ c

(1 ∧ δD(x)√

t

)×(t−d/2 ∧

(t−d/2e

− |x−y|2

4cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1)

)),

(3.3.16)

where the constants cU and aU are from (3.3.10).

Proof. We prove the proposition for the case that D is bounded, H satisfies

La(γ, CL) and Ua(δ, CU) with δ < 2 for some a > 0 only, because the proof

of the other case is almost identical.

Fix T > 0 and t ∈ (0, T ]. Let x, y ∈ D. We just consider the case

δD(x) < r0√t/(16

√T ) ≤ r0/16, if not, we can directly obtain (3.3.16) by

(3.3.10). Choose x0 ∈ ∂D and x1 ∈ D such that δD(x) = |x − x0| and

x1 = x0 + r0√t

16√Tn(x0), respectively, where n(x0) = (x − x0)/|x0 − x| be the

unit inward normal of D at the boundary point x0. Define

U1 := B(x0, r0√t/(8√T )) ∩D.

Since (3.3.14) holds, by [27, Lemma 4.3]

Ex[τU1 ] ≤ c1√tδD(x). (3.3.17)

32


Using Theorem 3.3.6 and δD(x1) = r0√t

16√T

, we have

Px(XτU1∈ D \ U1) ≤ c2Px1(XτU1

∈ D \ U1)δD(x)

δD(x1)

≤ c216√TδD(x)

r0√t

≤ c3

(1 ∧ δD(x)√

t

). (3.3.18)

Thus by (3.3.18) and (3.3.17) we have

Px(τD > t/2) ≤ Px(τU1 > t/2) + Px(XτU1

∈ D \ U1

)≤(

1 ∧(2

tEx[τU1 ]

))+ Px(XτU1

∈ D \ U1) ≤ c4

(1 ∧ δD(x)√

t

).

(3.3.19)

Now we estimate pD(t, x, y) considering two cases separately. Let c5 :=

(d/2)∨ ((dC−1/γL T 1/γ−1φ−1(1)−1)/(2cUaU)) ∨ (r20/(4cUT )) where aU and cU

are the constants in (3.3.10).

Case (1): |x− y| ≤ 2(cUc5)1/2√t. By the semigroup property,

pD(t, x, y) =

ˆD

pD(t/2, x, z)pD(t/2, z, y)dz

≤

(supz,w∈D

p(t/2, z, w)

)ˆD

pD(t/2, x, z)dz.

Using Theorem 3.1.1 and (3.3.19) in the above display, we have

pD(t, x, y) ≤ c6(t/2)−d/2Px(τD > t/2) ≤ c6c42d/2t−d/2

(1 ∧ δD(x)√

t

).

Since |x− y|2/(4cU t) ≤ c5, we have

pD(t, x, y) ≤ c6c42d/2ec5t−d/2e−|x−y|

2/(4cU t)

(1 ∧ δD(x)√

t

). (3.3.20)

33


Case (2): |x− y| ≥ 2(cUc5)1/2√t. Define

U3 := {z ∈ D : |z − x| > |x− y|/2} and U2 := D \ (U1 ∪ U3). (3.3.21)

For z ∈ U2,

3

2|x− y| ≥ |x− y|+ |x− z| ≥ |z− y| ≥ |x− y| − |x− z| ≥ |x− y|

2. (3.3.22)

By our choice of c5 and (3.2.5) (we can assume that a ≤ φ−1(T−1) by

Remark 2.0.2), we have t ≤ |x− y|2/(2dcU) and

φ−1(t−1)−1 ≤ C−1/γL T 1/γ−1tφ−1(1)−1 ≤ aU |x− y|2

2d.

Using this and the fact that s → s−d/2e−β/s is increasing on the interval

(0, 2β/d], we get that for s ≤ t

s−d/2e− |x−y|

2

4cUs + φ−1(s−1)d/2e−aU4|x−y|2φ−1(s−1)

≤t−d/2e−|x−y|24cU t + φ−1(t−1)d/2e−

aU4|x−y|2φ−1(t−1).

Thus by (3.3.10) and (3.3.22),

sups≤t,z∈U2

p(s, z, y)

≤ c7 sups≤t,

2|z−y|≥|x−y|

(s−d/2e

− |z−y|2

cUs +sH(|z − y|−2)|z − y|d

+ φ−1(s−1)d/2e−aU |z−y|2φ−1(s−1)

)

≤ c7 sups≤t

(s−d/2e

− |x−y|2

4cUs + 2dsH(4|x− y|−2)|x− y|d

+ φ−1(s−1)d/2e−aU4|x−y|2φ−1(s−1)

)≤ c7

(t−d/2e

− |x−y|2

4cU t + 2d+4 tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1)

)≤ c8

(t−d/2 ∧

(t−d/2e

− |x−y|2

4cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1)

)).

(3.3.23)

34


For the last inequality, we argue as follows: by the proof of [29, Corollary

1.3]

2d+4 tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1) ≤ c9

tφ(|x− y|−2)|x− y|d

.

(3.3.24)

On the other hand, by Lemma 2.0.3

tφ(|x− y|−2)|x− y|d

≤ tφ((4cUc5)−1t−1)

|x− y|d≤ Tφ((4cUc5)

−1T−1)

|x− y|d≤ c10t

−d/2.

Therefore

2d+4 tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1) ≤ c11t

−d/2. (3.3.25)

For u ∈ U1 and z ∈ U3, since |z − x|/2 > |x− y|/4 ≥ r0√t

4√T

,

|u−z| ≥ |z−x|−|x−x0|−|x0−u| ≥ |z−x|−r0√t

4√T≥ |z − x|

2>|x− y|

4≥ r0√t

4√T.

Thus, dist(U1, U3) > 0 and by (3.2.4) and Lemma 2.0.3, we have

supu∈U1,z∈U3

J(u, z) ≤ sup|u−z|≥ 1

4|x−y|

c12H(|u− z|−2)|u− z|d

≤ c124dH(16|x− y|−2)|x− y|d

≤ c124d+4H(|x− y|−2)|x− y|d

. (3.3.26)

By the same argument in (3.3.18), we can apply Theorem 3.3.6 to get

Px(XτU1∈ U2) ≤ c13Px1(XτU1

∈ U2)δD(x)

δD(x1)≤ c14

δD(x)√t. (3.3.27)

Applying (3.3.17), (3.3.23), (3.3.26) and (3.3.27) in the inequality (3.3.9), we

35


obtain

pD(t, x, y) ≤ c15δD(x)√

t

(t−d/2 ∧

(t−d/2e

− |x−y|2

4cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1)

))+ c16

√tδD(x)

H(|x− y|−2)|x− y|d

≤ c17δD(x)√

t

(t−d/2 ∧

(t−d/2e

− |x−y|2

4cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU4|x−y|2φ−1(t−1)

)).

In the last inequality, (3.3.25) is used. Combining above two cases, we have

completed the proof of the proposition. 2

Proof of Theorem 3.1.3 (1) and Theorem 3.1.4 (1). We only prove

Theorem 3.1.3 (1), because both proofs are almost identical. Using (3.3.8)

and (3.3.16) instead of (3.3.9) and (3.3.10) respectively, we will follow the

proof of Proposition 3.3.7.

Fix T > 0. Let t ∈ (0, T ] and x, y ∈ D. By Proposition 3.3.7, The-

orem 3.1.1 and symmetry, we only need to prove Theorem 3.1.3 (1) when

δD(x)∨δD(y) < r0√t/(16

√T ) ≤ r0/16. Thus we assume that δD(x)∨δD(y) <

r0√t/(16

√T ) ≤ r0/16. Define x0, x1 and U1 in the same way as in the proof of

Proposition 3.3.7 and let c1 := ((d+1)/2)∨((dC−1/γL T 1/γ−1φ−1(1)−1)/(cUaU))∨

(r20/(16cUT )) where aU and cU are constants in (3.3.16). Now we estimate

pD(t, x, y) by considering the following two cases separately.

Case (1): |x−y| ≤ 4(cUc1)1/2√t. By the semigroup property and symmetry,

pD(t, x, y) =

ˆD

pD(t/2, x, z)pD(t/2, z, y)dz

≤(

supz∈D

pD(t/2, y, z))ˆ

D

pD(t/2, x, z)dz.

36


Using Proposition 3.3.7 and (3.3.19) in the above inequality , we have

pD(t, x, y) ≤ c2t−d/2

(1 ∧ δD(y)√

t

)Px(τD > t/2)

≤ c2t−d/2

(1 ∧ δD(y)√

t

)(1 ∧ δD(x)√

t

)≤ c2e

c1t−d/2e−|x−y|2/(16cU t)

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

). (3.3.28)

Case (2): |x − y| ≥ 4(cUc1)1/2√t. Define U2 and U3 in the same way as in

(3.3.21). Then by the same way (3.3.22) and (3.3.26) hold.

By our choice of c1, we have t ≤ |x− y|2/(8(d+ 1)cU). Using this and the

fact that s → s−(d+1)/2e−β/s is increasing on the interval (0, 2β/(d + 1)], we

get for s ≤ t,

s−(d+1)/2e− |x−y|

2

16cUs ≤ t−(d+1)/2e− |x−y|

2

16cU t .

37


Thus by (3.3.16) and (3.3.22),

sups≤t,z∈U2

pD(s, z, y)

≤ c3 sups≤t,z∈U2

(1 ∧ δD(y)√

s

)(s−d/2 ∧

(s−d/2e

− |z−y|2

4cUs +sH(|z − y|−2)|z − y|d

+ φ−1(s−1)d/2e−aU4|z−y|2φ−1(s−1)

))≤ c3δD(y) sup

s≤t,|z−y|≥ |x−y|2

(s−(d+1)/2e

− |z−y|2

4cUs +

√sH(|z − y|−2)|z − y|d

+φ−1(s−1)d/2√

se−

aU4|z−y|2φ−1(s−1)

)≤ c4δD(y) sup

s≤t

(s−(d+1)/2e

− |x−y|2

16cUs +2d√tH(4|x− y|−2)|x− y|d

+ s−1/2φ−1(s−1)d/2e−aU16|x−y|2φ−1(s−1)

)≤ c5

δD(y)√t

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

)+ c4δD(y)

(sups≤t

s−1/2e−aU32|x−y|2φ−1(s−1)

)(sups≤t

φ−1(s−1)d/2e−aU32|x−y|2φ−1(s−1)

).

(3.3.29)

We find upper bounds of s−1/2e−aU32|x−y|2φ−1(s−1) and φ−1(s−1)d/2e−

aU32|x−y|2φ−1(s−1)

for s ≤ t. By our choice of c1 and (3.2.5), we have

t ≤ aU16d

C1/γL T 1−1/γφ−1(1)|x− y|2

and φ−1(t−1)−1 ≤ C−1/γL T 1/γ−1tφ−1(1)−1 ≤ aU

16d|x− y|2.

Using this and the fact that s → s−d/2e−β/s is increasing on the interval

38


(0, 2β/d], we get(sups≤t

s−1/2e−aU32|x−y|2φ−1(s−1)

)(sups≤t

φ−1(s−1)d/2e−aU32|x−y|2φ−1(s−1)

)≤(

sups≤t

s−1/2e−aU32|x−y|2C1/γ

L T 1−1/γφ−1(1)s−1)(sups≤t

φ−1(s−1)d/2e−aU32|x−y|2φ−1(s−1)

)≤ t−1/2e−

aU32C

1/γL T 1−1/γφ−1(1)t−1

φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

=1√te−

aU32C

1/γL T−1/γφ−1(1)φ−1(t−1)d/2e−

aU32|x−y|2φ−1(t−1). (3.3.30)

Therefore combine (3.3.29) with (3.3.30) we get

sups≤t,z∈U2

pD(s, z, y)

≤ c6δD(y)√

t

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

)≤ c7

δD(y)√t

(t−d/2 ∧

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

)). (3.3.31)

In the last inequality we have used similar argument as the one leading

(3.3.25). On the other hand by (3.3.19) we have

ˆ t

0

Px(τU1 > s)Py(τD > t− s)ds ≤ˆ t

0

Px(τD > s)Py(τD > t− s)ds

≤ c8

ˆ t

0

δD(x)√s

δD(y)√t− s

ds = c8δD(x)δD(y)

ˆ 1

0

1√r(1− r)

dr = c9δD(x)δD(y).

(3.3.32)

Using (3.3.26), (3.3.27), (3.3.31) and (3.3.32) in the inequality (3.3.8), we

39


conclude that

pD(t, x, y)

≤ c10δD(x)δD(y)

t

(t−d/2 ∧

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

))+ c11

δD(x)δD(y)

t× tH(|x− y|−2)

|x− y|d

≤ c12δD(x)δD(y)

t

×(t−d/2 ∧

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

))= c12

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)×(t−d/2 ∧

(t−d/2e

− |x−y|2

16cU t +tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU32|x−y|2φ−1(t−1)

)).

In the second inequality above, we also used similar argument as the one

leading (3.3.25). This combined with (3.3.28) completes the proof. 2

Proof of Theorem 3.1.3 (3). The proof is same as [14, Theorem 1.1(iii),

(iv)]. We should consider Theorem 3.1.3(1) and Theorem 3.1.3(2) instead of

[14, Theorem 1.1(ii)] and [14, Theorem 2.6], respectively. We omit the proof.

2

3.4 Green function estimates

In this section we give the proof of Corollaries 3.1.2 and 3.1.5.

Proof of Corollary 3.1.2. Note that by Lemma 2.0.3,

if |x| ≤ 1, φ(1)φ(|x|−2)−1 ≥ |x|2, and if |x| > 1, φ(1)φ(|x|−2)−1 < |x|2. We

split the integral

G(x) =

ˆ φ(1)φ(|x|−2)−1∧|x|2

0

p(t, x)dt+

ˆ ∞φ(1)φ(|x|−2)−1∧|x|2

p(t, x)dt.

40


By Remark 3.2.3 and (3.3.24),

ˆ φ(1)φ(|x|−2)−1∧|x|2

0

p(t, x)dt

≤ c1

ˆ φ(1)φ(|x|−2)−1∧|x|2

0

(t−d/2e−c2|x|

2/t +tH(|x|−2)|x|d

+ φ−1(t−1)d/2e−c3|x|2φ−1(t−1)

)dt

≤ c4

( ˆ φ(1)φ(|x|−2)−1∧|x|2

0

t−d/2e−c2|x|2/tdt+

ˆ φ(1)φ(|x|−2)−1∧|x|2

0

tφ(|x|−2)|x|d

dt

)≤ c5

( ˆ φ(1)φ(|x|−2)−1∧|x|2

0

t−d/2( t

|x|2)d/2

dt+φ(|x|−2)|x|d

ˆ φ(1)φ(|x|−2)−1∧|x|2

0

tdt

)≤ c6

(|x|−d

(φ(|x|−2)−1 ∧ |x|2

)+ |x|−d

(φ(|x|−2)−1 ∧ |x|4φ(|x|−2)

))≤ c7

(|x|−d+2 ∧ |x|−dφ(|x|−2)−1

).

For |x| ≤ 1, using Remark 3.2.3,

ˆ ∞φ(1)φ(|x|−2)−1∧|x|2

p(t, x)dt =

ˆ ∞|x|2

p(t, x)dt ≤ c8

ˆ ∞|x|2

t−d/2dt =2c8d− 2

|x|−d+2.

For |x| > 1, using Remark 3.2.3 and change of variables,

ˆ ∞φ(1)φ(|x|−2)−1∧|x|2

p(t, x)dt =

ˆ ∞φ(1)φ(|x|−2)−1

p(t, x)dt

≤ c9

ˆ ∞φ(1)φ(|x|−2)−1

φ−1(t−1)d/2dt = c9

ˆ c10|x|−2

0

sd/2(− 1

φ(s)

)′ds

= c9

ˆ c10|x|−2

0

sd/2−1yφ′(s)

φ(s)2ds ≤ c9

ˆ c10|x|−2

0

sd/2−1

φ(s)ds,

in the last inequality we use λφ′(λ) ≤ φ(λ) because φ is represented by

(2.0.1). Since d ≥ 3, we have d2− 2 > −1. Hence

ˆ c10|x|−2

0

sd/2−1

φ(s)ds ≤ c11

φ(|x|−2)

ˆ c10|x|−2

0

sd/2−1(|x|−2

s

)ds = c12|x|−dφ(|x|−2)−1.

41


On the other hand, for |x| > 1, by Lemma 3.2.6 and the condition of L0(γ, CL)

on φ, we have

G(x) ≥ˆ ∞φ(|x|−2)−1

p(t, x)dt ≥ c13

ˆ 2φ(|x|−2)−1

φ(|x|−2)−1

φ−1(t−1)d/2dt

≥ c13φ−1(φ(|x|−2)

2

)d/2 1

φ(|x|−2)≥ c13(CL/2)d/(2γ)

|x|dφ(|x|−2).

When |x| ≤ 1, by Lemma 3.2.6 and Lemma 3.2.7, we have

G(x) ≥ˆ 2|x|2

|x|2p(t, x)dt ≥ c14

ˆ 2|x|2

|x|2t−d/2 ∧ φ−1(t−1)d/2dt

≥ c15

ˆ 2|x|2

|x|2t−d/2dt ≥ c15

|x|−d

2d/2|x|2.

Third inequality holds because for t ≤ 2, t−d/2 ≤ cφ−1(t−1)d/2 for some

c16 > 0. Hence we conclude that

G(x) � |x|−d+2 ∧ |x|−dφ(|x|−2)−1.

2

Proof of Corollary 3.1.5. Recall that gD(x, y) is defined in Corollary 3.1.5.

Let T :=diam(D)2. Then we have (see the proof of [14, Corollary 1.3])

gD(x, y)

�ˆ T

0

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)p(2)(t, c(x− y))dt+

ˆ ∞T

e−λ1tδD(x)δD(y)dt.

By Theorem 3.1.3(2) and (3), we can easily obtain GD(x, y) ≥ c1gD(x, y).

Next we consider the upper bound for GD(x, y). By Theorem 3.1.3 (1),

42


for the bounded C1,1 open set D,

GD(x, y) =

ˆ ∞0

pD(t, x, y)dt

≤ c1

ˆ T

0

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧

(p(2)(t, c2(x− y))

+tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1)

))dt

+ c1

ˆ ∞T

e−λ1tδD(x)δD(y)dt

≤ c1

ˆ T

0

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(p(2)(t, c2(x− y))

+(t−d/2 ∧

(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1)

)))dt

+ c1

ˆ ∞T

e−λ1tδD(x)δD(y)dt

≤ c2

(gD(x, y) +

ˆ T

0

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧ t

|x− y|d+2

)dt

).

For the last inequality, we use (3.3.24), boundedness of D and Lemma 2.0.3:

tH(|x− y|−2)|x− y|d

+φ−1(t−1)d/2e−aU |x−y|2φ−1(t−1) ≤ c3tφ(|x− y|−2)

|x− y|d≤ c4t

|x− y|d+2.

To complete the proof, it suffices to show that

ˆ T

0

(1 ∧ δD(x)√

t

)(1 ∧ δD(y)√

t

)(t−d/2 ∧ t

|x− y|d+2

)≤ c5gD(x, y),

which is [14, (4.1)]. Thus the remaining proof is same as the part of proof

starting on the page 135 in [14, Corollary 1.3]. So we omit it. 2

43

Chapter 4

Feynman-Kac perturbation of

Dirichlet heat kernels of

subordinate Brownian motions

4.1 Setting

Let φ be a Bernstein function defined in (2.0.1). Recall H(λ) = φ(λ)−λφ′(λ).

Let D be a Borel set in Rd. For c > 0, η ≥ 0, we define following functions:

qc(t, x, y) := φ−1(t−1)d/2 ∧(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c|x−y|2φ−1(t−1)

),

t > 0, x, y ∈ Rd,

ψη(t, x) :=

(1 ∧ φ(δD(x)−2)−1

t

)η/2, t > 0, x ∈ D,

qc,η(t, x, y) := ψη(t, x)qc(t, x, y)ψη(t, y), t > 0, x, y ∈ D. (4.1.1)

Let X be a Hunt process on D with transition semigroup {Pt : t ≥ 0}that admits a jointly continuous transition density pD(t, x, y) and there exist

constants C0 ≥ 1, a1, a2 > 0 and η ∈ [0, 2) such that

C−10 qa1,η(t, x, y) ≤ pD(t, x, y) ≤ C0qa2,η(t, x, y) (4.1.2)

44

CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM

for (t, x, y) ∈ (0, 1]×D×D. Since H(λ) ≤ φ(λ), simple calculation gives us

that if tφ(|x− y|−2) ≤ 1,

1

2

(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c|x−y|2φ−1(t−1)

)≤ qc(t, x, y) ≤ tH(|x− y|−2)

|x− y|d+ φ−1(t−1)d/2e−c|x−y|

2φ−1(t−1),

and

e−cφ−1(t−1)d/2 ≤ qc(t, x, y) ≤ φ−1(t−1)d/2 for tφ(|x− y|−2) > 1. (4.1.3)

Using above observations, we see that, for any c > 0 there exits C depending

on c such thatˆRdqc(t, x, y) dy ≤ C for all (t, x) ∈ (0,∞)× Rd. (4.1.4)

Throughout this chapter, we assume that X has a Levy system (N, t),

where N = N(x, dy) has a kernel with respect to dy so that

N(x, dy) =c(x, y)H(|x− y|−2)

|x− y|ddy,

That is, for any x ∈ D, any stopping time S and any non-negative measurable

function f on [0,∞) ×D ×D with f(t, y, y) = 0 for all (t, y) ∈ [0,∞) ×Dthat is extended to be zero off D ×D,

Ex

[∑t≤S

f(t,Xt−, Xt)

]= Ex

[ˆ S

0

(ˆD

f(t,Xt, y)c(Xt, y)H(|Xt − y|−2)

|Xt − y|ddy

)dt

].

(4.1.5)

We further assume that c(x, y) is a measurable function such that

C−10 ≤ c(x, y) ≤ C0. (4.1.6)

For example, if φ(λ) = λα/2 where 0 < α < 2 then qc(t, x, y) is the sharp

45


upper and lower bound for the heat kernel of symmetric α-stable process in

Rd by [29, Cor1.3], and Dirichlet heat kernel for symmetric α-stable process

in a C1,1 open set D satisfies (4.1.2) with η = 1 and c(x, y) = c1.

Recall that a measure µ on D is said to be a smooth measure of X with

respect to the Lebesgue measure if there is a positive continuous additive

functional A of X such that for any bounded non-negative Borel function f

on D, ˆD

f(x)µ(dx) = limt↓0

ˆD

Ex[

1

t

ˆ t

0

f(Xs)dAs

]dx. (4.1.7)

The additive functional A is called the positive continuous additive functional

(PCAF) of X with Revuz measure µ.

For a signed measure ν, we use ν+ and ν− to denote its positive and

negative parts of ν respectively. A signed measure ν is called smooth if both

ν+ and ν− are smooth. For a signed smooth measure ν, if A+ and A− are

the PCAFs of XD with Revuz measures ν+ and ν− respectively, the additive

functional A := A+−A− of is called the continuous additive functional (CAF

in abbreviation) of XD with (signed) Revuz measure ν.

It is known (see [22]) that for any x ∈ D, α ≥ 0 and bounded non-negative

Borel function f on D,

Exˆ ∞0

e−αtf(Xt)dAt =

ˆ ∞0

e−αtˆD

pD(t, x, y)f(y)dydt,

and we have for any x ∈ D, t > 0 and non-negative Borel function f on D,

Exˆ t

0

f(Xs)dAs =

ˆ t

0

ˆD

pD(s, x, y)f(y)dyds. (4.1.8)

We now define our Kato type classes. For a smooth (signed) Radon

measure µ on D and t > 0, we define

Nφ,ηµ,c (t) := sup

x∈D

ˆ t

0

ˆD

ψη(s, z)qc(s, x, z)|µ|(dz)ds. (4.1.9)

Here we denote the total variation measure of µ by |µ|.

Definition 4.1.1. A smooth (signed) Radon measure µ on D is said to be

46


in the class Kφ,η if limt↓0Nφ,ηµ,c (t) = 0 for some and hence for all c > 0.

For any measurable function F on D ×D vanishing on the diagonal, we

define

Nφ,ηF,c (t) := sup

x∈D

ˆ t

0

ˆD×D

ψη(s, z)qc(s, x, z)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|x− z|−2)−1

)η/2× |F (z, w)|+ |F (w, z)||z − w|dH(|z − w|−2)−1

dzdwds.

(4.1.10)

Definition 4.1.2. We say that a measurable function F on D×D vanishing

on the diagonal belongs to the class Jφ,η if F is bounded and limt↓0Nφ,ηF,c (t) = 0

for some and hence for all c > 0.

For any smooth (signed) Radon measure µ on D and any measurable

function F on D ×D vanishing on the diagonal, we define

Nφ,ηµ,F,c(t) := Nφ,η

µ,c (t) +Nφ,ηF,c (t).

When µ ∈ Kφ,η and F is a measurable function on D ×D vanishing on the

diagonal, we define

Aµ,Ft = Aµt +∑0<s≤t

F (Xs−, Xs), (4.1.11)

where Aµt is the continuous additive functional of X with Revuz measure µ.

For any nonnegative Borel function f on D, we define

T µ,Ft f(x) = Ex[exp(Aµ,Ft )f(Xt)

], t ≥ 0, x ∈ D.

Then {T µ,Ft : t ≥ 0} is called the Feynman-Kac semigroup ofX corresponding

to µ and F . Let Exp(K)t be the Stieltjes exponential of a right continuous

function Kt with left limits on R+ with K0 = 1 and ∆Kt := Kt −Kt− > −1

for every t > 0, and of finite variation on each compact time interval. In [13,

Section 1.2], third-named author, jointly with Zhen-Qing Chen and Renming

Song, provided an approach to the semigroup T µ,Ft by following identity which

47


uses Stieltjes exponential:

exp(Aµ,Ft

)= Exp

(Aµ +

∑0<s≤·

(eF (Xs−,Xs) − 1

))t

for t ≥ 0.

Let F1(x, y) = eF (x,y) − 1. Then formally, we can rewrite

exp(Aµ,Ft

)= Exp

(Aµ,F1

)t

= 1 +∞∑n=1

ˆ[0,t]

dAµ,F1sn

ˆ[0,sn)

dAµ,F1sn−1· · ·ˆ[0,s2)

dAµ,F1s1

so that for any nonnegative bounded measurable function f on D,

T µ,Ft f(x) = Ptf(x) + Ex

[f(Xt)

∞∑n=1

ˆ[0,t]

dAµ,F1sn

ˆ[0,sn)

dAµ,F1sn−1· · ·ˆ[0,s2)

dAµ,F1s1

].

Define p0(t, x, y) := pD(t, x, y) and for k ≥ 1,

pk(t, x, y) =

ˆ t

0

ˆD

pD(t− s, x, z)pk−1(s, z, y)µ(dz)ds

+

ˆ t

0

ˆD×D

pD(t− s, x, z)c(z, w)F1(z, w)

|z − w|dH(|z − w|−2)−1pk−1(s, w, y)dzdwds.

(4.1.12)

Let

qD(t, x, y) :=∞∑k=0

pk(t, x, y), (t, x, y) ∈ (0,∞)×D ×D.

One of the purpose of this thesis is, using the discussion in [13, Section 1.2],

to show that the above qD(t, x, y) is well-defined for small t and its extension

to t ∈ (0,∞) is the transition density of T µ,Ft so that

T µ,Ft f(x) =

ˆD

qD(t, x, y)f(y)dy, (t, x) ∈ (0,∞)×D.

48


4.2 Local and Non-local 3P inequalities

In this section we will explain how the method in [13] can be extended cor-

rectly for nonlocal operators with more general scaling conditions.

4.2.1 3P inequality for Rd

Let p(t, x, y) be a transition density of subordinate Brownian motion with

subordinator S whose Laplace exponent is φ.

Proposition 4.2.1. Suppose that φ satisfies La(γ, CL) for some a > 0. For

any c > 0 and T > 0, there exists a constant C = C(a, γ, CL, c, T ) > 0 such

that for t ≤ T and tφ(|x− y|−2) ≤ 1

p(t, x, y) ≤ C

(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c|x−y|2φ−1(t−1)

).

Proof. The proof of this proposition is similar to [29, Proposition 3.4]. By

Remark 2.0.2, without loss of generality we assume that a ≤ φ−1( 12eT

). Let

r > 0, t ≤ T satisfy tφ(r−2) ≤ 1 and k is a constant satisfying CLk2γ ≥ 2e.

Let B = (Bt)t≥0 be (d + 2)−dimensional Brownian motion and denote by

X = (Xt)t≥0 = (BSt) corresponding subordinate Brownian motion. Let Y

be a random variable with the chi-square distribution with parameter d+ 2.

Then by Lemma 2.0.3 and [29, (3.2) and Proposition 2.3],

P(|Xt| ≥ kr) =

ˆ(0,∞)

P(St ≥ (2y)−1k2r2)P0(Y ∈ dy)

≤ c1

ˆtφ(2yk−2r−2)≤ 1

2e

tH(2yk−2r−2)P0(Y ∈ dy) +

ˆtφ(2yk−2r−2)≥ 1

2e

P0(Y ∈ dy)

≤ c1tH(r−2)

ˆ(0,∞)

(1 + (2yk−2)2)P0(Y ∈ dy) +

ˆtφ(2yr−2)≥CLk

2γ

2e

P0(Y ∈ dy)

≤ c2tH(r−2) + P0

(Y ≥ 1

2r2φ−1(

CLk2γ

2et−1)

)≤ c3

(tH(r−2) + rdφ−1(

CLk2γ

2et−1)d/2e−

14r2φ−1(

CLk2γ

2et−1)

).

49


In the second inequality we use the condition La(γ, CL) on φ since 2yk−2r−2 ≥φ−1( 1

2et) ≥ φ−1( 1

2eT) ≥ a and in the last inequality we use [29, Lemma 3.3]

since 12r2φ−1(CLk

2γ

2et−1) ≥ 1

2. By using Lemma 2.0.5(i) and Lemma 2.0.5(ii)

we have

P(|Xt| ≥ kr) ≤ c4

(tH(r−2) + rdφ−1(t−1)d/2e−

CLk2γ

8er2φ−1(t−1)

). (4.2.1)

On the other hand, by the same way as in [29, 644p], we have

P(|Xt| ≥ kr) ≥ c5rdp(t, x, y), (4.2.2)

for |x− y| = r. Thus, using the last display (4.2.1) and (4.2.2) we have

p(t, x, y) ≤ c6

(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−CLk

2γ

8e|x−y|2φ−1(t−1)

).

Since k is arbitrary constant satisfying CLk2γ ≥ 2e, we get the proposition.

2

Proposition 4.2.2. Suppose that φ satisfies La(γ, CL) for some a > 0. For

c1, c2 > 0 there exist constants C1 = C1(a, γ, CL, c1, c2) > 0 and t0 = t0(a, c1)

such that for t ≤ t0, qc1(t, x, y) ≤ C1qc2(t, x, y).

Proof. Note that if c1 ≥ c2 or tφ(|x− y|−2) ≥ 1, by (4.1.3) we can easily see

that qc1(t, x, y) ≤ Cqc2(t, x, y).

We now assume that tφ(|x− y|−2) ≤ 1, c1 < 1/2, c2 > 0. Let c = 12c1

> 1.

Then by [29, Proposition 2.4] with α = 2 and β = 1, there exist constants

ρ ∈ (0, 1), τ > 0 such that

P(

1

2φ−1((ct)−1)≤ Sct ≤

1

φ−1(ρ(ct)−1)

)≥ τ. (4.2.3)

Note that ρc−1 < 1. For t < ρc−1φ(a)−1, using the condition La(γ, CL) on φ,

50


Lemma 2.0.5(i) and (4.2.3) we have

p(ct,x, y) = c3

ˆ(0,∞)

s−d/2e−|x−y|2

4s P(Sct ∈ ds)

≥ c3

ˆ[ 12φ−1((ct)−1)

, 1φ−1(ρt−1)

]

s−d/2e−|x−y|2

4s P(Sct ∈ ds)

≥ c3φ−1(ρt−1)d/2e−

2φ−1((ct)−1)|x−y|24 P

(1

2φ−1((ct)−1)≤ Sct ≤

1

φ−1(ρt−1)

)≥ c3τ(ρCL)d/2γφ−1(t−1)d/2e−

2|x−y|2φ−1(t−1)4c

≥ c4φ−1(t−1)d/2e−c1|x−y|

2φ−1(t−1).

By Proposition 4.2.1, for t < ρc−1φ(a)−1 there exists c5 > 0 such that

p(t, x, y) ≤ c5

(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−(c/CL)1/γc2|x−y|2φ−1(t−1)

).

By using the above two inequalities and the condition La(γ, CL) on φ, for

t < ρc−2φ(a)−1 = 4c21ρφ(a)−1 we get

qc1(t, x, y) = φ−1(t−1)d/2 ∧(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c1|x−y|2φ−1(t−1)

)≤ φ−1(t−1)d/2 ∧

(tH(|x− y|−2)|x− y|d

+ c−14 p(ct, x, y)

)≤ c6φ

−1(t−1)d/2 ∧(tH(|x− y|−2)|x− y|d

+ φ−1((ct)−1)d/2e−(cCL)1/γc2|x−y|2φ−1((ct)−1)

)≤ c7φ

−1(t−1)d/2 ∧(tH(|x− y|−2)|x− y|d

+ φ−1(t−1)d/2e−c2|x−y|2φ−1(t−1)

)= c7qc2(t, x, y).

If c1 ≥ 1/2, by the monotonicity of c → qc(t, x, y) and the above result, we

get qc1(t, x, y) ≤ q1/4(t, x, y) ≤ c8qc2(t, x, y) for t < ρφ(a)−1/4. 2

Remark 4.2.3. By Remark 2.0.2, for any T > 0, if we assume that T <

4c1ρφ(a)−1, Proposition 4.2.2 also holds for t ≤ T .

51


Lemma 4.2.4. Suppose that φ satisfies La(γ, CL) for some a > 0. For any

c1, c2 > 0 there exist C = C(a, γ, CL, c1, c2) > 0 and t0 = t0(a, c1, c2) > 0

such that the following 3P inequality holds: for 0 < s < t < t0,

qc1(s, x, z)qc2(t− s, z, y)

qc1∧c2(t, x, y)≤ C(qc1(s, x, z) + qc2(t− s, z, y)). (4.2.4)

Proof. We assume that c1 ≤ c2, because the proof of case c1 > c2 is the

same as the proof of case c1 ≤ c2. Let t < φ(a)−1. If |x − y|2φ−1(t−1) ≤ 4dc1

and t/2 ≤ s ≤ t, then by Lemma 2.0.5(ii)


qc1(t, x, y)≤ φ−1(s−1)d/2qc2(t− s, z, y)

e−4dφ−1(t−1)d/2

≤ e4d(

2

CL

)d/2γqc2(t− s, z, y),

and if |x− y|2φ−1(t−1) ≤ 4dc1

and s ≤ t/2, then by Lemma 2.0.5(ii)


qc1(t, x, y)≤ qc1(s, x, z)φ−1((t− s)−1)d/2

e−4dφ−1(t−1)d/2

≤ e4d(

2

CL

)d/2γqc1(s, x, z).

If |x− y|2φ−1(t−1) ≥ 4dc1

and |x− z| ≥ |z − y|, then |x− y| ≤ 2|x− z|. Since

the function s−d/2e−β/s increase on (0, 2β/d] we have,


qc1/4(t, x, y)

≤ c3(s|x− z|−dH(|x− z|−2) + φ−1(s−1)d/2e−c1|x−z|

2φ−1(s−1))qc2(t− s, z, y)

t|x− y|−dH(|x− y|−2) + φ−1(t−1)d/2e−c1/4|x−y|2φ−1(t−1)

≤ c3(s2d|x− y|−dH(22|x− y|−2) + φ−1(s−1)d/2e−c1/4|x−y|

2φ−1(s−1))qc2(t− s, z, y)


≤ (t2d+4|x− y|−dH(|x− y|−2) + φ−1(t−1)d/2e−c1/4|x−y|2φ−1(t−1))qc2(t− s, z, y)


≤ 2d+4qc2(t− s, z, y).

52


Similarly when |x− y|2φ−1(t−1) ≥ 4dc1

and |x− z| ≤ |z − y|, we have


qc1/4(t, x, y)≤ qc1(s, x, z)qc2(t− s, z, y)

qc2/4(t, x, y)≤ 2d+4qc1(s, x, z).

Therefore, combining above results and Proposition 4.2.2, there exists t0 such

that for t ≤ t0,


qc1(t, x, y)≤ C(qc1(s, x, z) + qc2(t− s, z, y)).

2

4.2.2 Integral 3P inequality for local perturbation

Lemma 4.2.5. For any s, t > 0 and (y, z) ∈ D ×D, we have

1 ∧ φ(δD(z)−2)−1

t=φ(δD(y)−2)−1

t

(t ∧ φ(δD(z)−2)−1

φ(δD(y)−2)−1

)(4.2.5)

and(1 ∧ φ(δD(y)−2)−1

s

)(1 ∧ φ(δD(z)−2)−1

t

)≤ 4

(1 +

φ(|y − z|−2)−1

s ∨ φ(δD(y)−2)−1

)(1 ∧ φ(δD(y)−2)−1

t

).

(4.2.6)

Proof. The identity 1∧ ba

= a∧ba

gives us (4.2.5). Observe that δD(z) ≤ |z−y|+δD(y) for z, y ∈ D, thus φ(δD(z)−2)−1 ≤ 4

(φ(|x−y|−2)−1+φ(δD(y)−2)−1

).

By the identity 1 ∧ ba

= ba∨b and Lemma 2.0.3, we get(

1 ∧ φ(δD(y)−2)−1

s

)(1 ∧ φ(δD(z)−2)−1

t

)≤ (φ(δD(y)−2)−1

4(φ(δD(y)−2)−1 + φ(|z − y|−2)−1) ∨ t· 4(φ(δD(y)−2)−1 + φ(|z − y|−2)−1)

φ(δD(y)−2)−1 ∨ s,

53


which follows (4.2.6). 2

Observe that for A,B > 0 and a, b ≥ 0 such that a+ b < 2,

A2−a−b(A ∧B)b

2− a≤ˆ A

0

s1−a(

1 ∧ Bs

)bds ≤ A2−a−b(A ∧B)b

2− a− b. (4.2.7)

This asymptotic equation is used in the proof of the following Lemma.

Lemma 4.2.6. Suppose that φ satisfies La(γ, CL) for some a > 0. For any

η ∈ [0, 4) and c > 0, there exist constants C2 = C2(a, γ, CL, c, η) ≥ 1 and

t0 = t0(a, γ, CL, c) > 0 such that for all (t, y, z) ∈ (0, t0)×D ×D,

ψη(t, z)

ˆ t/2

0

qc,η(s, z, y)ds ≤ C2ψη(t, y)

ˆ t/2

0

ψη(s, z)qc(s, z, y)ds. (4.2.8)

Proof. If η = 0, the lemma holds obviously with C2 = 1. Let t < 2φ(a)−1.

We assume that c < 1/2. Then d/(2c) ≥ 1. Thus by Lemma 2.0.3, we

get 2cdφ(|z − y|−2)−1 ≤ φ( d

2c|z − y|−2)−1. By (4.2.6), it is enough to assume

φ(δD(y)−2)−1 < 2cdφ(|z − y|−2)−1 and prove the result for the integral with

the interval (0, t/2) replaced by (0, 2cdφ(|z − y|−2)−1 ∧ (t/2)). We can also

assume φ(δD(y)−2)−1 ≤ t/2 because ψη(t, z) ≤ 2η/2ψη(t, y) otherwise. Let

A = 2cdφ(|z − y|−2)−1 ∧ (t/2). Applying (4.2.7) with a = 0 and b = η/2 we

54


have

ˆ A

0

qc(s, z, y)ψη(s, y)ds

=H(|z − y|−2)|z − y|d

ˆ A

0

s

(1 ∧ φ(δD(y)−2)−1

s

)η/2ds

+

ˆ A

0

φ−1(s−1)d/2e−c|z−y|2φ−1(s−1)

(1 ∧ φ(δD(y)−2)−1

s

)η/2ds

� A2−η/2H(|z − y|−2)|z − y|d

(A ∧ φ(δD(y)−2)−1

)η/2+

ˆ A

0

φ−1(s−1)d/2e−c|z−y|2φ−1(s−1)

(1 ∧ φ(δD(y)−2)−1

s

)η/2ds

=: I + II(c). (4.2.9)

First, we observe that for any β > 0, the function f(s) := s−d/2e−β/s is

increasing on the interval (0, 2β/d]. Using the observation and applying

(4.2.7) with a = 1 and b = η/2, we have

II(c) ≤ φ−1(A−1)d/2e−c|z−y|2φ−1(A−1)

ˆ A

0

(1 ∧ φ(δD(y)−2)−1

s

)η/2ds

≤ c1φ−1(A−1)d/2e−c|z−y|

2φ−1(A−1)A1−η/2(A ∧ φ(δD(y)−2)−1)η/2.

(4.2.10)

Thus by (4.2.9) and (4.2.10), we have the following upper bound.

ˆ A

0

qc(s, z, y)ψη(s, y)ds ≤ c2(A ∧ φ(δD(y)−2)−1

)η/2×(A2−η/2H(|z − y|−2)

|z − y|d+ φ−1(A−1)d/2e−c|z−y|

2φ−1(A−1)A1−η/2). (4.2.11)

Since t < 2φ(a)−1, we have φ(a) < A−1. Thus by using Lemma 2.0.5(ii), we

55


have that

II(c(CL/2)1/γ)

≥ˆ A

A/2

φ−1(s−1)d/2e−c(CL/2)1/γ |z−y|2φ−1(s−1)

(1 ∧ φ(δD(y)−2)−1

s

)η/2ds

≥ φ−1(A−1)d/2e−c(CL/2)1/γ |z−y|2φ−1(2A−1)A−η/2(A ∧ φ(δD(y)−2)−1)η/2

ˆ A

A/2

ds

≥ 1

2φ−1(A−1)d/2e−c|z−y|

2φ−1(A−1)A1−η/2(A ∧ φ(δD(y)−2)−1)η/2. (4.2.12)

By using (4.2.9) and (4.2.12) by replacing ψγ(s, z) with ψγ(s, y) we get

ˆ A

0

ψη(s, z)qc(CL/2)1/γ (s, z, y)ds ≥ c3(A ∧ φ(δD(z)−2)−1

)η/2×(A2−η/2H(|z − y|−2)

|z − y|d+ φ−1(A−1)d/2e−c|z−y|

2φ−1(A−1)A1−η/2). (4.2.13)

Thus, by using (4.2.11), (4.2.13) and the following inequalities φ(δD(y)−2)−1 ≤A and δD(z) ≤ δD(y) + |z − y|, we get

ˆ A

0

qc(s, z, y)ψη(s, y)ds

≤ c4

(φ(δD(y)−2)−1

t ∧ φ(δD(z)−2)−1

)η/2 ˆ A

0

ψη(s, z)qc(CL/2)1/γ (s, z, y)ds.

By combining this with (4.2.5) and Proposition 4.2.2, we obtain that there

exists 0 < t0 ≤ 2φ(a)−1 such that for t ≤ t0

ψη(t, z)

ˆ A

0

qc(s, z, y)ψη(s, y)ds ≤ c5ψη(t, y)

ˆ A

0

ψη(s, z)qc(s, z, y)ds

which implies the result. 2

Suppose that φ satisfies La(γ, CL) for some a > 0 and c > 0. Note that

from the monotonicity of φ, Lemma 2.0.5(ii), and Proposition 4.2.2, there

exist C = C(a, γ, CL, c) > 1 and t0 = t0(a, c) > 0 such that for (t, x, y) ∈

56


(0, t0]× Rd × Rd,

C−1qc(t, x, y) ≤ qc(s, x, y) ≤ Cqc(t, x, y), s ∈ [t/2, t]. (4.2.14)

Theorem 4.2.7. Suppose that φ satisfies La(γ, CL) for some a > 0. For ev-

ery η ∈ [0, 2) and c1, c2 > 0, there exist constants C4 = C4(a, γ, CL, c1, c2, η) >

0 and t0 = t0(a, γ, CL, c1, c2) > 0 such that for any measure µ on D and any

(t, x, y) ∈ (0, t0]×D ×D,

ˆ t

0

ˆD

qc1,η(t− s, x, z)qc2,η(s, z, y)

qc1∧c2,η(t, x, y)µ(dz)ds

≤ C4 supu∈D

ˆ t

0

ˆD

ψη(s, z)qc1(s, u, z)µ(dz)ds.

Proof. We will show that there exist constants C3 = C3(a, γ, CL, c1, c2, η) >

0 and t0 = t0(a, γ, CL, c1, c2) > 0 such that for all (t, x, y, z) ∈ (0, t0] ×D ×D ×D,

ˆ t

0

qc1,η(t− s, x, z)qc2,η(s, z, y)

qc1∧c2,η(t, x, y)ds ≤ C3

ˆ t

0

ψη(s, z)(qc1(s, x, z) + qc2(s, z, y))ds.

(4.2.15)

Then by integrating the above integral over the region D with respect to the

variable z, we get this theorem.

When η = 0, (4.2.15) follows from (4.2.4). Thus, we assume η ∈ (0, 2)

and define

J(t, x, y, z) :=

ˆ t

0

qc1,η(t− s, x, z)qc2,η(s, z, y)ds.

By dividing the integral into two parts according to the intervals [0, t/2] and

[t/2, t] and using (4.2.14), there exists t1 = t1(a, c1, c2) > 0 such that for

57


t ∈ (0, t1], we get

J(t, x, y, z)

≤ c3

(qc1,η(t, x, z)

ˆ t/2

0

qc2,η(s, z, y)ds+ qc2,η(t, z, y)

ˆ t

t/2

qc1,η(t− s, x, z)ds

).

Using Lemma 4.2.6 in the above expression, there exists t2 = t2(a, γ, CL, c1, c2) ∈(0, t1] such that for t ∈ (0, t2],

J(t, x, y, z) ≤ c4

(ψη(t, x)qc1(t, x, z)ψη(t, y)

ˆ t/2

0

ψη(s, z)qc2(s, z, y)ds

+ ψη(t, y)qc2(t, z, y)ψη(t, x)

ˆ t

t/2

ψη(t− s, z)qc1(t− s, x, z)ds

).

Thus, by using (4.2.14) and (4.2.4) in turn, there exists t3 = t3(a, c1, c2) ∈(0, t2] such that for t ∈ (0, t3],

J(t, x, y, z) ≤c5qc1∧c2,η(t, x, y)

( ˆ t/2

0

ψη(s, z)(qc1(t− s, x, z) + qc2(s, z, y))ds

+

ˆ t

t/2

ψη(t− s, z)(qc1(t− s, x, z) + qc2(s, z, y))ds

).

(4.2.16)

From the inequality (4.2.7), we have

ˆ t

0

(1 ∧ a

sβ

)ds ≤ t

1− β

(1 ∧ a

tβ

).

Thus, we get

ˆ t/2

0

ψη(s, z)ds ≤2

2− η(t/2)ψη(t, z) ≤

4

2− η

ˆ t

t/2

ψη(s, z)ds. (4.2.17)

58


Therefore, for t ∈ (0, t3], by using (4.2.14) and (4.2.17), we have

ˆ t/2

0

ψη(s, z)qc1(t− s, x, z)ds ≤ c6

ˆ t

t/2

ψη(s, z)qc1(s, x, z)ds,

ˆ t

t/2

ψη(t− s, z)qc2(s, z, y)ds ≤ c7

ˆ t

t/2

ψη(s, z)qc2(s, z, y)ds.

Applying this inequalities to (4.2.16), we obtain (4.2.15).

2

4.2.3 Integral 3P inequality for nonlocal perturbation

In this subsection we always assume that φ satisfies La(γ, CL) for some a > 0.

Lemma 4.2.8. For every η ∈ [0, 2) and c > 0, there exist constants C5 =

C5(a, γ, CL, c, η) ≥ 1 and t0 = t0(a, c) > 0 such that for all (t, y, z, w) ∈(0, t0]×D ×D ×D,

ψη(t, z)

ˆ t/2

0

qc,η(s, w, y)ds ≤ C5ψη(t, y)

×(

1 +φ(|y − z|−2)−1 ∧ φ(|z − w|−2)−1 ∧ t

φ(|y − w|−2)−1

)η/2 ˆ t/2

0

ψη(s, w)qc(s, w, y)ds.

(4.2.18)

Proof. When η = 0, it is trivial, so we only consider the case η ∈ (0, 2).

Since the result is obvious for the case φ(δD(y)−2)−1 ≥ t or 2δD(y) ≥ δD(z)

we may and will assume δD(y) < φ−1(t−1)−1/2 ∧ (δD(z)/2). Note that

|y − z| ≥ δD(z)− δD(y) ≥ δD(z)

2≥ δD(y). (4.2.19)

59


Since φ(δD(y)−2)−1 < t, (4.2.5) implies(1 ∧ φ(δD(z)−2)−1

t

)(1 ∧ φ(δD(y)−2)−1

s

)=φ(δD(y)−2)−1

t

( t ∧ φ(δD(z)−2)−1

s ∨ φ(δD(y)−2)−1

)≤ 2(

1 ∧ φ(δD(y)−2)−1

t

) t ∧ φ(δD(z)−2)−1

s+ φ(δD(y)−2)−1. (4.2.20)

When φ(|w − y|−2)−1 ≤ s, by (4.2.19), Lemma 2.0.3 and the fact

|z − y| ≤ |z − y| ∧ |z − w|+ |z − y| ∧ |y − w|≤ 2(|z − y| ∧ |z − w|) ∨ 2(|z − y| ∧ |y − w|), (4.2.21)

we have

t ∧ φ(δD(z)−2)−1

s+ φ(δD(y)−2)−1≤ 4 · t ∧ φ(|z − y|−2)−1

φ(|w − y|−2)−1

≤ 16

(1 +

t ∧ φ(|z − y|−2)−1 ∧ φ(|z − w|−2)−1

φ(|w − y|−2)−1

).

(4.2.22)

To simplify the notation, we define A := (t/2) ∧ φ(|w − y|−2)−1. Plugging

(4.2.22) into (4.2.20) gives rise to

ψη(t, z)

ˆ t/2

A

qc,η(s, w, y)ds ≤ 25η/2ψη(t, y)

×(

1 +t ∧ φ(|z − y|−2)−1 ∧ φ(|z − w|−2)−1

φ(|w − y|−2)−1

)η/2 ˆ t/2

A


(4.2.23)

60


On the other hand, by (4.2.19),

ψη(t, y)

ˆ A

0

(t ∧ φ(δD(z)−2)−1

s+ φ(δD(y)−2)−1

)η/2ψη(s, w)qc(s, w, y)ds

≤ 2ηψη(t, y)

ˆ A

0

(t ∧ φ(|y − z|−2)−1

s

)η/2ψη(s, w)

sH(|w − y|−2)|w − y|d

ds

+ 2ηψη(t, y)

ˆ A

0

(t ∧ φ(|y − z|−2)−1

s

)η/2ψη(s, w)φ−1(s−1)d/2

× e−c|w−y|2φ−1(s−1)ds

= c1ψη(t, y)(t ∧ φ(|y − z|−2)−1)η/2H(|w − y|−2)

|w − y|d

ˆ A

0

s1−η/2ψη(s, w)ds

+ c1ψη(t, y)(t ∧ φ(|y − z|−2)−1)η/2ˆ A

0

s−η/2φ−1(s−1)d/2e−c|w−y|2φ−1(s−1)

× ψη(s, w)ds

=: I + II. (4.2.24)

We estimate the integral parts of I and II. By using (4.2.7) with B =

φ(δD(w)−2)−1, and a = b = η/2 firstly and then using (4.2.7) with the same

B, b and a = 0 secondly, we get

ˆ A

0

s1−η/2ψη(s, w)ds � A−η/2ˆ A

0

sψη(s, w)ds. (4.2.25)

61


Using Lemma 2.0.3 and inequality x(d+η)/2e−cx ≤ c2e− c

2x for x ≥ 1, we have

ˆ A

0

s−η/2φ−1(s−1)d/2e−c|w−y|2φ−1(s−1)ψη(s, w)ds

=

ˆ A

0

(φ(|w − y|−2)−1

s

)η/2 |w − y|d

φ−1(s−1)−d/2e−c|w−y|

2φ−1(s−1)φ(|w − y|−2)η/2

× |w − y|−dψη(s, w)ds

≤ˆ A

0

(|w − y|2

φ−1(s−1)−1

)η/2 |w − y|d

φ−1(s−1)−d/2e−c|w−y|

2φ−1(s−1)φ(|w − y|−2)η/2

× |w − y|−dψη(s, w)ds

≤ c2

ˆ A

0

e−c2|w−y|2φ−1(s−1)φ(|w − y|−2)

η2 |w − y|−dψη(s, w)ds

≤ c2

ˆ A

0

e−c2|w−y|2φ−1(s−1)φ(|w − y|−2)

η2φ−1(s−1)d/2ψη(s, w)ds. (4.2.26)

Therefore, by (4.2.24), (4.2.25), and (4.2.26), we have

ψη(t, y)

ˆ A

0

(t ∧ φ(δD(z)−2)−1

s+ φ(δD(y)−2)−1

)η/2ψη(s, w)qc(s, w, y)ds

≤ c3ψη(t, y)

(t ∧ φ(|y − z|−2)−1

t ∧ φ(|w − y|−2)−1

)η/2 ˆ A

0

ψη(s, w)sH(|w − y|−2)|w − y|d

ds

+ c3ψη(t, y)

(t ∧ φ(|y − z|−2)−1

φ(|w − y|−2)−1

)η/2 ˆ A

0

ψη(s, w)φ−1(s−1)d/2e−c2|w−y|2φ−1(s−1)ds

≤ c3ψη(t, y)

(1 +

t ∧ φ(|y − z|−2)−1

φ(|w − y|−2)−1

)η/2 ˆ A

0

ψη(s, w)qc/2(s, w, y)ds.

(4.2.27)

By using (4.2.21) and Proposition 4.2.2, there exists t0 = t0(a, c) > 0 such

that for t ≤ t0, (4.2.27) has the following upper bound:

c4ψη(t, y)

(1 +

t ∧ φ(|y − z|−2)−1 ∧ φ(|z − w|−2)−1

φ(|w − y|−2)−1

)η/2 ˆ A

0


(4.2.28)

62


Combining (4.2.23) and (4.2.28) gives rise to the result. 2

In Theorem 4.2.9 and Next section, we use the following notation: For

any (x, y) ∈ D ×D,

Vx,y :={(z, w) ∈ D ×D : |x− y| ≥ 4(|y − w| ∨ |x− z|)}Ux,y :=(D ×D) \ Vx,y.

Theorem 4.2.9. For every η ∈ [0, 2) and c1, c2 > 0, there exist C6 =

C6(a, γ, CL, c1, c2, η) > 0 and t0 = t0(a, c1, c2) > 0 such that for any non-

negative bounded function F (x, y) on D × D, the following are true for

(t, x, y) ∈ (0, t0]×D ×D.

(a) If tφ(|x− y|−2) ≥ 1, then

ˆ t

0

ˆD×D

qc1,η(t− s, x, z)qc2,η(s, w, y)

qc1∧c2,η(t, x, y)

F (z, w)H(|z − w|−2)|z − w|d

dzdwds

≤ C6

ˆ t

0

ˆD×D

ψη(s, z)qc1(s, x, z)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|x− z|−2)−1

)η/2× F (z, w)H(|z − w|−2)

|z − w|ddzdwds

+ C6

ˆ t

0

ˆD×D

ψη(s, w)qc2(s, y, w)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|y − w|−2)−1

)η/2× F (z, w)H(|z − w|−2)

|z − w|ddzdwds.

63


(b) If tφ(|x− y|−2) < 1, then

ˆ t

0

Ûx,y



F (z, w)H(|z − w|−2)|z − w|d

dzdwds

≤ C6

ˆ t

0

Ûx,y

ψη(s, z)qc1(s, x, z)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|x− z|−2)−1

)η/2× F (z, w)H(|z − w|−2)

|z − w|ddzdwds

+ C6

ˆ t

0

Ûx,y

ψη(s, w)qc2(s, y, w)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|y − w|−2)−1

)η/2× F (z, w)H(|z − w|−2)

|z − w|ddzdwds.

(c) If tφ(|x− y|−2) < 1, then

ˆ t

0

ˆVx,y



F (z, w)H(|z − w|−2)|z − w|d

dzdwds ≤ C6‖F‖∞.

Proof. To prove (a), we assume that tφ(|x − y|−2) ≥ 1. By (4.2.14) and

Lemma 4.2.8, there exists t1 = t1(a, c1, c2) > 0 such that for t ∈ (0, t1],

ˆ t/2

0

qc1,η(t− s, x, z)qc2,η(s, w, y)ds ≤ c3qc1,η(t, x, z)

ˆ t/2

0

qc2,η(s, w, y)ds

≤ c4ψη(t, x)qc1(t, x, z)ψη(t, y)

(1 +

t ∧ φ(|z − w|−2)−1

φ(|w − y|−2)−1

)η/2×ˆ t/2

0

ψη(s, w)qc2(s, w, y)ds

≤ ec1c4qc1,η(t, x, y)

(1 +

t ∧ φ(|z − w|−2)−1

φ(|w − y|−2)−1

)η/2 ˆ t/2

0


where we used the fact that qc1(t, x, z) ≤ φ−1(t−1)d/2 = ec1qc1(t, x, y) if tφ(|x−

64


y|−2) ≥ 1. Similarly,

ˆ t

t/2

qc1,η(t− s, x, z)qc2,η(s, w, y)ds

≤ c5qc2,η(t, x, y)

(1 +

t ∧ φ(|z − w|−2)−1

φ(|x− z|−2)−1

)η/2 ˆ t

t/2

ψη(t− s, z)qc1(t− s, x, z)ds.

These two inequalities gives (a).

For (b), we denote F (z,w)H(|z−w|−2)|z−w|d by J(z, w) for the sake of brevity. We

first consider the integral

I1 =

ˆ t/2

0

ˆD

ˆ|x−z|> |x−y|

4

qc1,η(t− s, x, z)qc2,η(s, w, y)J(z, w)dzdwds.

Since there exists t2 = t2(a, c1, c2) > 0 such that

qc1,η(t− s, x, z) ≤ c6ψη(t, x)qc1(t, x, y)ψη(t, z)

for t ∈ (0, t2] and (s, z, w) ∈ (0, t/2]× {z : |x− z| > |x− y|/4} ×D (4.2.14)

and Proposition 4.2.2, it follows from Lemma 4.2.8 that for t ∈ (0, t2]

I1 ≤ c6ψη(t, x)qc1(t, x, y)

ˆD

ˆ|x−z|> |x−y|

4

ψη(t, z)

ˆ t/2

0

qc2,η(s, w, y)dsJ(z, w)dzdw

≤ c7qc1,η(t, x, y)

ˆD

ˆ|x−z|> |x−y|

4

ˆ t/2

0


×(

1 +φ(|z − w|−2)−1 ∧ tφ(|w − y|−2)−1

)η/2J(z, w)dzdw.

Now, we estimate

I2 =

ˆ t/2

0

ˆD

ˆ|w−y|> |x−y|

4

qc1,η(t− s, x, z)qc2,η(s, w, y)J(z, w)dwdzds.

Note that qc1,η(t− s, x, z) ≤ c8qc1,η(t, x, z) and

qc2,η(s, w, y) ≤ c9ψη(s, w)ψη(s, y)qc2(t, x, y)

65


for t ∈ (0, t2] and (s, z, w) ∈ (0, t/2] × D × {w ∈ D : |w − y| > |x − y|/4}.Indeed, the first inequality comes from (4.2.14) and the second inequal-

ity follows from the following inequality: Since |w − y|2φ−1(s−1) ≥ |x −y|2φ−1(t−1)/8 ≥ 1/8 and xd/2e−c2x ≤ c10e

− c22x for x ≥ 1/8,

qc2(s, w, y) ≤ sH(|w − y|−2)|w − y|d

+ φ−1(s−1)d/2e−c2|w−y|2φ−1(s−1)

≤ c11tH(|x− y|−2)|x− y|d

+ |w − y|−d|w − y|dφ−1(s−1)d/2e−c2|w−y|2φ−1(s−1)

≤ c11tH(|x− y|−2)|x− y|d

+ 4dc10|x− y|−de−c22|w−y|2φ−1(s−1)

≤ c11tH(|x− y|−2)|x− y|d

+ 4dc10φ−1(t−1)d/2e−

c216|x−y|2φ−1(t−1)

and Proposition 4.2.2. Therefore by (4.2.7) for t ∈ (0, t2]

I2 ≤ c12qc2(t, x, y)

ˆD

ˆ|w−y|> |x−y|

4

qc1,η(t, x, z)

ˆ t/2

0

ψη(s, w)ψη(s, y)ds

× J(z, w)dwdz

≤ c12ψη(t, x)qc2(t, x, y)

ˆD

ˆ|w−y|> |x−y|

4

ψη(t, z)qc1(t, x, z)

ˆ t/2

0

ψη(s, y)ds

× J(z, w)dwdz

≤ c13qc2,η(t, x, y)

ˆD

ˆ|w−y|> |x−y|

4

tψη(t, z)qc1(t, x, z)J(z, w)dwdz

≤ c14qc2,η(t, x, y)

ˆ t/2

0

ˆD

ˆ|w−y|> |x−y|

4

ψη(t− s, z)qc1(t− s, x, z)J(z, w)dwdzds,

where we used (4.2.14) in the last inequality. Hence we obtain that for

66


t ∈ (0, t2],

ˆ t/2

0

Ûx,y

qc1,η(t− s, x, z)qc2,η(s, w, y)J(z, w)dzdwds ≤ I1 + I2

≤ c15qc1,η(t, x, y)

(Ûx,y

ˆ t/2

0

ψη(t− s, z)qc1(t− s, x, z)ds

×(

1 +φ(|z − w|−2)−1 ∧ tφ(|x− z|−2)−1

)η/2J(z, w)dzdw

+

Ûx,y

ˆ t/2

0


(1 +

φ(|z − w|−2)−1 ∧ tφ(|w − y|−2)−1

)η/2J(z, w)dzdw

).

By exchanging the role of x for y, z for w, and s for t−s and using Proposition

4.2.2, we can obtain the remaining part.

For (c), we first observe that for (z, w) ∈ Vx,y,

F (z, w)H(|z − w|−2)|z − w|d

≤ c16‖F‖∞H(|x− y|−2)|x− y|d

.

We claim that there exists t3 = t3(a, c1, c2) > 0 such that

ˆ t

0

ˆVx,y

qc1,η(t− s, x, z)qc2,η(s, w, y)dzdwds ≤ c7tψγ(t, x)ψγ(t, y).

By the symmetry of qc,η it suffices to show that

I3 :=

ˆ t/2

0

ˆVx,y

qc1,η(t− s, x, z)qc2,η(s, w, y)dzdwds ≤ c17tψη(t, x)ψη(t, y).

By (4.2.14) and Lemma 4.2.8, there exists t3 = t3(a, c1, c2) > 0 such that for

t ∈ (0, t3], qc1,η(t− s, x, z) ≤ c18qc1,η(t, x, z) for s ∈ (0, t/2] and

ˆ t/2

0

qc1,η(t− s, x, z)qc2,η(s, w, y)ds ≤ c19ψη(t, x)qc1(t, x, z)ψη(t, y)

×(

1 +t

φ(|w − y|−2)−1

)η/2 ˆ t/2

0

ψη(s, w)qc2(s, w, y)ds.

67


By integrating both sides in z and w, we get

I3 ≤ c19ψη(t, x)ψη(t, y)

ˆ t/2

0

ˆRdqc2(s, w, y)

(1 +

t

φ(|w − y|−2)−1

)η/2dwds.

Since we can see easily that´ t/20

´Rd qc2(s, w, y)dwds ≤ c20t, we only need to

check that

ˆ t/2

0

ˆRdqc2(s, w, y)

(t

φ(|w − y|−2)−1

)η/2dwds ≤ c21t,

which follows from the following two estimates:

ˆ|w|≤φ−1(s−1)−1/2

φ−1(s−1)d/2(

t

φ(|w|−2)−1

)η/2dw ≤ c22

ˆ|w|≤1

(1

|w|

)ηdw

(t

s

)η/2≤ c23(t/s)

η/2

and

ˆ|w|≥φ−1(s−1)−1/2

s tη/2φ(|w|−2)1+η/2

|w|ddw ≤ c24

ˆ ∞φ−1(s−1)−1/2

s tη/2φ(r−2)1+η/2

rdr

≤ c25(t/s)η/2.

We have finished the proof. 2

4.3 Main Result

In this section we always assume that φ satisfies La(γ, CL) for some a > 0

and η ∈ [0, 2). Now we follow [13]. Using (4.1.2), (4.1.6), Proposition 4.2.2,

Theorem 4.2.7 and Theorem 4.2.9, we can choose constants

M = M(C0, C1, C2, C4, C5, η, d) >2d+5+η/2

2− ηC3

0C1(C2 ∨ C4 ∨ C5 ∨ C6)

(4.3.1)

68


and 0 < t0 = t0(a, γ, CL, a2) < 1 such that for any µ in Kφ,η, any measurable

function F with F1 = eF − 1 ∈ Jφ,η and any (t, x, y) ∈ (0, t0]×D ×D,

ˆ t

0

ˆD

qa2,η(t− s, x, z)qa2,η(s, z, y)|µ|(dz)ds

≤ C4qa2,η(t, x, y) supu∈D

ˆ t

0

ˆD

ψη(s, z)qa2(s, u, z)|µ|(dz)ds

≤ C4C1qa1,η(t, x, y)Nφ,ηµ,a2

(t) ≤MpD(t, x, y)Nφ,ηµ,a2

(t), (4.3.2)ˆ t

0

ˆD×D

qa2,η(t− s, x, z)qa2,η(s, w, y)c(z, w)|F1|(z, w)H(|z − w|−2)

|z − w|ddzdwds

≤MpD(t, x, y)(Nφ,ηF1,a2

(t) + ‖F1‖∞1{tφ(|x−y|−2)<1}

)(4.3.3)

and

ˆ t

0

ˆUx,y

qa2,η(t− s, x, z)qa2,η(s, w, y)c(z, w)|F1|(z, w)H(|z − w|−2)

|z − w|ddzdwds

≤MpD(t, x, y)Nφ,ηF1,a2

(t). (4.3.4)

In the remainder of this section, we fix a locally finite signed measure µ ∈Kφ,η, a measurable function F with F1 = eF − 1 ∈ Jφ,η and the constant

M > 0 and t0 > 0 in (4.3.1). We define pk(t, x, y) as shown in (4.1.12) for

non-negative integer k.

Lemma 4.3.1. For every k ≥ 0 and (t, x) ∈ (0, t0]×D,

ˆD

|pk(t, x, y)|dy ≤ C20ψη(t, x)

(MNφ,η

µ,F1,a2(t))k, (4.3.5)

ˆD

|pk(t, y, x)|dy ≤ C20ψη(t, x)

(MNφ,η

µ,F1,a2(t))k. (4.3.6)

69


Proof. We use induction on k ≥ 0. It is clear when k = 0. Suppose (4.3.5)

is true for k − 1 ≥ 0. Then we have

ˆD

|pk(t, x, y)|dy ≤ˆ t

0

ˆD

p0(t− s, x, z)

ˆD

|pk−1(s, z, y)|dy|µ|(dz)ds

+

ˆ t

0

ˆD

ˆD

p0(t− s, x, z)|c(z, w)F1(z, w)|

|z − w|dH(|z − w|−2)−1

ˆD

|pk−1(s, w, y)|dydzdwds

=: I1 + I2.

We first estimate I1 by using (4.1.2), the induction hypothesis and Lemma

4.2.6 as follows.

I1 ≤ C30(MNφ,η

µ,F1,a2(t))k−1

ˆ t

0

ˆD

qa2,η(t− s, x, z)ψη(s, z)|µ|(dz)ds

≤ 2η/2C30(MNφ,η

µ,F1,a2(t))k−1

(ψη(t, x)

ˆD

ˆ t/2

0

qa2(t− s, x, z)ψη(t− s, z)ds|µ|(dz)

+

ˆD

ψη(t, z)

ˆ t

t/2

qa2,η(t− s, x, z)ds|µ|(dz)

)≤ 2η/2C3

0(MNφ,ηµ,F1,a2

(t))k−1(ψη(t, x)Nφ,η

µ,a2(t) + C2ψη(t, x)Nφ,η

µ,a2(t))

≤ C20ψη(t, x)Mk

(Nφ,ηµ,F1,a2

(t))k−1

Nφ,ηµ,a2

(t).

70


Secondly, we estimate I2 by using the induction hypothesis and Lemma 4.2.8.

In the following estimate, we denote |F1(z,w)|H(|z−w|−2)|z−w|d by J1(z, w).

I2 ≤ C40(MNφ,η

µ,F1,a2(t))k−1

ˆD×D

ˆ t

0

qa2,η(t− s, x, z)ψη(s, w)dsJ1(z, w)dzdw


µ,F1,a2(t))k−1

ˆD×D

(ψη(t, x)

ˆ t/2

0

qa2(t− s, x, z)ψη(t− s, z)ds

+ ψη(t, w)

ˆ t

t/2

qa2,η(t− s, x, z)ds

)J1(z, w)dzdw


µ,F1,a2(t))k−1ψη(t, x)

( ˆ t/2

0

ˆD×D

ψη(t− s, z)qa2(t− s, x, z)

× J1(z, w)dzdwds+ C5

ˆ t

t/2

ˆD×D

ψη(t− s, z)qa2(t− s, x, z)

×(

1 +φ(|z − w|−2)−1 ∧ tφ(|z − w|−2)−1

)η/2J1(z, w)dzdwds

)≤ C2

0ψη(t, x)Mk(Nφ,ηµ,F1,a2

(t))k−1Nφ,ηF1,a2

(t).

Summing up the above estimates for I1 and I2, we get

ˆD

|pk(t, x, y)|dy ≤ C20ψη(t, x)

(MNφ,η

µ,F1(t))k.

Repeating the induction argument for pk(t, y, x), together with the symmetry

of qa2,η(t, ·, ·), follows the remaining part. 2

Lemma 4.3.2. For every k ≥ 0 and (t, x, y) ∈ (0, t0]×D ×D,

ˆ t

0

ˆD

p0(t− s, x, z)dz

ˆD

|pk(s, w, y)|dwds

≤ t2

2− η2η/2C4

0Mkψη(t, x)ψη(t, y)

(Nφ,ηµ,F1,a2

(t))k.

71


Proof. By increasing the value C0 to appear in (4.1.2) if necessary, we can

choose C0 so that ˆRdqa2(t, x, y)dy ≤ C0.

Therefore, by (4.3.6) and our choice of C0, we have

ˆ t

0

ˆD

p0(t− s, x, z)dz

ˆD

|pk(s, w, y)|dwds

≤ˆ t

0

ˆD

C0qa2,η(t− s, x, z)dzC20ψη(s, y)(MNφ,η

µ,F1,a2(s))kds

≤ C30(MNφ,η

µ,F1,a2(t))k

ˆ t

0

ψη(t− s, x)ψη(s, y)

ˆD

qa2(t− s, x, z)dzds


µ,F1,a2(t))k

(ψη(t, x)

ˆ t/2

0

ψη(s, y)ds

+ ψη(t, y)

ˆ t

t/2

ψη(t− s, x)ds

)≤ 2η/2+1C4

0

2− ηt ψη(t, x)ψη(t, y)(MNφ,η

µ,F1,a2(t))k,

where we used (4.2.7) in the last inequality. 2

Lemma 4.3.3. For k ≥ 0 and (t, x, y) ∈ (0, t0]×D ×D we have

(1) The following inequality holds true.

|pk(t, x, y)|

≤ p0(t, x, y)(

(C20MNφ,η

µ,F1,a2(t))k + k‖F1‖∞C2

0M(C20MNφ,η

µ,F1,a2(t))k−1

)(4.3.7)

(2) pk(t, x, y) is continuous in (t, y) for each fixed x, and is also continuous

in (t, x) for each fixed y.

Proof. For the sake of brevity, we define B(t) = C20MNφ,η

µ,F1,a2(t), and

Ak(t) = B(t)k + k‖F1‖∞C20MB(t)k−1.

72


Note that B(t) is increasing in t, so Ak(t) is. We also denote the first term

and second term of the left hand side in (4.1.12) by Ik(t, x, y) and Jk(t, x, y)

respectively. We need to show that |pk(t, x, y)| ≤ p0(t, x, y)Ak(t). Since

A0(t) = 1, it is obvious when k = 0. Suppose that (4.3.7) holds true for

k − 1 ≥ 0, i.e.

|pk−1(t, x, y)| ≤ p0(t, x, y)Ak−1(t). (4.3.8)

Firstly, from (4.3.8) and (4.3.2) we have

|Ik(t, x, y)| ≤ Ak−1(t)

ˆ t

0

ˆD

p0(t− s, x, z)p0(s, z, y)|µ|(dz)ds

≤ Ak−1(t)C20

ˆ t

0

ˆD

qa2,η(t− s, x, z)qa2,η(s, z, y)|µ|(dz)ds

≤ p0(t, x, y)Ak−1(t)C20MNφ,η

µ,a2(t). (4.3.9)

When tφ(|x− y|−2) ≥ 1, it follows from (4.3.8) and (4.3.3) that

|Jk(t, x, y)|

≤ Ak−1(t)

ˆ t

0

ˆD×D

p0(t− s, x, z)c(z, w)F1(z, w)

|z − w|dH(|z − w|−2)−1p0(s, w, y)dzdwds

≤ Ak−1(t)C20Mp0(t, x, y)Nφ,η

F1,a2(t). (4.3.10)

Summing up (4.3.9) and (4.3.10), the result is obtained as follows:

|pk(t, x, y)| ≤ p0(t, x, y)Ak−1(t)B(t) ≤ p0(t, x, y)Ak(t).

73


When tφ(|x− y|−2) < 1, it follows from (4.3.8) and (4.3.4) that

|Jk(t, x, y)|

≤ Ak−1(t)

ˆ t

0

ˆUx,y

p0(t− s, x, z)c(z, w)|F1(z, w)|

|z − w|dH(|z − w|−2)−1p0(s, w, y)dzdwds

+

ˆ t

0

ˆVx,y

p0(t− s, x, z)c(z, w)|F1(z, w)|

|z − w|dH(|z − w|−2)−1|pk−1(s, w, y)|dzdwds

≤ p0(t, x, y)Ak−1(t)C20MNφ,η

F1,a2(t)

+2d+4C0‖F1‖∞H(|x− y|−2)

|x− y|d

ˆ t

0

ˆD×D

p0(t− s, x, z)|pk−1(s, w, y)|dzdwds.

(4.3.11)

Applying Lemma 4.3.2 and using (4.3.1), the last term of (4.3.11) is bounded

by

ψη(t, x)ψη(t, y)tH(|x− y|−2)|x− y|d

‖F1‖∞2d+5+η/2C5

0

2− ηB(t)k−1

≤ C20Mp0(t, x, y)‖F1‖∞B(t)k−1.

Thus when tφ(|x− y|−2) < 1, we have from (4.3.9) and (4.3.11) that

|pk(t, x, y)| ≤ p0(t, x, y)(Ak−1(t)B(t) + C2

0M‖F1‖∞B(t)k−1)

= p0(t, x, y)Ak(t).

Obviously, (4.3.3) holds true for k = 0, by the assumption on the joint

continuity of pD. Suppose that (4.3.3) holds true for k − 1 ≥ 0. For fixed

x ∈ D and 0 ≤ ε < δ/2, define the function

ϕε(t, y) =

ˆ 1

0

ˆD

p0(t− s, x, z)pk−1(s, z, y)1[ε,t](s)µ(dz)ds

+

ˆ 1

0

ˆD×D

p0(t− s, x, z)c(z, w)F1(z, w)

|z − w|dH(|z − w|−2)−1pk−1(s, w, y)1[ε,t](s)dzdwds

where δ > 0 is a small number. It follows from (4.3.7), (4.1.9), and (4.1.10)

74


that

supδ<t<1−δ,y∈D

|ϕε(t, y)− ϕ0(t, y)|

≤ˆ 1

0

ˆD

p0(t− s, x, z)p0(s, z, y)Ak−1(s)1[0,ε)(s)|µ|(dz)ds

+

ˆ 1

0

ˆD×D

p0(t− s, x, z)c(z, w)|F1(z, w)|

|z − w|dH(|z − w|−2)−1p0(s, w, y)Ak−1(s)

× 1[0,ε)(s)dzdwds

≤ φ−1(2/δ)d/2Ak−1(ε)Nφ,ηµ,a2

(ε) + C0φ−1(2/δ)d/2Ak−1(ε)N

φ,ηF1,a2

(ε)

≤ φ−1(2/δ)d/2Ak(ε),

which means that ϕε converges to ϕ0 uniformly in (δ, 1 − δ) × D. Since

ϕ0(t, y) = pk(t, x, y), it suffices to show that ϕε is continuous. To this end we

note that by (4.3.7)

sup{|pk−1(t− s, w, y)1[0,t−ε](s) : δ < t < 1− δ, and w, y ∈ D}≤ φ−1(1/ε)d/2Ak−1(1− δ) <∞.

Therefore using dominated convergence theorem, ϕε is continuous provided

pk−1(t, x, y) is continuous in (t, y). The same argument is applied for the

continuity of pk(t, x, y) in (t, x) for each fixed y ∈ D. 2

Theorem 4.3.4. Suppose that µ ∈ Kφ,η and F1 := eF − 1 ∈ Jφ,η. Then the

series∑∞

k=0 pk(t, x, y) converges absolutely to a jointly continuous function

qD(t, x, y) on (0,∞)×D×D. Moreover, qD(t, x, y) is the transition density

of the Feynman-Kac semigroup (T µ,Ft ; t ≥ 0) and for any T > 0 there exists

a positive constant C7 = C7(a, γ, CL, a2, C0,M, ||F1||∞, T ) > 0 such that

qD(t, x, y) ≤ C7qa2,η(t, x, y)

for every (t, x, y) ∈ (0, T ]×D ×D.

75


Proof. Since F1 ∈ Jφ,η, there is t1 ∈ (0, 1) so that

Nφ,ηµ,F1,a2

(t1) ≤ (3C20M)−1.

It follows from Lemma 4.3.3 that for every (t, x, y) ∈ (0, t1]×D ×D,

qD(t, x, y) ≤ p0(t, x, y) +∞∑k=1

|pk(t, x, y)|

≤ p0(t, x, y) + p0(t, x, y)

( ∞∑k=1

(C20MNφ,η

µ,F1,a2(t))k

+ ‖F1‖∞C20M

∞∑k=1

k(C20MNφ,η

µ,F1,a2(t))k−1

)≤ pD(t, x, y) + pD(t, x, y)

(1

2+

9

4‖F1‖∞C2

0M

)≤ c1qa2,η(t, x, y). (4.3.12)

Here c1 = C0(32

+ 94‖F1‖∞C2

0M). Using the semigroup property of T µ,Ft , we

see that for small δ > 0,

qD(t, x, y) =

ˆD

qD(t− δ, x, z)qD(δ, z, y)dz for all (t, x, y) ∈ (0, t1]×D ×D.

Since the integrand is controlled by

c2qa2(t− δ, x, z)qa2(δ, z, y),

and is continuous in (t, x, y) by Lemma 4.3.3, qD(t, x, y) is jointly continuous

on (0, t1] × D × D by the dominated convergence theorem. The semigroup

property of T µ,Ft allows to define qD(t, x, y) for (t, x, y) ∈ (0, 2t1]×D×D as

follows:

qD(t, x, y) =

ˆD

qD(t/2, x, z)qD(t/2, z, y)dz.

76


Thus, by (4.3.12) and Remark 4.2.3, we have for (t, x, y) ∈ (0, 2t1]×D ×D

qD(t, x,y) ≤ c21

ˆD

qa2,η(t/2, x, z)qa2,η(t/2, z, y)dz

≤ c21ψη(t, x)ψη(t, y)

ˆRdqa2(t/2, x, z)qa2(t/2, z, y)dz ≤ c21c2qa2,η(t, x, y).

Iterate the above argument one can deduce that there is a constant c3 > 0

so that

qD(t, x, y) ≤ c3qa2,η(t, x, y) for every t ≤ T and x, y ∈ D.

Moreover, by using the discussion in [13, Section 1.2] and the semigroup

property of T µ,Ft , we can extend qD(t, x, y) on (0,∞) ×D ×D which is the

transition density of T µ,Ft . 2

For the lower bound estimate, we need to assume that F is a function in

Jφ,η. The following theorem is our main result.

Theorem 4.3.5. Suppose that µ ∈ Kφ,η and F is a function in Jφ,η. Then for

every T > 0 there exists a positive constant C8 := C8(a, γ, CL, a2, φ, η, C0,M,

Nφ,ηµ,F,a2

, ‖F‖∞, T ) ≥ 1 such that

C−18 qa1,η(t, x, y) ≤ qD(t, x, y) ≤ C8qa2,η(t, x, y) (4.3.13)

for every (t, x, y) ∈ (0, T ]×D ×D.

Proof. Note that if F ∈ Jφ,η, since F is bounded, F1 := eF − 1 also belongs

to Jφ,η. Thus, Theorem 4.3.4 implies the upper bound estimates in (4.3.13).

The proof of lower bound estimates is the same as that of [13, Theorem 3.5].

We just write the last parts of the proof of [13, Theorem 3.5] in detail.

Note that for n ≥ 1 and (t, x, y) ∈ (0, n/φ(a)∧4n2ρa22/φ(a)∧n]×D×D,

where ρ is the constant in the proof of Proposition 4.2.2, by using (4.1.2),

Lemma 2.0.5(i), Lemma 2.0.5(ii), and Proposition 4.2.2, we have

pD(t/n, x, y) ≥ C−10 qa1,η(t/n, x, y) ≥ c1qa1,η(t/n2, x, y) (4.3.14)

≥ c2qa2,η(t/n2, x, y) ≥ c3pD(t/n2, x, y) (4.3.15)

77


and for (t, x, y) ∈ (0, 1/φ(a) ∧ 4a21ρ/φ(a) ∧ n] × D × D, by using (4.1.2),

Lemma 2.0.5(i), Lemma 2.0.5(ii), and Proposition 4.2.2, we have

pD(t/n, x, y) ≥ C−10 qa1,η(t/n, x, y) ≥ c4qa1C−1/γL n1/γ ,η

(t, x, y) ≥ c5qa1,η(t, x, y).

(4.3.16)

Let T > 0. We may assume that a < φ−1(1/(2T )) ∧ φ−1(4a21ρ/(2T )) ∧φ−1(4a22ρ/(2T )) by [24, Remark 2.2]. By the same way as in the proof of [13,

Theorem 3.5], we have that for (t, x, y) ∈ (0, t0]×D ×D,

qD(t, x, y) ≥ c6pD(t, x, y). (4.3.17)

Choose n ≥ 1 satisfying T ≤ nt0 < 2T . Then, for (t, x, y) ∈ (0, T ]×D ×D,

by (4.3.17), (4.3.14) and (4.3.16) we have

qD(t, x, y) =

ˆD

· · ·ˆD

qD(t/n, x, z1) · · · qD(t/n, zn−1, y)dz1 · · · dzn−1

≥ cn6

ˆD

· · ·ˆD

pD(t/n, x, z1) · · · pD(t/n, zn−1, y)dz1 · · · dzn−1

≥ cn6cn3

ˆD

· · ·ˆD

pD(t/n2, x, z1) · · · pD(t/n2, zn−1, y)dz1 · · · dzn−1

= cn6cn3pD(t/n, x, y)

≥ c5cn6cn3qa1,η(t, x, y).

2

4.4 Large time heat kernel estimates and Green

function estimates

In this section we always assume that φ satisfies La(γ, CL) for some a > 0 and

η ∈ [0, 2). Recall that X be a Hunt process on D with the corresponding

transition semigroup Pt which has a jointly continuous density pD(t, x, y).

78


Let {Pt : t ≥ 0} be the dual semigroup of {Pt : t ≥ 0} defined by

Ptf(x) =

ˆD

pD(t, y, x)f(y)dy.

Note that by (4.1.2) and (4.1.4),

ˆD

pD(t, y, x)dy ≤ C0

ˆD

qa2,η(t, y, x) ≤ c1, (4.4.1)

where c1 = c1(C0, a2). Therefore for any f ∈ L2(D; dx), by using Holder

inequality, Fubini theorem and (4.4.1), we have

||Ptf ||22 ≤ c1||f ||22. (4.4.2)

Lemma 4.4.1. Pt and Pt are strongly continuous semigroups in L2(D; dx).

Proof. Recall the general results on the first exit time for the process with

right continuous path that for any open subset D of Rd and any x ∈ D,

limt→0

Px(τD ≤ t) = 0. (4.4.3)

We will show first that Pt is strongly continuous on Cc(D). Let f be a

function in Cc(D) and fix x ∈ D. Given ε > 0, choose δ > 0 such that

|f(y)− f(x)| < ε/2 for y ∈ B(x, δ) ⊂ D. Then for x ∈ D,

|Ptf(x)− f(x)|

=

∣∣∣∣ˆD

pD(t, x, y)(f(y)− f(x))dy − f(x)(1− Px(τD > t))

∣∣∣∣≤ ε

2

ˆB(x,δ)

pD(t, x, y)dy + 2‖f‖∞ˆD\B(x,δ)

pD(t, x, y)dy + ‖f‖∞Px(τD ≤ t)

≤ ε/2 + 2‖f‖∞Px(|Xt − x| ≥ δ) + ‖f‖∞Px(τD ≤ t)

≤ ε/2 + 2‖f‖∞Px(τB(x,δ) ≤ t) + ‖f‖∞Px(τD ≤ t).

79


Applying (4.4.3) to D and B(x, δ), we get

limt↓0|Ptf(x)− f(x)| = 0 for x ∈ D.

Since Cc(D) is dense in L2(D; dx) with respect to ‖ · ‖L2(D;dx) and (4.4.2)

holds, it follows from [28, Proposition II.4.3] that Pt is strongly continuous

on L2(D; dx). Finally, by [30, Corollary 1.10.6], Pt is strongly continuous on

L2(D; dx). 2

Recall that Aµ,Ft is an additive functional defined by

Aµ,Ft = Aµt +∑0<s≤t

F (Xs−, Xs)

where Aµt is the corresponding continuous additive functional associated with

Revuz measure µ and the semigroup T µ,Ft , t ≥ 0 given by

T µ,Ft f(x) = Ex[exp

(Aµ,Ft

)f(XD

t )].

The following lemmas prove that T µ,Ft is a strongly continuous semigroup in

L2(D; dx).

Lemma 4.4.2. Suppose that µ is a nonnegative measure on D and F is a

nonnegative measurable function on D ×D. Then we have

supx∈D

Ex[(Aµ,Ft )2] ≤ 2 supx∈D

(Ex[Aµ,Ft

])2+ sup

x∈DEx

[∑0<s≤t

F (Xs−, Xs)2

].

(4.4.4)

Proof. The Stieltjes integral calculus implies that for a right continuous

function A(s) of bounded variation on [0, t],

A(t)2 − A(0)2 =

ˆ(0,t]

A(s−) dA(s) +

ˆ(0,t]

A(s) dA(s)

= 2

ˆ(0,t]

A(s) dA(s)−∑0<s≤t

(A(s)− A(s−))2. (4.4.5)

80


By applying (4.4.5) to A(s) = Aµ,Ft − Aµ,Fs on [0, t], we have

(Aµ,Ft )2 = 2

ˆ(0,t]

(Aµ,Ft − Aµ,Fs ) dAµ,Fs +∑0<s≤t

F (Xs−, Xs)2.

Using the additive property of Aµ,Ft (see [23, 245p]), we get

Ex[(Aµ,Ft )2

]≤ 2Ex

[ˆ(0,t]

EXs[Aµ,Ft

]dAµ,Fs

]+ Ex

[∑0<s≤t

F (Xs−, Xs)2

]

≤ 2

(supx∈D

Ex[Aµ,Ft

])Ex[Aµ,Ft

]+ Ex

[∑0<s≤t

F (Xs−, Xs)2

].

Taking supremum for x ∈ D gives the result. 2

Corollary 4.4.3. For µ ∈ Kφ,η and F ∈ Jφ,η, we have

limt↓0

supx∈D

Ex[(Aµ,F1t

)2]= 0. (4.4.6)

Proof. Note that if F ∈ Jφ,η since F is bounded, F1 := eF − 1 and F 21 also

belong to Jφ,η. Since the total variational process of Aµ,F1t is A

|µ|,|F1|t , the

result follows. 2

Lemma 4.4.4. T µ,Ft is a strongly continuous semigroup in L2(D; dx).

Proof. Since qD(t, x, y) = p0(t, x, y) + p1(t, x, y) +∑∞

k=2 pk(t, x, y), it follows

81


from (4.3.7) that

|T µ,Ft f(x)− PDt f(x)|2

=

∣∣∣∣∣ˆD

(p1(t, x, y) +∞∑k=2

pk(t, x, y))f(y)dy

∣∣∣∣∣2

≤ 4∣∣∣Ex [Aµ,F1

t f(Xt)]∣∣∣2 + 4

∣∣∣∣∣ˆD

∞∑k=2

|pk(t, x, y)||f(y)|dy

∣∣∣∣∣2

≤ 4Ex[(Aµ,F1

t )2]Ex[f(Xt)

2]

+ 4

(∞∑k=2

Ak(t)

)2(ˆD

p0(t, x, y)|f(y)|dy)2

≤ 4

supx∈D

Ex[(Aµ,F1

t )2]

+ supx∈D

(∞∑k=2

Ak(t)

)2Ex

[f(Xt)

2].

Integrating both sides on D, we have

‖(T µ,Ft − Pt)f‖L2(D;dx)

≤ 4

supx∈D

Ex[(Aµ,F1

t )2]

+ supx∈D

(∞∑k=2

Ak(t)

)2ˆ

D

ˆD

pD(t, x, y)f(y)2dydx

≤ 4

supx∈D

Ex[(Aµ,F1

t )2]

+ supx∈D

(∞∑k=2

Ak(t)

)2 ‖f‖L2(D;dx).

Since Ak(t) = B(t)k + k‖F1‖∞C20MB(t)k−1, we have

∞∑k=2

Ak(t) =B(t)2

1−B(t)+ ‖F1‖∞C2

0M(2−B(t))B(t)

(1−B(t))2.

Together with (4.4.6) we have

limt↓0‖(T µ,Ft − Pt)f‖L2(D;dx) = 0

for any f ∈ L2(D; dx). Since Pt is strongly continuous semigroup in L2(D; dx),

82


the result follows from

‖T µ,Ft f − f‖L2(D;dx) ≤ ‖(T µ,Ft − Pt)f‖L2(D;dx) + ‖Ptf − f‖L2(D;dx).

2

In the remainder of this section, we assume that D is a bounded domain in

Rd. Since (x, y) 7→ qD(t, x, y) is bounded in D×D and the Lebesgue measure

ofD is finite, T µ,Ft is compact operator in L2(D; dx) for each t ∈ (0,∞). It fol-

lows from the above lemmas that T µ,Ft f(x) =´DqD(t, x, y)f(y)dy is strongly

continuous semigroup of compact operators on L2(D; dx). Let {T µ,Ft : t ≥ 0}be the adjoint semigroup of {T µ,Ft : t ≥ 0}. Then

T µ,Ft f(x) =

ˆD

qD(t, y, x)f(y)dy

and it is well known that {T µ,Ft : t ≥ 0} is a strongly continuous semigroup

on L2(D; dx). (See [30, Corollary 1.10.6, Lemma 1.10.1].) Let L and L be the

infinitesimal generators of the semigroups {T µ,Ft } and {T µ,Ft } on L2(D; dx),

respectively. It follows from Jentzsch’s Theorem ([32, Theorem V.6.6, 337p])

and the strong continuity of {T µ,Ft } and {T µ,Ft } on L2(D; dx) that the com-

mon value

−λ0 := sup{Reλ : λ ∈ σ(L)} = sup{Reλ : λ ∈ σ(L)} (4.4.7)

is an eigenvalue of multiplicity one for both L and L, and that an eigen-

function φ0 of L associated with −λ0 can be chosen to be strictly positive

almost everywhere with ‖φ0‖L2(D;dx) = 1 and an eigenfuction φ0 of L associ-

ated with −λ0 can be chosen to be strictly positive almost everywhere with

‖φ0‖L2(D;dx) = 1. Thus for x ∈ D,

e−λ0tφ0(x) =

ˆD

qD(t, x, z)φ0(z)dz, e−λ0tφ0(x) =

ˆD

qD(t, z, x)φ0(z)dz

(4.4.8)

Lemma 4.4.5. φ0(x) is strictly positive and continuous in D.

83


Proof. Note that by [32, Theorem V.6.6], φ0(x) is strictly positive a.e. in

D. Fix x ∈ D and consider a sequence xn ∈ D converging to x. Since

φ0(xn) = eλ0

ˆD

qD(1, xn, y)φ0(y)dy

and qD(1, xn, y) ≤ c1qa2,η(1, xn, y) ≤ c2qa2,η(1, x, y) for any |xn−x| ≤ (δD(x)∧|x−y|)/2, it follows from the dominated convergence theorem and continuity

of qD(t, x, y) that φ0 is continuous in D, and so φ0 is strictly positive in D.

2

Recall that ψη is defined in (4.1.1).

Lemma 4.4.6. There exists a constant C9 > 1 such that for every x ∈ D,

C−19 ψη(1, x) ≤ φ0(x) ≤ C9ψη(1, x), (4.4.9)

Proof. By (4.4.8) and (4.3.13) we have

φ0(x) � eλ0

ˆD

ψη(1, x)ψη(1, y)qc(1, x, y)φ0(y)dy

= ψη(1, x)eλ0

ˆD

ψη(1, y)qc(1, x, y)φ0(y)dy,

where c depends on whether the upper or lower bound is taken into ac-

count. For the upper bound, the facts that ψη(1, x)2qc(1, x, y) ≤ φ−1(1)d/2

and ‖φ0‖L2(D;dx) = 1 give us

ˆD

ψη(1, y)qc(1, x, y)φ0(y)dy ≤(ˆ

D

ψη(1, y)2qc(1, x, y)2dy

)1/2

‖φ0‖L2(D;dx)

≤ c1

ˆRdqc(1, x, y)dy ≤ c2.

84


For the lower bound we observe that

infx,y∈D

qc(1, x, y) = infx,y∈D

(φ−1(1)d/2 ∧ H(|x− y|−2) + e−c|x−y|

2

|x− y|d

)

≥

(φ−1(1)d/2 ∧ H(diam(D)−2) + e−c·diam(D)2

diam(D)d

)=: c3 > 0.

(4.4.10)

Thus we have that for every x ∈ D,

ˆD

ψη(1, y)qc(1, x, y)φ0(y)dy ≥ c3

ˆD

ψη(1, y)φ0(y)dy > 0.

2

Theorem 4.4.7. Suppose that µ ∈ Kφ,η, F is a function in Jφ,η, φ satisfies

La(γ, CL) for some a > 0, η ∈ [0, 2) and D is a bounded domain in Rd. Then

there exists a constant C > 1 such that for t ∈ (2,∞),

C−1e−λ0tψη(1, x)ψη(1, y) ≤ qD(t, x, y) ≤ Ce−λ0tψη(1, x)ψη(1, y), (4.4.11)

where −λ0 is eigenvalue of infinitesimal gernerator of T µ,Ft .

Proof. By the semigroup property, we have

qD(t, x, y) =

ˆD

ˆD

qD(1, x, z)qD(t− 2, z, w)qD(1, w, y)dzdw.

Since D is a bounded domain, Theorem 4.3.5, (4.4.10) and Lemma 4.4.6

85


implies that

qD(t, x, y) � ψη(1, x)ψη(1, y)

ˆD

ˆD

ψη(1, z)ψη(1, w)qD(t− 2, z, w)dzdw

� ψη(1, x)ψη(1, y)

ˆD

φ0(z)

ˆD

φ0(w)qD(t− 2, z, w)dwdz

= ψη(1, x)ψη(1, y)

ˆD

φ0(z)φ0(z)e−λ0(t−2)dz

= e−λ0(t−2)ψη(1, x)ψη(1, y).

2

We assume that λ0 > 0 where −λ0 is the eigenvalue of infinitesimal generator

of T µ,Ft in (4.4.7). Let GD(x, y) be a Green function of {T µ,Ft } defined by

GD(x, y) =

ˆ ∞0

qD(t, x, y)dt.

As an application of Theorem 4.3.5 and Theorem 4.4.7, we obtain the sharp

two sided estimates on GD(x, y). Define

a(x, y) := φ(δD(x)−2)−η/2φ(δD(y)−2)−η/2

and

g(x, y) :=

φ(|x− y|−2)−1

|x− y|d

(1 ∧ φ(δD(x)−2)−1

φ(|x− y|−2)−1

)η/2(1 ∧ φ(δD(y)−2)−1

φ(|x− y|−2)−1

)η/2when d > 2,

a(x, y)

|x− y|d∧(

a(x, y)

φ−1(a(x, y)−1)−d/2

+( ˆ φ−1(a(x,y)−1)−1/2

|x−y|

1

sd+1φ(s−2)ds)+)

when d ≤ 2.

Corollary 4.4.8. Suppose that µ ∈ Kφ,η, F is a function in Jφ,η, φ satisfies

La(γ, CL) for some a > 0, η ∈ [0, 2) and D is a bounded domain in Rd. If the

eigenvalue −λ0 of infinitesimal generator of T µ,Ft in (4.4.7) is negative, then

86


the Green function GD(x, y) of {T µ,Ft } has the following two sided estimates:

GD(x, y) � g(x, y), x, y ∈ D.

Proof. The proof is the same as that of [24, Theorem 7.3] by using Theorem

4.3.5 and Theorem 4.4.7. We skip the proof.

4.5 Examples

In this section, we first show some sufficient conditions. A signed measure µ

on Rd, d ≥ 3, is said to be in Kato class Kd,φ if

limr→0

supx∈Rd

ˆB(x,r)

1

|x− y|dφ(|x− y|−2)|µ|(dy) = 0.

A function g on Rd is said to be in Kd,φ if g(x)dx ∈ Kd,φ.

Note that

qc(s, y, w) ≤ c1

(φ−1(s−1)d/2 ∧ sφ(|y − w|−2)

|y − w|d

). (4.5.1)

Proposition 4.5.1. Suppose that d ≥ 3, φ satisfies La(γ, CL) and D is a

bounded Lipschitz open set in Rd and η ∈ (0, 2γ). Let g be a function on D.

If there exist c > 0, β ∈ (0, η/2+(γ−η/2)/d) and a compact set K ⊂ D such

that g1K ∈ Kd,φ and |g(x)| ≤ cφ(δD(x)−2)β for x ∈ D \K, then g ∈ Kφ,η.

Proof. It is enough to show that β ∈ (η/2, η/2 + (γ − η/2)/d). Note that

since γ < 1, β < η/2+(1−η/2)/d = ((d/2−1/2)η+1)/d < 1. By the similar

arguments in [13, Proposition 4.1(ii)], it suffices to show that g1D\K ∈ Kφ,η.

87


By (4.5.1) we have that

supx∈D

ˆ t

0

ˆD\K

(1 ∧ φ(δD(y)−2)−1

s

)η/2qc(s, x, y)|g(y)|dyds

≤ c1 supx∈D

ˆ t

0

ˆD\K

(1 ∧ φ(δD(y)−2)−1

s

)η/2φ(δD(y)−2)βqc(s, x, y)dyds

≤ c2 supx∈D

ˆD\K

(ˆ φ(δD(y)−2)−1∧t

0

(φ−1(s−1)d/2 ∧ sφ(|x− y|−2)

|x− y|d

)ds

)× φ(δD(y)−2)βdy

+ c2 supx∈D

ˆD\K

(ˆ t

φ(δD(y)−2)−1∧ts−η/2

(φ−1(s−1)d/2 ∧ sφ(|x− y|−2)

|x− y|d

)ds

)× φ(δD(y)−2)β−η/2dy

=: I + II.

Note that by change of variables, inequality λφ′(λ) ≤ φ(λ), and Lemma 2.0.3,

we have

ˆ ∞φ(|x−y|−2)−1

φ−1(s−1)d/2ds =

ˆ |x−y|−2

0

λd/2−1λφ′(λ)

φ(λ)2dλ ≤

ˆ |x−y|−2

0

λd/2−1

φ(λ)dλ

≤ 1

φ(|x− y|−2)

ˆ |x−y|−2

0

λd/2−1|x− y|−2

λdλ =

c

|x− y|dφ(|x− y|−2).

(4.5.2)

88


Observe that by (4.5.2), we have

I ≤ c2 supx∈D

( ˆD

ˆ φ(δD(y)−2)−1∧φ(|x−y|−2)−1∧t

0

sφ(|x− y|−2)|x− y|d

dsφ(δD(y)−2)βdy

+

ˆD

ˆ φ(δD(y)−2)−1∧t

φ(δD(y)−2)−1∧φ(|x−y|−2)−1∧tφ−1(s−1)d/2dsφ(δD(y)−2)βdy

)≤ c2 sup

x∈D

ˆD

((φ(δD(y)−2)−1 ∧ φ(|x− y|−2)−1 ∧ t)2φ(δD(y)−2)β

|x− y|dφ(|x− y|−2)−1

+ 1{|x−y|<δD(y)∧φ−1(t−1)−1/2}φ(δD(y)−2)β

|x− y|dφ(|x− y|−2)

)dy

≤ c2 supx∈D

ˆD

((φ(|x− y|−2)−1 ∧ t)2−β

|x− y|dφ(|x− y|−2)−1

+ 1{|x−y|<δD(y)∧φ−1(t−1)−1/2}φ(|x− y|−2)β

|x− y|dφ(|x− y|−2)

)dy

≤ c2 supx∈D

ˆD

(t(1−β)/2φ(|x− y|−2)(β−3)/2

|x− y|dφ(|x− y|−2)−1

+t(1−β)/2φ(|x− y|−2)(1−β)/2φ(|x− y|−2)β

|x− y|dφ(|x− y|−2)

)dy

≤ 2c2t(1−β)/2 sup

x∈D

ˆD

1

|x− y|dφ(|x− y|−2)1/2−β/2dy

= c3t(1−β)/2.

Since β < 1, the second last line is finite and I → 0 as t→ 0. On the other

89


hand, by (4.5.2)

II ≤ c2 supx∈D

(ˆD

(ˆ t

φ(δD(y)−2)−1∧t1{s<φ(|x−y|−2)−1}

s1−η/2

|x− y|dφ(|x− y|−2)−1ds


+

ˆD

(ˆ t

0

1{s≥φ(|x−y|−2)−1}s−η/2φ−1(s−1)d/2ds

)φ(δD(y)−2)β−η/2dy

)

≤ c2 supx∈D

(ˆD

1

|x− y|dφ(|x− y|−2)−1

(ˆ t∧φ(|x−y|−2)−1

0

s1−η/2ds


+

ˆD

φ(|x− y|−2)η/2(ˆ ∞

φ(|x−y|−2)−1

φ−1(s−1)d/2ds

)1{|x−y|≤φ−1(t−1)−1/2}

× φ(δD(y)−2)β−η/2dy

)

≤ c4 supx∈D

(ˆD

(t ∧ φ(|x− y|−2)−1)2−η/2

|x− y|dφ(|x− y|−2)−1φ(δD(y)−2)β−η/2dy

+

ˆD

1

|x− y|dφ(|x− y|−2)1−η/21{|x−y|≤φ−1(t−1)−1/2}φ(δD(y)−2)β−η/2dy

)≤ c4t

δ supx∈D

ˆD

1

|x− y|dφ(|x− y|−2)(1−ε)φ(δD(y)−2)η/2−βdy,

where ε = (γ + η/2− d(β − η/2))/2 and δ = (γ − η/2− d(β − η/2))/2. Note

that δ > 0 since β < η/2 + (γ − η/2)/d, ε < 1 since β > η/2, ε − δ = η/2

and ε + δ = γ − d(β − η/2). We choose p = d/(d − 2γ(1 − ε − δ/2)) > 1

and q = d/(2γ(1 − ε − δ/2)) > 1. Then 1/p + 1/q = 1. By using Young’s

90


inequality, we have

supx∈D

ˆD

1

|x− y|dφ(|x− y|−2)(1−ε)φ(δD(y)−2)η/2−βdy

≤ supx∈D

ˆD

(1

p|x− y|dpφ(|x− y|−2)p(1−ε)+

1

qφ(δD(y)−2)(η/2−β)q

)dy

≤ c6 supx∈D

ˆD

(1

p|x− y|p(d−2γ(1−ε))+

1

qδD(y)2q(β−η/2)

)dy <∞,

sinceD is a bounded Lipschitz open set, p(d−2γ(1−ε)) < d and 2q(β−η/2) <

1. Thus we conclude that limt→0Nφ,ηg1D\K ,c

(t) = 0. 2

Proposition 4.5.2. Suppose that η < d∧ 2 and |F (z, w)| ≤ A(|z −w|β ∧ 1)

for some A > 0 and β ≥ 2. Then there exists C > 0 such that for every

Borel subset D ⊂ Rd,

Nφ,ηF,c (t) ≤ CAt.

Thus we have F ∈ Jφ,η.

Proof. Since

φ(λ)

λ=

ˆ ∞0

H(s)

s2ds =

ˆ λ1/2

0

2rH(r−2)dr,

we have that for β ≥ 2,

ˆB(0,1)

H(|z|−2)dz|z|d−β

=

ˆ 1

0

H(r−2)

r1−βdr ≤ φ(1)

2<∞. (4.5.3)

First, by using (4.5.3), we have

ˆRd

|z|β ∧ 1

|z|dH(|z|−2)−1dz ≤

ˆB(0,1)

H(|z|−2)dz|z|d−β

+

ˆB(0,1)c

H(|z|−2)dz|z|d

<∞.

By using this, (4.5.1), Lemma 2.0.3 and the condition La(γ, CL) on φ, we

91


have that for η < d ∧ 2,

ˆ t

0

ˆRd×Rd

qc(s, y, w)

(1 +

φ(|z − w|−2)−1 ∧ tφ(|y − w|−2)−1

)η/2 |F (z, w)|+ |F (w, z)||z − w|dH(|z − w|−2)−1

× dzdwds

≤ 2A

(ˆRd

|z|β ∧ 1

|z|dH(|z|−2)−1dz

) ˆ t

0

ˆRdqc(s, y, w)

(1 +

t

φ(|y − w|−2)−1

)η/2× dwds

≤ c2A

ˆ t

0

(1 +

ˆRd

tη/2qc(s, y, w)

φ(|y − w|−2)−η/2dw

)ds

≤ c3A

ˆ t

0

(1 +

ˆsφ(|y−w|−2)≥1

φ−1(s−1)d/2tη/2

φ(|y − w|−2)−η/2dw

+

ˆsφ(|y−w|−2)<1

tη/2sφ(|y − w|−2)1+η/2

|y − w|ddw

)ds

≤ c3A

ˆ t

0

(1 + φ−1(s−1)d/2tη/2

ˆsφ(r−2)≥1

rd−1

φ(r−2)−η/2dr

+ tη/2s

ˆsφ(r−2)<1

φ(r−2)1+η/2

rdr

)ds

≤ c4At+ c4Atη/2

ˆ t

0

s−η/2ds ≤ c5At.

In the last inequality, we use η < 2 and in the last second inequality we use

η < d. Indeed, by Lemma 2.0.3

ˆsφ(r−2)≥1

rd−1

φ(r−2)−η/2dr ≤

ˆsφ(r−2)≥1

rd−1r−η

sη/2φ−1(s−1)η/2dr

=1

sη/2φ−1(s−1)η/2

ˆsφ(r−2)≥1

rd−1−ηdr

= c6φ−1(s−1)(η−d)/2

sη/2φ−1(s−1)η/2.

2

Example 4.5.3. Let S = (St)t≥0 be a subordinator with zero drift whose

92


Laplace exponent is φ and let X = (Xt)t≥0 be the corresponding subordinate

Brownian motion in Rd. Assume that [24, (1.8)] holds and that H satisfies

La(γ, CL) and Ua(δ, CU) with δ < 2 and γ > 2−11δ≥1 for some a ≥ 0. Suppose

that D is a C1,1 open set in Rd. If D is unbounded, we further assume that

H satisfies L0(γ0, CL) and U0(δ, CU) with δ < 2 and that the path distance

in each connected component of D is comparable to the Euclidean distance.

Then by [24, Theorem 1.3] for any T > 0 there exist constants c, aL, aU such

that for (t, x, y) ∈ (0, T ]×D ×D,

c−1qaL,1(t, x, y) ≤ pD(t, x, y) ≤ cqaU ,1(t, x, y),

where pD(t, x, y) is the transition density of killed subordinate Brownian

motion XD. Assume that µ ∈ Kφ,1 and F ∈ Jφ,1. Let qD be the density

of the Feynman-Kac semigroup of XD corresponding to Aµ,F . Then there

exists C > 1 such that for all (t, x, y) ∈ (0, T ]×D ×D,

C−1qaL,1(t, x, y) ≤ qD(t, x, y) ≤ CqaU ,1(t, x, y).

Example 4.5.4. Let ψ : (0,∞) → (0,∞) is a non-decreasing function sat-

isfying L0(β1, CL), U0(β2, CU), and

ˆ 1

0

s

ψ(s)ds <∞.

Let J : Rd × Rd \ diag → [0,∞] be a symmetric function satisfying

C−1

|x− y|dψ(|x− y|)≤ J(x, y) ≤ C

|x− y|dψ(|x− y|), (x, y) ∈ Rd × Rd

for some C ≥ 1. For u, v ∈ L2(Rd, dx), define

E(u, v) :=

ˆRd×Rd

(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy

and D(E) = {f ∈ L2(Rd) : E(f, f) < ∞}. Then, by [1, Theorem 1.2], there

is a conservative Feller process X = (Xt,Px, x ∈ Rd, t ≥ 0) associated with

(E ,F) that starts every point in Rd. Moreover, X has a continuous transition

93


density function p(t, x, y) on (0,∞)× Rd × Rd. Define

Φ(r) :=r2

2´ r0

sψ(s)

ds.

Suppose Φ satisfies the following lower scaling condition: for γ > 1

Φ(λx)

Φ(λ)≥ CLx

γ for all λx < a and x ≥ 1

and r → Φ(r−1/2)−1 is a Bernstein function. Then by [1, Corollary 1.5] for

all (t, x, y) ∈ (0, T )× Rd × Rd,

c−1(

1

Φ−1(t)d∧( t

|x− y|dψ(|x− y|)+

1

Φ−1(t)de−aL |x−y|

2

Φ−1(t)2

))≤ p(t, x, y) ≤ c

(1

Φ−1(t)d∧( t

|x− y|dψ(|x− y|)+

1

Φ−1(t)de−aU |x−y|

2

Φ−1(t)2

)).

Define φ(r) := Φ(r−1/2)−1, then φ satisfies L1/a2(γ/2, CL) andH(r) = ψ(r−1/2)−1.

Thus for (t, x, y) ∈ (0, T )× Rd × Rd

c−1qaL,0(t, x, y) ≤ p(t, x, y) ≤ qaU ,0(t, x, y).

Assume that µ ∈ Kφ,0 and F ∈ Jφ,0. Let q be the density of the Feynman-

Kac semigroup of X corresponding to Aµ,F . Then there exists C > 1 such

that for all (t, x, y) ∈ (0, T ]× Rd × Rd,

C−1qaL,0(t, x, y) ≤ q(t, x, y) ≤ CqaU ,0(t, x, y),

i.e.,

C−1(

1

Φ−1(t)d∧( t

|x− y|dψ(|x− y|)+

1

Φ−1(t)de−aL |x−y|

2

Φ−1(t)2

))≤ q(t, x, y) ≤ C

(1

Φ−1(t)d∧( t

|x− y|dψ(|x− y|)+

1

Φ−1(t)de−aU |x−y|

2

Φ−1(t)2

)).

94


Example 4.5.5. For u, v ∈ Rd, define

E(u, v) :=1

2

ˆRd

ˆRd

(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy

and F := {u ∈ L2(Rd) : E(u, u) < ∞}, where J(x, y) = ϕ(x, y)J(x, y),

J(x, y) is the Levy density(Jumping kernel) of subordinate Brownian mo-

tion X in Example 4.5.3 and ϕ : Rd × Rd → R+ is a function satisfying

| lnϕ(x, y)| ≤ |x− y|β ∧ 1 for some β ≥ 2. Since

supx∈Rd

ˆRd

(|x− y|2 ∧ 1)J(x, y) <∞,

by [33, Theorem 2.1] and [34, Theorem 2.4], (E ,F) is a regular Dirichlet form

on L2(Rd). Thus there is a Hunt process X associated with (E ,F) on Rd,

starting from quasi-everywhere point in Rd. Let D be a C1,1 open set in Rd

satisfying the condition in Example 4.5.3 and XD be a killed process of X.

Then associated Dirichlet form for XD is (E ,FD), where

FD := {u ∈ F : u = 0 on Dc except for a set of zero capacity }.

Thus for u, v ∈ FD,

E(u, v) =1

2

ˆD

ˆD

(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy +

ˆD

u(x)v(x)kD(x)dx,

where kD(x) =´DcJ(x, y)dy.

Let (E ,D(E)D) be a Dirichlet form associated with killed subordinate

Brownian motion XD in Example 4.5.3, where

E(u, v) =1

2

ˆD

ˆD


ˆD

u(x)v(x)kD(x)dx,

D(E)D := {u ∈ L2(Rd) : E(u, u) <∞ and u = 0 on Dc except for a set of zero

capacity} = FD,

95


and kD(x) =´DcJ(x, y)dy. Define q(x) :=

´Rd J(x, y)− J(x, y)dy,

Kt := exp

(∑0≤s≤t

lnϕ(XDs−, X

Ds )−

ˆ t

0

q(XDs )ds

)

and Qtf(x) := Ex[Ktf(XDt )]. Since | lnϕ(x, y)| ≤ |x−y|β∧1, by Proposition

4.5.2, lnϕ ∈ Jφ,1 and since q is bounded, q ∈ Kφ,1. Indeed, by (4.5.3) we

have that

|q(x)| ≤ˆB(x,1)

|1− ϕ(x, y)|J(x, y)dy + e

ˆB(x,1)c

J(x, y)dy

≤ c1

ˆB(0,1)

|y|βH(|y|−2)|y|d

dy + c1

ˆB(0,1)c

H(|y|−2)|y|d

dy <∞.

Thus by [18, Theorem 4.8], Qt is a strongly continuous semigroup in L2(D)

whose associated quadratic form is (E ,FD). Indeed,

E(u, v) =1

2

ˆD

ˆD


ˆD

u(x)v(x)kD(x)dx

−ˆD

u(x)v(x)q(x)dx−ˆD

ˆD

u(x)v(y)(ϕ(x, y)− 1)J(x, y)dxdy

=1

2

ˆD

ˆD


ˆD

u(x)v(x)kD(x)dx.

Let qD be the transition density of XD. Then by Theorem 4.3.4 and

Theorem 4.3.5, qD(t, x, y) is jointly continuous, hence XD is a Hunt process

associated with (E ,FD) that starts every point in Rd and

C−1qaL,1(t, x, y) ≤ qD(t, x, y) ≤ CqaU ,1(t, x, y).

96

Bibliography

[1] J. Bae, J. Kang, P. Kim, J. Lee, Heat kernel estimates for symmetric

jump processes with mixed polynomial growths, To appear Annals of

probability, (2018).

[2] J. Bertoin, Levy Processes, Cambridge University Press, Cambridge

(1996).

[3] K. Bogdan, T. Grzywny, M. Ryznar, Density and tails of unimodal

convolution semigroup. J. Funct. Anal. 266(6) (2014), 3543-3571.

[4] K. Bogdan, T. Grzywny, M. Ryznar, Dirichlet heat kernel for uni-

modal Levy processes. Stoch. Processes. Appl. 124(11) (2014), 3612-

3650.

[5] K. Bogdan, T. Grzywny, M. Ryznar, Barriers, exit time and sur-

vival probability for unimodal Levy processes. Probab. Theory Relat.

Fields 162 (2015), 155-198.

[6] Z.-Q. Chen, P. Kim, T. Kumagai, Weighted Poincare Inequality and

Heat Kernel Estimates for Finite Range Jump Processes. Math. Ann.

342(4) (2008), 833-883.

[7] Z.-Q. Chen, P. Kim, T. Kumagai, Global heat kernel estimates for

symmetric jump processes. Trans. Amer. Math. Soc. 363 (2011),

5021-5055.

[8] Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimates for Dirichlet

fractional Laplacian. J. European Math. Soc., 12 (2010), 1307-1329.

97

BIBLIOGRAPHY

[9] Z.-Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for cen-

sored stable-like processes. Probab. Theory Relat. Fields, 146 (2010),

361-399.

[10] Z.-Q. Chen, P. Kim, R. Song, Dirichlet heat kernel estimates for

∆α/2 + ∆β/2. Ill. J. Math. 54 (2010), 1357-1392.

[11] Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimate for ∆ + ∆α/2 in

C1,1 open sets. J. London Math. Soc. 84(1) (2011), 58-80.

[12] Z.-Q. Chen, P. Kim, R. Song, Sharp heat kernel estimates for rela-

tivistic stable processes in open sets. Ann. Probab. 40 (2012), 213-

244.

[13] Z.-Q. Chen, P. Kim, R. Song, Stability of Dirichlet heat kernel

estimates for non-local operators under Feynman-Kac perturbation,

Trans. Amer. Math. Soc. 367 (2015), 5237-5270.

[14] Z.-Q. Chen, P. Kim, R. Song, Dirichlet Heat Kernel Estimates for

Subordinate Brownian Motion with Gaussian Components. J. Reine

Angew. Math. 711 (2016), 111-138.

[15] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like pro-

cesses on d-sets. Stochastic Process. Appl. 108 (2003), 27-62.

[16] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for jump processes of

mixed types on metric measure spaces. Probab. Theory Relat. Fields,

140 (2008), 277-317.

[17] Z.-Q. Chen, T. Kumagai, A priori Holder estimate, parabolic Harnack

principle and heat kernel estimates for diffusions with jumps. Rev.

Mat. Iberoamericana 26 (2010), 551-589.

[18] Z.-Q. Chen and R. Song, Conditional gauge theorem for non-local

Feynman-Kac transforms, Probab. Theory Relat Fields. 125 (2003),

45-72.

[19] S. Cho, Two-sided global estimates of the Green’s function of

parabolic equations. Potential Anal. 25(4) (2006), 387-398.

98

BIBLIOGRAPHY

[20] K.L. Chung, Z. Zhao, From Brownian Motion to Schrodinger’s Equa-

tion, Springer, Berlin, (1995).

[21] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts

in Mathematics, vol. 92, Cambridge University Press, Cambridge,

(1990).

[22] P. J. Fitzsimmons, R. K. Getoor, Smooth measures and continuous

additive functionals of right Markov processes, Ito’s stochastic calcu-

lus and probability theory, Springer, Tokyo, (1996), 31-49.

[23] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmet-

ric Markov processes. 2nd Edition. Walter de Gruyter, (2011).

[24] P. Kim, A. Mimica, Estimates of Dirichlet Heat kernel for subordinate

Brownian motions. Electron. J. Probab. 23 (2018), no. 64, 1-45.

[25] P. Kim, R. Song, Intrinsic ultracontractivity for non-symmetric Levy

processes, Forum Math. 21(1) (2009), no. 1, 43-66.

[26] P. Kim, R. Song, Z. Vondracek, On the potential theory of one-

dimensional subordinate Brownian motions with continuous compo-

nents. Potential Anal. 33 (2010), 153-173.

[27] P. Kim, R. Song, Z. Vondracek, Potential theory of subordinate

Brownian motions with Gaussian components. Stoch. Processes Appl.

123(3) (2013), 764-795.

[28] Z.-M. Ma, M. Rockner, Introduction to the Theory of (Non-

Symmetric) Dirichlet Forms, Springer, Berlin.

[29] A. Mimica, Heat kernel estimates for subordinate brownian motions.

Proc. London Math. Soc., 113(5) (2016), 627-648.

[30] A. Pazy, Semigroups of Linear Operators and Applications to Partial

Differential Equations, Applied Mathematical Sciences 44. Springer,

New York.

99

BIBLIOGRAPHY

[31] K.-i. Sato, Levy processes and infinitely divisible distributions, Cam-

bridge University Press, (1999).

[32] H. H. Schaefer, Banach lattices and positive operators, Springer-

Verlag, New York-Heidelberg, 1974, Die Grundlehren der mathema-

tischen Wissenschaften, Band 215.

[33] R. L. Schilling, T. Uemura, On the Feller property of Dirichlet forms

generated by pseudo differential operators, Tohoku Math. J. (2),

59(3) (2007), 401-422.

[34] R. L. Schilling, T. Uemura, On the structure of the domain of a sym-

metric jump-type Dirichlet form, Publ. Res. Inst. Math. Sci., 48(1)

(2012), 1-20.

[35] R. Song, Z. Vondracek, Parabolic Harnack inequality for the mixture

of Brownian motion and stable process. Tohoku Math. J. (2) 59(1)

(2007), 1-19.

[36] Q. S. Zhang, The boundary behavior of heat kernels of Dirichlet

Laplacians. J. Differential Equations 182 (2002), 416-430.

[37] Z. Zhao, A probabilistic principle and generalized Schrodinger per-

turbation, J. Funct. Anal. 101 (1991), 162-176.

100

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dirichlet heat kernel estimates for subordinate brownian · 2019. 11. 14. · we obtain dirichlet...

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