· introduction the kernel of a jump process jump processes are studied under various assumptions,...

On the potential theory of jump processes in open sets

Zoran Vondracek

University of Zagreb, CroatiaSupported in part by the Croatian Science Foundation under the project 4197

Bedlewo, May 24, 2019

Zoran Vondracek (University of Zagreb) Potential theory of jump processes Bedlewo, May 24, 2019 1 / 33

Table of Contents

1 Introduction

2 Subordinate killed processes

3 Jump kernels decaying at the boundary

Joint work with P. Kim (SNU) and R. Song (UIUC)

Table of Contents

1 Introduction

Joint work with P. Kim (SNU) and R. Song (UIUC)

Introduction

1 Introduction

Introduction

The kernel of a jump process

Jump processes are studied under various assumptions, but most of thestudies assume that the kernel J(x , y) which describes the intensity ofjumps from x to y depends solely on the distance between x and y (orsome function of the distance).

A typical example is a Levy process in Rd where J(x , y) = j(|x − y |) withj the density of the Levy measure. The most familiar case is the isotropicα-stable process with J(x , y) = |x − y |−d−α.Suppose that the underlying state space E is a metric space with distanced . There are various modifications of the above kernels, usually of thefollowing type: There exists c ≥ 1 such that

c−1f (d(x , y)) ≤ J(x , y) ≤ cf (d(x , y)) , x , y ∈ E ,

with f : [0,∞)→ (0,∞] bounded away from zero on compact sets.This includes stable-like processes where J(x , y) = c(x , y)|x − y |−d−αwhere c(x , y) is bounded and bounded away from zero.

Introduction

Jump processes are studied under various assumptions, but most of thestudies assume that the kernel J(x , y) which describes the intensity ofjumps from x to y depends solely on the distance between x and y (orsome function of the distance).A typical example is a Levy process in Rd where J(x , y) = j(|x − y |) withj the density of the Levy measure. The most familiar case is the isotropicα-stable process with J(x , y) = |x − y |−d−α.

Suppose that the underlying state space E is a metric space with distanced . There are various modifications of the above kernels, usually of thefollowing type: There exists c ≥ 1 such that

Introduction

Jump processes are studied under various assumptions, but most of thestudies assume that the kernel J(x , y) which describes the intensity ofjumps from x to y depends solely on the distance between x and y (orsome function of the distance).A typical example is a Levy process in Rd where J(x , y) = j(|x − y |) withj the density of the Levy measure. The most familiar case is the isotropicα-stable process with J(x , y) = |x − y |−d−α.Suppose that the underlying state space E is a metric space with distanced . There are various modifications of the above kernels, usually of thefollowing type: There exists c ≥ 1 such that

with f : [0,∞)→ (0,∞] bounded away from zero on compact sets.

This includes stable-like processes where J(x , y) = c(x , y)|x − y |−d−αwhere c(x , y) is bounded and bounded away from zero.

Introduction

Jump processes are studied under various assumptions, but most of thestudies assume that the kernel J(x , y) which describes the intensity ofjumps from x to y depends solely on the distance between x and y (orsome function of the distance).A typical example is a Levy process in Rd where J(x , y) = j(|x − y |) withj the density of the Levy measure. The most familiar case is the isotropicα-stable process with J(x , y) = |x − y |−d−α.Suppose that the underlying state space E is a metric space with distanced . There are various modifications of the above kernels, usually of thefollowing type: There exists c ≥ 1 such that

Introduction

More general kernels of jump processes

Bogdan, Kumagai and Kwasnicki (2015) studied jump processes in ametric space E with the kernel J(x , y) satisfying the following assumption:For every x0 ∈ E and all radii 0 < r < R < R0 (R0 ∈ (0,∞] is thelocalization radius), there exists a constant c ≥ 1 such that for allx ∈ B(x0, r) and all y ∈ E \ B(x0,R)

c−1J(x0, y) ≤ J(x , y) ≤ cJ(x0, y) .

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.

The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.What happens to the jumping kernel J(x , y)? Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |. Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

Hence, the Dirichlet form of the killed process looks like

E(u, v) =1

∫∫D×D

(u(x)− u(y))(v(x)− v(y))J(x , y)dydx

∫Du(x)v(x)κ(x) dx

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.

What happens to the jumping kernel J(x , y)? Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |. Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

E(u, v) =1

∫∫D×D

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.What happens to the jumping kernel J(x , y)?

Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |. Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

E(u, v) =1

∫∫D×D

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.What happens to the jumping kernel J(x , y)? Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |.

Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

E(u, v) =1

∫∫D×D

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.What happens to the jumping kernel J(x , y)? Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |. Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

E(u, v) =1

∫∫D×D

Introduction

Killed process

Suppose now that D ⊂ Rd is an open set. Then one can study jumpprocesses with the state space D.The most common way to do that is to kill the jump process X when itexits D for the first time. The obtained process is denoted by XD andcalled the killed process, or the part of X in D. Analytically, itsinfinitesimal generator is, roughly, the generator of X with zero exteriorcondition.What happens to the jumping kernel J(x , y)? Nothing! The jumpingkernel of XD is JD(x , y) = J(x , y) and depends solely on |x − y |. Thechange is in introducing the killing function κ(x) =

∫Dc J(x , y) dy .

E(u, v) =1

∫∫D×D

Introduction

Censored process

Another type of processes that were studied in open subsets D arecensored ones. One way to get a censored processes is to remove thekilling part from the Dirichlet form of the killed process and get the form

C(u, v) =1

∫∫D×D

(u(x)− u(y))(v(x)− v(y))J(x , y)dydx .

The jumping kernel is still J(x , y) and still depends only on |x − y |.For example, with J(x , y) = |x − y |−d−α one gets the censored α-stableprocess.Note that the censored process is intrinsically defined in D and its not partof some larger process.

Introduction

Censored process

C(u, v) =1

∫∫D×D

The jumping kernel is still J(x , y) and still depends only on |x − y |.

For example, with J(x , y) = |x − y |−d−α one gets the censored α-stableprocess.Note that the censored process is intrinsically defined in D and its not partof some larger process.

Introduction

Censored process

C(u, v) =1

∫∫D×D

The jumping kernel is still J(x , y) and still depends only on |x − y |.For example, with J(x , y) = |x − y |−d−α one gets the censored α-stableprocess.

Note that the censored process is intrinsically defined in D and its not partof some larger process.

Introduction

Censored process

C(u, v) =1

∫∫D×D

The jumping kernel is still J(x , y) and still depends only on |x − y |.For example, with J(x , y) = |x − y |−d−α one gets the censored α-stableprocess.Note that the censored process is intrinsically defined in D and its not partof some larger process.

Introduction

Harmonic functions

Various potential-theoretic results were proved for such jump processes inthe last twenty years including the boundary Harnack principle.

Harmonic functions. Let X = (Xt ,Px) be a strong Markov process withthe state space E . A function f : E → [0,∞) is said to be harmonic in anopen subset D of E (with respect to X ) if for every open B ⊂ B ⊂ D andall x ∈ B,

f (x) = Ex [f (XτB )].

Here τB = inf{t > 0 : Xt /∈ B} is the first exit time from B.

f : E → [0,∞) is said to be regular harmonic in D if for all x ∈ D,

f (x) = Ex [f (XτD )].

Introduction

Harmonic functions

f (x) = Ex [f (XτB )].

Here τB = inf{t > 0 : Xt /∈ B} is the first exit time from B.

f : E → [0,∞) is said to be regular harmonic in D if for all x ∈ D,

f (x) = Ex [f (XτD )].

Introduction

Harmonic functions

f (x) = Ex [f (XτB )].

Here τB = inf{t > 0 : Xt /∈ B} is the first exit time from B.f : E → [0,∞) is said to be regular harmonic in D if for all x ∈ D,

f (x) = Ex [f (XτD )].

Introduction

Boundary Harnack principle

BHP for diffusions (i.e., local operators): X = (Xt ,Px) a continuousstrong Markov process in Rd . The scale invariant BHP (or a local BHP)holds in an open set D ⊂ Rd if there exist C > 0 and r0 > 0 such that forall Q ∈ ∂D, all r ∈ (0, r0) and all non-negative functions f and g harmonicin B(Q, r) ∩ D that vanish continuously on B(Q, r) ∩ ∂D it holds that

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

It is well known that the BHP holds for Brownian motion (more generally,elliptic diffusions) in Lipschitz domains (Ancona 1978, Wu 1978, Dahlberg1977 global BHP), NTA domains (Jerison-Kenig 1982), uniform domains(Aikawa 2001). In case of smooth boundary (say C 1,1) one gets the BHPwith explicit decay rate:

δD(x)≤ C

δD(y), for all x , y ∈ B(Q, r/2) ∩ D.

Introduction

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

δD(x)≤ C

Introduction

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

δD(x)≤ C

Introduction

Boundary Harnack principle, cont.

BHP for jump processes (i.e, non-local operators):

X = (Xt ,Px) a jumpprocess in a metric space (E , d). The scale invariant BHP holds in anopen set D ⊂ E if there exist C > 0 and r0 > 0 such that for all Q ∈ ∂D,all r ∈ (0, r0) and all non-negative functions f , g : E → [0,∞) that areregular harmonic in B(Q, r) ∩ D and vanish in B(Q, r) ∩ Dc it holds that

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

The BHP holds for isotropic α-stable process in Rd in Lipschitz domain(Bogdan 1977), open κ-fat set (Song-Wu 1999), any open set (Bogdan,Kulczycki, Kwasnicki 2008). Explicit decay rate in C 1,1 open sets isδD(x)α/2.Generalizations to various Levy processes (Kim, Song, V) and to jumpprocesses in metric spaces (Bogdan, Kumagai, Kwasnicki).

Introduction

BHP for jump processes (i.e, non-local operators): X = (Xt ,Px) a jumpprocess in a metric space (E , d). The scale invariant BHP holds in anopen set D ⊂ E if there exist C > 0 and r0 > 0 such that for all Q ∈ ∂D,all r ∈ (0, r0) and all non-negative functions f , g : E → [0,∞) that areregular harmonic in B(Q, r) ∩ D and vanish in B(Q, r) ∩ Dc it holds that

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

Introduction

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

The BHP holds for isotropic α-stable process in Rd in Lipschitz domain(Bogdan 1977), open κ-fat set (Song-Wu 1999), any open set (Bogdan,Kulczycki, Kwasnicki 2008).

Explicit decay rate in C 1,1 open sets isδD(x)α/2.Generalizations to various Levy processes (Kim, Song, V) and to jumpprocesses in metric spaces (Bogdan, Kumagai, Kwasnicki).

Introduction

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

The BHP holds for isotropic α-stable process in Rd in Lipschitz domain(Bogdan 1977), open κ-fat set (Song-Wu 1999), any open set (Bogdan,Kulczycki, Kwasnicki 2008). Explicit decay rate in C 1,1 open sets isδD(x)α/2.

Generalizations to various Levy processes (Kim, Song, V) and to jumpprocesses in metric spaces (Bogdan, Kumagai, Kwasnicki).

Introduction

g(x)≤ C

g(y), for all x , y ∈ B(Q, r/2) ∩ D.

Introduction

The BHP for censored α-stable process, α ∈ (1, 2):

Let D be a C 1,1 openset in Rd and X a censored α-stable process. There exists C > 0 andr0 > 0 such that for all Q ∈ ∂D, all r ∈ (0, r0) and all non-negativefunctions f : D → [0,∞) that are harmonic (wrt X ) in D ∩ B(Q, r) andvanish continuously on B(Q, r) ∩ ∂D it holds that

δD(x)α−1≤ C

δD(y)α−1, for all x , y ∈ B(Q, r/2) ∩ D

Bogdan, Burdzy, Chen 2003

Introduction

The BHP for censored α-stable process, α ∈ (1, 2): Let D be a C 1,1 openset in Rd and X a censored α-stable process. There exists C > 0 andr0 > 0 such that for all Q ∈ ∂D, all r ∈ (0, r0) and all non-negativefunctions f : D → [0,∞) that are harmonic (wrt X ) in D ∩ B(Q, r) andvanish continuously on B(Q, r) ∩ ∂D it holds that

δD(x)α−1≤ C

Introduction

The BHP for censored α-stable process, α ∈ (1, 2): Let D be a C 1,1 openset in Rd and X a censored α-stable process. There exists C > 0 andr0 > 0 such that for all Q ∈ ∂D, all r ∈ (0, r0) and all non-negativefunctions f : D → [0,∞) that are harmonic (wrt X ) in D ∩ B(Q, r) andvanish continuously on B(Q, r) ∩ ∂D it holds that

δD(x)α−1≤ C

Introduction

Carleson’s estimate

An essential ingredient of most of the proofs of the BHP is Carleson’sestimate: Let D ⊂ Rd be an open set. There exist constants C > 0 andr0 > 0 such that for all Q ∈ ∂D, all r ∈ (0, r0) and all non-negativefunctions f : Rd → [0,∞) (f : D → [0,∞)) that are harmonic inB(Q, r) ∩ D and vanish in B(Q, r) ∩ Dc (vanish continuously onB(Q, r) ∩ ∂D) it holds that

f (x) ≤ Cf (x0) , for all x ∈ B(Q, r/2) ∩ D,

where x0 ∈ B(Q, r) ∩ D with δD(x0) ≥ r/2.

Subordinate killed processes

1 Introduction

Killed stable process

Let Z = (Zt ,Px) be the isotropic α-stable process in Rd , α ∈ (0, 2),(Qt)t≥0 the corresponding semigroup: Qt f (x) := Ex f (Zt) t ≥ 0,f : Rd → R.

−(−∆)α/2f := limt→0

Qt f − f

tis the fractional Laplacian.

For D ⊂ Rd open, let τD := inf{t > 0 : Zt /∈ D}, ZDt := Zt if t < τD , ∂

(cemetary) otherwise, QDt f (x) := Ex f (ZD

t ) = Ex(f (Zt), t < τD) thecorresponding semigroup.

L1f := limt→0

QDt f − f

a possible definition of fractional Laplacian in D; usually called fractional

Laplacian in D with zero exterior condition:. Notation: −(−∆)α/2∣∣D .

−(−∆)α/2f := limt→0

Qt f − f

L1f := limt→0

QDt f − f

−(−∆)α/2f := limt→0

Qt f − f

L1f := limt→0

QDt f − f

−(−∆)α/2f := limt→0

Qt f − f

L1f := limt→0

QDt f − f

KSBM and SKBM

Let W = (Wt ,Px) be a Brownian motion in Rd , S = (St)t≥0 anindependent α/2-stable subordinator. Then WSt is a subordinate

Brownian motion and (Zt)d= (WSt ).

Hence, ZD is a killed subordinate Brownian motion (KSBM).

WD Brownian motion killed upon exiting D, Y Dt := WD

Stis a subordinate

killed Brownian motion (SKBM).

If (PDt )t≥0 is the semigroup of WD ,

then the infinitesimal generator of Y D is

L0f = −(−∆∣∣D)α/2f :=1

|Γ(−α/2)|

∫ ∞0

(PDt f − f )t−α/2−1dt

Another possible definition of a fractional Laplacian in D: the fractionalpower of the Dirichlet Laplacian. L0 6= L1

KSBM and SKBM

Stis a subordinate

L0f = −(−∆∣∣D)α/2f :=1

|Γ(−α/2)|

∫ ∞0

KSBM and SKBM

Stis a subordinate

L0f = −(−∆∣∣D)α/2f :=1

|Γ(−α/2)|

∫ ∞0

KSBM and SKBM

Stis a subordinate

killed Brownian motion (SKBM). If (PDt )t≥0 is the semigroup of WD ,

L0f = −(−∆∣∣D)α/2f :=1

|Γ(−α/2)|

∫ ∞0

Another possible definition of a fractional Laplacian in D: the fractionalpower of the Dirichlet Laplacian.

L0 6= L1

KSBM and SKBM

Stis a subordinate

killed Brownian motion (SKBM). If (PDt )t≥0 is the semigroup of WD ,

L0f = −(−∆∣∣D)α/2f :=1

|Γ(−α/2)|

∫ ∞0

If (QDt )t≥0 is the semigroup of Y D , then QD

t f ≤ QDt f , f ≥ 0. Y D is a

”smaller” process then ZD .

Let S be a δ/2-stable subordinator, δ ∈ (0, 2], T a γ/2-stablesubordinator, γ ∈ (0, 2), so that (δ/2)(γ/2) = α/2. Let Zt = WSt be aSBM, ZD the KSBM (=killed δ-stable), and Y D

t = ZDTt

the subordinate

killed Z .

Let (RDt )t≥0 be the semigroup of Y D . Then

QDt f ≤ RD

t f ≤ QDt f , f ≥ 0, and the infinitesimal generator of (RD

t ) canbe written as

L = −((−∆)δ/2|D)γ/2 .

Since (δ/2)(γ/2) = α/2, also a version of α-fractional Laplacian in D.

t = ZDTt

the subordinate

killed Z .

Let (RDt )t≥0 be the semigroup of Y D . Then

QDt f ≤ RD

L = −((−∆)δ/2|D)γ/2 .

t = ZDTt

the subordinate

killed Z . Let (RDt )t≥0 be the semigroup of Y D . Then

QDt f ≤ RD

L = −((−∆)δ/2|D)γ/2 .

t = ZDTt

the subordinate

killed Z . Let (RDt )t≥0 be the semigroup of Y D . Then

QDt f ≤ RD

L = −((−∆)δ/2|D)γ/2 .

In two recent papers, Potential theory of subordinate killed Brownianmotion Trans.Amer. Math. Soc. (2019) and On the boundary theory ofsubordinate killed Levy processes, to appear in Pot. Anal. (2019), Kim,Song, V studied potential theory of operators like L (i.e. potential theoryof Y D) in order to understand how it depends on δ and γ.

These subordinate processes have Dirichlet forms

E(u, v) =1

∫∫D×D

(u(x)− v(x))(u(y)− v(y))JD(x , y) dy dx

∫Du(x)v(x)κ(x) dx ,

JD(x , y) = c(γ)

∫ ∞0

qD(t, x , y)t−γ/2−1 dt, x , y ∈ D,

κ(x) = c(γ)

∫ ∞0

(1− QDt 1(x))t−γ/2−1 dt, x ∈ D.

Here qD(t, x , y) are transition densities of ZD (killed δ-stable).

In two recent papers, Potential theory of subordinate killed Brownianmotion Trans.Amer. Math. Soc. (2019) and On the boundary theory ofsubordinate killed Levy processes, to appear in Pot. Anal. (2019), Kim,Song, V studied potential theory of operators like L (i.e. potential theoryof Y D) in order to understand how it depends on δ and γ.These subordinate processes have Dirichlet forms

E(u, v) =1

∫∫D×D

JD(x , y) = c(γ)

∫ ∞0

qD(t, x , y)t−γ/2−1 dt, x , y ∈ D,

κ(x) = c(γ)

∫ ∞0

(1− QDt 1(x))t−γ/2−1 dt, x ∈ D.

E(u, v) =1

∫∫D×D

JD(x , y) = c(γ)

∫ ∞0

qD(t, x , y)t−γ/2−1 dt, x , y ∈ D,

κ(x) = c(γ)

∫ ∞0

(1− QDt 1(x))t−γ/2−1 dt, x ∈ D.

E(u, v) =1

∫∫D×D

JD(x , y) = c(γ)

∫ ∞0

qD(t, x , y)t−γ/2−1 dt, x , y ∈ D,

κ(x) = c(γ)

∫ ∞0

(1− QDt 1(x))t−γ/2−1 dt, x ∈ D.

Here qD(t, x , y) are transition densities of ZD (killed δ-stable).Zoran Vondracek (University of Zagreb) Potential theory of jump processes Bedlewo, May 24, 2019 17 / 33

There were two rather surprising results.

In case δ = 2 (then Y D is a subordinate killed Brownian motion),

JD(x , y) �(δD(x)

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

In case δ ∈ (0, 2) (then Y D is subordinate killed δ-stable process),

GD(x , y) �(δD(x)

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

Somewhat surprisingly

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),

(δD (x)∧δD (y)|x−y| ∧ 1

((δD (x)∨δD (y)∧|x−y|(δD (x)∧δD (y))∧|x−y|

)δ)|x − y |−d−α, γ = 1,(

δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),

(δD (x)∧δD (y)|x−y| ∧ 1

)δ)|x − y |−d−α, γ = 1,(

δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),

(δD (x)∧δD (y)|x−y| ∧ 1

)δ)|x − y |−d−α, γ = 1,(

δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),

(δD (x)∧δD (y)|x−y| ∧ 1

)δ)|x − y |−d−α, γ = 1,(

δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),

(δD (x)∧δD (y)|x−y| ∧ 1

)δ)|x − y |−d−α, γ = 1,

(δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

|x − y |∧ 1

)(δD(y)

|x − y |∧ 1

)|x − y |−d−α .

|x − y |∧ 1

)δ/2( δD(y)

|x − y |∧ 1

|x − y |−d+α .

JD(x , y) �

(δD (x)∧δD (y)|x−y| ∧ 1

)δ(1−γ/2)

|x − y |−d−α, γ ∈ (1, 2),(δD (x)∧δD (y)|x−y| ∧ 1

)δ)|x − y |−d−α, γ = 1,(

δD (x)∧δD (y)|x−y| ∧ 1

)δ/2 (δD (x)∨δD (y)|x−y| ∧ 1

)1−γ/2

|x − y |−d−α, γ ∈ (0, 1).

Define B(x , y) by JD(x , y) = B(x , y)|x − y |−d−α. We think of B(x , y),x , y ∈ D, as a boundary part of the jumping kernel JD(x , y). Displays onthe previous slide give sharp two-sided estimates of B(x , y) for thesubordinate killed process Y D .

Assume γ ∈ (1, 2) or δ = 2. If D is a bounded C 1,1 open set in Rd , then the boundaryHarnack principle (with explicit decay rate) holds for Y D :

There exist C > 0 and r0 > 0such that for any r ∈ (0, r0], any Q ∈ ∂D and any non-negative function f in D which isharmonic in D ∩ B(Q, r) (wrt Y D) and vanishes continuously on ∂D ∩ B(Q, r), we have

δD(x)δ/2≤ C

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

If γ ∈ (0, 1] and δ ∈ (0, 2), then the (non-scale invariant) boundary Harnack principlefails.

The non-scale-invariant boundary Harnack principle holds near the boundary of D ifthere is a constant R ∈ (0, 1) such that for any r ∈ (0, R ], there exists a constantc = c(r) ≥ 1 such that for every Q ∈ ∂D and any non-negative functions f , g in Dwhich are harmonic in D ∩ B(Q, r) with respect to Y D and vanish continuously on∂D ∩ B(Q, r), we have

f (y)≤ c

g(y)for all x , y ∈ D ∩ B(Q, r/2).

Assume γ ∈ (1, 2) or δ = 2. If D is a bounded C 1,1 open set in Rd , then the boundaryHarnack principle (with explicit decay rate) holds for Y D : There exist C > 0 and r0 > 0such that for any r ∈ (0, r0], any Q ∈ ∂D and any non-negative function f in D which isharmonic in D ∩ B(Q, r) (wrt Y D) and vanishes continuously on ∂D ∩ B(Q, r), we have

δD(x)δ/2≤ C

f (y)≤ c

δD(x)δ/2≤ C

f (y)≤ c

δD(x)δ/2≤ C

If γ ∈ (0, 1] and δ ∈ (0, 2), then the (non-scale invariant) boundary Harnack principlefails.The non-scale-invariant boundary Harnack principle holds near the boundary of D ifthere is a constant R ∈ (0, 1) such that for any r ∈ (0, R ], there exists a constantc = c(r) ≥ 1 such that for every Q ∈ ∂D and any non-negative functions f , g in Dwhich are harmonic in D ∩ B(Q, r) with respect to Y D and vanish continuously on∂D ∩ B(Q, r), we have

f (y)≤ c

Jump kernels decaying at the boundary

1 Introduction

Assumptions on B

Let D be an open subset of Rd and α ∈ (0, 2). Let Y = Y D,κ be a Huntprocess with the Dirichlet form whose jumping kernel has the form

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

and the killing function κ : D → [0,∞) which satisfies κ(x) � δD(x)−α.

The boundary term B(x , y) depends on δD(x), δD(y) and |x − y |.

Assumptions on B(x , y):

(B1) B(x , y) = B(y , x) for all x , y ∈ D.(B2) There exists C2 > 0 such that B(x , y) ≤ C2.(B3) For every a ∈ (0, 1) there exists C3 = C3(a) > 0 such thatB(x , y) ≥ C3 whenever δD(x) ∧ δD(y) ≥ a|x − y |.

Assumptions on B

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

and the killing function κ : D → [0,∞) which satisfies κ(x) � δD(x)−α.The boundary term B(x , y) depends on δD(x), δD(y) and |x − y |.

Assumptions on B

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

Assumptions on B

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

Assumptions on B(x , y):(B1) B(x , y) = B(y , x) for all x , y ∈ D.

(B2) There exists C2 > 0 such that B(x , y) ≤ C2.(B3) For every a ∈ (0, 1) there exists C3 = C3(a) > 0 such thatB(x , y) ≥ C3 whenever δD(x) ∧ δD(y) ≥ a|x − y |.

Assumptions on B

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

Assumptions on B(x , y):(B1) B(x , y) = B(y , x) for all x , y ∈ D.(B2) There exists C2 > 0 such that B(x , y) ≤ C2.

(B3) For every a ∈ (0, 1) there exists C3 = C3(a) > 0 such thatB(x , y) ≥ C3 whenever δD(x) ∧ δD(y) ≥ a|x − y |.

Assumptions on B

JD(x , y) = B(x , y)|x − y |−α−d , x , y ∈ D,

Assumptions on B(x , y):(B1) B(x , y) = B(y , x) for all x , y ∈ D.(B2) There exists C2 > 0 such that B(x , y) ≤ C2.(B3) For every a ∈ (0, 1) there exists C3 = C3(a) > 0 such thatB(x , y) ≥ C3 whenever δD(x) ∧ δD(y) ≥ a|x − y |.

Assumptions on B , cont.

(B4) There exists δ > 0 and C4 > 0 such that

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

)δwhenever δD(x) ∧ δD(y) ≥ |x − y |.

(B5) For every ε ∈ (0, 1) there exists C5 = C5(ε) ≥ 1 with the following property: Forevery x0 ∈ D and every r > 0 such that B(x0, (1 + ε)r) ⊂ D, we have for allx1, x2 ∈ B(x0, r) and all z ∈ D \ B(x0, (1 + ε)r),

C−15 B(x1, z) ≤ B(x2, z) ≤ C5B(x1, z) .

(B6) There exist β > 0 and C7 > 0 such that if δD(x) ≤ δD(z) and |y − z | ≤ M|y − x |with M ≥ 1, then

B(x , y) ≤ C7MβB(z , y) .

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

C−15 B(x1, z) ≤ B(x2, z) ≤ C5B(x1, z) .

B(x , y) ≤ C7MβB(z , y) .

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

C−15 B(x1, z) ≤ B(x2, z) ≤ C5B(x1, z) .

B(x , y) ≤ C7MβB(z , y) .

Examples of B

Let β1, β2, β3 ≥ 0 such that β1 > 0 if β3 > 0. Let

L(x , y) :=log(

1 + (δD (x)∨δD (y))∧|x−y|δD (x)∧δD (y)∧|x−y|

)log 2

The following functions satisfy assumptions (B1)-(B6).

B(x , y) =

(δD(x) ∧ δD(y)

|x − y | ∧ 1

)β1(δD(x) ∨ δD(y)

|x − y | ∧ 1

L(x , y)β3 .

B(x , y) :=

((δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2∧ 1

)L(x , y)β3 .

B(x , y) :=(δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2 + (δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2L(x , y)β3 .

Examples of B

L(x , y) :=log(

)log 2

B(x , y) =

(δD(x) ∧ δD(y)

|x − y | ∧ 1

L(x , y)β3 .

B(x , y) :=

((δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2∧ 1

)L(x , y)β3 .

Examples of B

L(x , y) :=log(

)log 2

B(x , y) =

(δD(x) ∧ δD(y)

|x − y | ∧ 1

L(x , y)β3 .

B(x , y) :=

((δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2∧ 1

)L(x , y)β3 .

Examples of B

L(x , y) :=log(

)log 2

B(x , y) =

(δD(x) ∧ δD(y)

|x − y | ∧ 1

L(x , y)β3 .

B(x , y) :=

((δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2∧ 1

)L(x , y)β3 .

Examples of B

L(x , y) :=log(

)log 2

B(x , y) =

(δD(x) ∧ δD(y)

|x − y | ∧ 1

L(x , y)β3 .

B(x , y) :=

((δD(x) ∧ δD(y))β1 (δD(x) ∨ δD(y))β2

|x − y |β1+β2∧ 1

)L(x , y)β3 .

Examples of B , cont.

All three boundary functions are comparable:

B(x , y) � B(x , y) � B(x , y), x , y ∈ D .

For certain choices of parameters β1, β2, β3, they are comparable toboundary terms of subordinate killed stable processes.Note that comparability implies that if the assumptions (B2), (B3), (B5)and (B6) hold for one of these functions, then they hold for the others aswell.

The boundary terms of subordinate killed stable processes satisfyassumptions (B1)-(B6).

B(x , y) � B(x , y) � B(x , y), x , y ∈ D .

For certain choices of parameters β1, β2, β3, they are comparable toboundary terms of subordinate killed stable processes.

Note that comparability implies that if the assumptions (B2), (B3), (B5)and (B6) hold for one of these functions, then they hold for the others aswell.

B(x , y) � B(x , y) � B(x , y), x , y ∈ D .

Interior results

Assume (B1)-(B5). We first prove that for a sufficiently small open set U ⊂ D which issufficiently far away from the boundary ∂D, the Green function of Y killed upon exitingU, GY

U , is comparable to the Green function of the α-stable process when it exits U.

This allows us to estimate the expected exit time from the ball B(x0, r) (away from theboundary): ExτB(x0,r) � rα for x close to the center x0.

Theorem (Harnack inequality):(a) There exists a constant C > 0 such that for any r ∈ (0, 1] and B(x0, r) ⊂ D and anyBorel function f which is non-negative in D and harmonic in B(x0, r) with respect to Y ,we have

f (x) ≤ Cf (y), for all x , y ∈ B(x0, r/2).

(b) For every K > 0 and every L > 0 there exists a constant C = C(K , L) > 0 such thatfor any r < K/L and any Borel function f which is non-negative in D and harmonic inB(x1, r) ∪ B(x2, r) with respect to Y we have

f (x2) ≤ CLd+αf (x1) .

Interior results

U , is comparable to the Green function of the α-stable process when it exits U.

This allows us to estimate the expected exit time from the ball B(x0, r) (away from theboundary): ExτB(x0,r) � rα for x close to the center x0.

f (x2) ≤ CLd+αf (x1) .

Interior results

U , is comparable to the Green function of the α-stable process when it exits U.This allows us to estimate the expected exit time from the ball B(x0, r) (away from theboundary): ExτB(x0,r) � rα for x close to the center x0.

f (x2) ≤ CLd+αf (x1) .

Interior results

f (x2) ≤ CLd+αf (x1) .

Interior results

f (x2) ≤ CLd+αf (x1) .

Carleson’s inequality

Theorem: Suppose that D ⊂ Rd is a κ-fat open set with characteristics(R, κ) and assume that (B1)-(B6) hold true. There exists a constantC = C (R, κ) > 0 such that for every Q ∈ ∂D, 0 < r < R/2, and everynon-negative function f in D that is harmonic in D ∩ B(Q, r) with respectto Y and vanishes continuously on ∂D ∩ B(Q, r), we have

f (x) ≤ Cf (x0) for x ∈ D ∩ B(Q, r/2),

where x0 ∈ D ∩ B(Q, r) with δD(x0) ≥ κr/2.

The half-space case

From now on we assume that D = Rd+ = {x = (x , xd) : xd > 0}.

Assumptions on B: We assume that B satisfies (B1), (B4), and

(B7) There exist a constant C8 ≥ 1 and parameters β1, β2, β3 ≥ 0 withβ1 > 0 if β3 > 0 such that

C−18 B(x , y) ≤ B(x , y) ≤ C8B(x , y) , x , y ∈ Rd

(B8) For all x , y ∈ Rd+ and all a > 0, B(ax , ay) = B(x , y).

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

Under (B8), Y enjoys α-scaling: If Y(r)t = rYr−αt , then (Y (r),Prx) has

the same law as (Y ,Px).

The half-space case

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

The half-space case

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

The half-space case

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

The half-space case

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

The half-space case

B(x , y) =

(δD(x) ∧ δD(y)

|x − y |∧ 1

L(x , y)β3 .

The operator LB

For f : Rd+ → [0,∞), we set for x ∈ Rd

LBαf (x) := p.v.

f (y)− f (x)

|y − x |d+αB(x , y) dy = p.v.

(f (y)− f (x))JRd+ (x , y) dy ,

whenever the principal value integral on the right hand side makes sense.

Let gp : Rd+ → R be defined by gp(y) = yp

d . Set ed = (0, 1).

Lemma: Assume (B1), (B7) and (B8). Let p ∈ ((α− 1)+, α + β1). Then

LBαgp(x) = C(α, p,B)xp−α

where C(α, p,B) is given by∫Rd−1

(|u|2 + 1)(d+α)/2

(∫ 1

(wp − 1)(1− wα−p−1)

(1− w)1+αB((1− w)u, 1),wed

The function p 7→ C(α, p,B) is increasing and maps ((α− 1)+, α + β1) onto (0,∞).

The operator LB

LBαf (x) := p.v.

f (y)− f (x)

|y − x |d+αB(x , y) dy = p.v.

(f (y)− f (x))JRd+ (x , y) dy ,

d . Set ed = (0, 1).

Lemma: Assume (B1), (B7) and (B8). Let p ∈ ((α− 1)+, α + β1). Then

(|u|2 + 1)(d+α)/2

(∫ 1

(wp − 1)(1− wα−p−1)

(1− w)1+αB((1− w)u, 1),wed

The operator LB

LBαf (x) := p.v.

f (y)− f (x)

|y − x |d+αB(x , y) dy = p.v.

(f (y)− f (x))JRd+ (x , y) dy ,

d . Set ed = (0, 1).Lemma: Assume (B1), (B7) and (B8). Let p ∈ ((α− 1)+, α + β1). Then

(|u|2 + 1)(d+α)/2

(∫ 1

(wp − 1)(1− wα−p−1)

(1− w)1+αB((1− w)u, 1),wed

The operator LB

LBαf (x) := p.v.

f (y)− f (x)

|y − x |d+αB(x , y) dy = p.v.

(f (y)− f (x))JRd+ (x , y) dy ,

d . Set ed = (0, 1).Lemma: Assume (B1), (B7) and (B8). Let p ∈ ((α− 1)+, α + β1). Then

(|u|2 + 1)(d+α)/2

(∫ 1

(wp − 1)(1− wα−p−1)

(1− w)1+αB((1− w)u, 1),wed

Operator LB

Recall gp(x) = xpd with p ∈ ((α− 1)+, α + β1).

SetLB f (x) := LBα f (x)− C (α, p,B)x−αd f (x) , x ∈ Rd

so that LBgp(x) = 0.

Operator LB

Recall gp(x) = xpd with p ∈ ((α− 1)+, α + β1).Set

LB f (x) := LBα f (x)− C (α, p,B)x−αd f (x) , x ∈ Rd+ ,

so that LBgp(x) = 0.

Exit probabilities

For a, b > 0 define D(a, b) := {x = (x , xd) ∈ Rd : |x | < a, 0 < xd < b}and U(r) = D( r

2 ,r2 ). Write U for U(1).

Lemma: For every x = (0, xd) with 0 < xd < 1/4,

∫ τU

t )β1 | logY dt |β3 dt � xpd .

If p ∈ ((α− 1)+, α), then ExτU � xpd .

Px(YτU ∈ D(1/2, 1) \ D(1/2, 3/4)) � xpd .

The key to the proofs is finding a good testing function φ and thenestimating LBφ.

Exit probabilities

∫ τU

Px(YτU ∈ D(1/2, 1) \ D(1/2, 3/4)) � xpd .

Exit probabilities

∫ τU

Px(YτU ∈ D(1/2, 1) \ D(1/2, 3/4)) � xpd .

Exit probabilities

∫ τU

Px(YτU ∈ D(1/2, 1) \ D(1/2, 3/4)) � xpd .

Exit probabilities

∫ τU

Px(YτU ∈ D(1/2, 1) \ D(1/2, 3/4)) � xpd .

Recall (B4): There exist δ > 0 and C4 > 0 such that

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

)δwhenever δD(x)∧ δD(y) ≥ |x − y | and note that if B = B or B = B, then any δ > 0 willdo (in particular δ = 1), while for B = B, δ = β1 + β2.

Theorem (BHP): Suppose that δ > (α− 1)+ and either(a) β1 = β2 = β > 0 and β3 = 0, or (b) p < α.There exists C > 0 such that for any non-negative function f in Rd

+ which is harmonic

in D(2, 2) with respect to Y = Y Rd+,κ and vanishes continuously on B(0, 2) ∩ ∂Rd

≤ Cf (y)

, x , y ∈ D(1/2, 1/2).

In case (a)

B(x , y) =

(δD(x)

|x − y | ∧ 1

)β (δD(y)

|x − y | ∧ 1

)βand p ∈ ((α− 1)+, α + β).

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

)δwhenever δD(x)∧ δD(y) ≥ |x − y | and note that if B = B or B = B, then any δ > 0 willdo (in particular δ = 1), while for B = B, δ = β1 + β2.Theorem (BHP): Suppose that δ > (α− 1)+ and either(a) β1 = β2 = β > 0 and β3 = 0, or (b) p < α.

There exists C > 0 such that for any non-negative function f in Rd+ which is harmonic

≤ Cf (y)

, x , y ∈ D(1/2, 1/2).

In case (a)

B(x , y) =

(δD(x)

|x − y | ∧ 1

)β (δD(y)

|x − y | ∧ 1

)βand p ∈ ((α− 1)+, α + β).

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

)δwhenever δD(x)∧ δD(y) ≥ |x − y | and note that if B = B or B = B, then any δ > 0 willdo (in particular δ = 1), while for B = B, δ = β1 + β2.Theorem (BHP): Suppose that δ > (α− 1)+ and either(a) β1 = β2 = β > 0 and β3 = 0, or (b) p < α.There exists C > 0 such that for any non-negative function f in Rd

+ which is harmonic

≤ Cf (y)

, x , y ∈ D(1/2, 1/2).

In case (a)

B(x , y) =

(δD(x)

|x − y | ∧ 1

)β (δD(y)

|x − y | ∧ 1

)βand p ∈ ((α− 1)+, α + β).

0 ≤ 1− B(x , y) ≤ C4

(|x − y |

δD(x) ∧ δD(y)

)δwhenever δD(x)∧ δD(y) ≥ |x − y | and note that if B = B or B = B, then any δ > 0 willdo (in particular δ = 1), while for B = B, δ = β1 + β2.Theorem (BHP): Suppose that δ > (α− 1)+ and either(a) β1 = β2 = β > 0 and β3 = 0, or (b) p < α.There exists C > 0 such that for any non-negative function f in Rd

+ which is harmonic

≤ Cf (y)

, x , y ∈ D(1/2, 1/2).

In case (a)

B(x , y) =

(δD(x)

|x − y | ∧ 1

)β (δD(y)

|x − y | ∧ 1

)βand p ∈ ((α− 1)+, α + β).

Failure of BHP

Theorem: Assume that α < p < α+ β1, α+ β2 < p and β3 ≥ 0. Then thenon-scale-invariant boundary Harnack principle is not valid for Y = Y Rd

· introduction the kernel of a jump process jump processes are studied under various assumptions,...

Documents