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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
Dirichlet heat kernel estimatesfor subordinate Brownian
motion and its perturbation:stable and beyond
(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)
Dirichlet heat kernel estimatesfor subordinate Brownian
motion and its perturbation:stable and beyond
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
c© 2018 Joohak Bae
All rights reserved.
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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.
Theorem 3.1.3. Let X = (Xt)t≥0 be a subordinate Brownian motion whose
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
(2) For any (t, x, y) ∈ (0, T ]×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))
).
(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.
Theorem 3.1.4. Let X = (Xt)t≥0 be a subordinate Brownian motion whose
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
(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))
).
(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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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− y)q(t, y)dy
≤ 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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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)
p(2)(t, x− y)q(t, y)dy
≥ 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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
p(2)(t, x− y)q(t, y)dy
≥ 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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
p(2)(t, x− y)q(t, y)dy
≥ 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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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)(
infw∈B(u,rt/8)z∈B(v,rt/8)
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)(
infw∈B(u,rt/8)z∈B(v,rt/8)
j(|w − z|))ˆ t
0
P0(τB(0,rt/8) > s)
≥ t(P0(τB(0,rt/8) > t))2(
infw∈B(u,rt/8)z∈B(v,rt/8)
j(|w − z|))
≥ c5t(
infw∈B(u,rt/8)z∈B(v,rt/8)
j(|w − z|)).
In the last inequality we have used Lemma 3.3.2. Note that if w ∈ B(u, rt/8)
22
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
))
׈B(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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
[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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
(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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 3. SBM WITH GAUSSIAN COMPONENTS
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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(s, x, z)qc2(t− s, z, y)
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(s, x, z)qc2(t− s, z, y)
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(s, x, z)qc2(t− s, z, y)
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)
t|x− y|−dH(|x− y|−2) + φ−1(t−1)d/2e−c1/4|x−y|2φ−1(t−1)
≤ (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)
t|x− y|−dH(|x− y|−2) + φ−1(t−1)d/2e−c1/4|x−y|2φ−1(t−1)
≤ 2d+4qc2(t− s, z, y).
52
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
Similarly when |x− y|2φ−1(t−1) ≥ 4dc1
and |x− z| ≤ |z − y|, we have
qc1(s, x, z)qc2(t− s, z, y)
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(s, x, z)qc2(t− s, z, y)
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
(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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
ψη(s, w)qc(s, w, y)ds.
(4.2.23)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
ψη(s, w)qc(s, w, y)ds.
(4.2.28)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
(b) If tφ(|x− y|−2) < 1, then
ˆ t
0
ˆUx,y
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
ˆUx,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
ˆUx,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
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‖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׈ 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
ψη(s, w)qc2(s, w, y)ds
where we used the fact that qc1(t, x, z) ≤ φ−1(t−1)d/2 = ec1qc1(t, x, y) if tφ(|x−
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
ψη(s, w)qc2(s, w, y)ds
×(
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
t ∈ (0, t2],
ˆ t/2
0
ˆUx,y
qc1,η(t− s, x, z)qc2,η(s, w, y)J(z, w)dzdwds ≤ I1 + I2
≤ c15qc1,η(t, x, y)
(ˆUx,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
+
ˆUx,y
ˆ t/2
0
ψη(s, w)qc2(s, w, y)ds
(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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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).
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
≤ 2η/2C40(MNφ,η
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
≤ 2η/2C40(MNφ,η
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
≤ 2η/2C40(MNφ,η
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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).
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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).
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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).
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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),
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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.
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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)
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
)× φ(δD(y)−2)β−η/2dy
+
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
)× φ(δD(y)−2)β−η/2dy
+
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
)).
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CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy +
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
CHAPTER 4. FEYNMAN-KAC PERTURBATION OF DHK OF SBM
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
(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy +
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
(u(x)− u(y))(v(x)− v(y))J(x, y)dxdy +
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
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