martin boundary for subordinate brownian...

Martin boundary for subordinate Brownian motion

Zoran Vondracek(joint work with Panki Kim and Renming Song)

Department of MathematicsUniversity of Zagreb

Croatia

Wroc law, 15-19.7.2013.

Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 1 / 42

Motivation

1 Motivation

2 Description of the class of processes - subordinate BM

3 Boundary Harnack principle

4 Boundary Harnack principle at infinity

5 Martin boundary of unbounded sets

6 Minimal thinness

Motivation

Representation of harmonic functions on the halfspace

H = {x = (x , xd) : x ∈ Rd−1, xd > 0} – halfspace in Rd .

If h : H→ [0,∞) harmonic in H, then

h(x) = cxd +

∫∂H

xd|x − z |d

µ(dz) = cxd +

∫∂H

xd|x − z |d

(1 + |z |2)d/2 ν(dz) ,

c ≥ 0 and a measure µ on ∂H.

1− 1 correspondence: ∂H 3 z ←→ M(x , z) :=xd

|x − z |d(1 + |z |2)d/2,

∞←→ M(x ,∞) := xd .

Martin boundary of H: ∂H ∪ {∞}:

h(x) =

∫∂H∪{∞}

M(x , z) ν(dz) .

Motivation

h(x) = cxd +

∫∂H

xd|x − z |d

µ(dz) = cxd +

∫∂H

xd|x − z |d

(1 + |z |2)d/2 ν(dz) ,

|x − z |d(1 + |z |2)d/2,

∞←→ M(x ,∞) := xd .

h(x) =

∫∂H∪{∞}

M(x , z) ν(dz) .

Motivation

h(x) = cxd +

∫∂H

xd|x − z |d

µ(dz) = cxd +

∫∂H

xd|x − z |d

(1 + |z |2)d/2 ν(dz) ,

|x − z |d(1 + |z |2)d/2,

∞←→ M(x ,∞) := xd .

h(x) =

∫∂H∪{∞}

M(x , z) ν(dz) .

Motivation

h(x) = cxd +

∫∂H

xd|x − z |d

µ(dz) = cxd +

∫∂H

xd|x − z |d

(1 + |z |2)d/2 ν(dz) ,

|x − z |d(1 + |z |2)d/2,

∞←→ M(x ,∞) := xd .Martin boundary of H: ∂H ∪ {∞}:

h(x) =

∫∂H∪{∞}

M(x , z) ν(dz) .

Motivation

Representation of α/2-harmonic functions

h : Rd → [0,∞) is α/2-harmonic in H, 0 < α < 2, if “∆α/2h = 0 in H”.

Probabilistic interpretation: X = (Xt ,Px) isotropic α-stable process. Thenh is harmonic in H (wrt to X ) if for every relatively compact open G ⊂ H

h(x) = Exh(XτG ) , ∀x ∈ G ,

and regular harmonic in H if

h(x) = Exh(XτH) , ∀x ∈ H .

If P(x , z) dz = Px(XτH ∈ dz) - Poisson kernel, then regular harmonic h hasa representation

h(x) =

P(x , z)h(z) dz .

x 7→ P(x , z) is not harmonic in H.

Motivation

h : Rd → [0,∞) is α/2-harmonic in H, 0 < α < 2, if “∆α/2h = 0 in H”.Probabilistic interpretation: X = (Xt ,Px) isotropic α-stable process. Thenh is harmonic in H (wrt to X ) if for every relatively compact open G ⊂ H

h(x) = Exh(XτG ) , ∀x ∈ G ,

h(x) = Exh(XτH) , ∀x ∈ H .

h(x) =

P(x , z)h(z) dz .

Motivation

h(x) = Exh(XτG ) , ∀x ∈ G ,

h(x) = Exh(XτH) , ∀x ∈ H .

h(x) =

P(x , z)h(z) dz .

Motivation

h(x) = Exh(XτG ) , ∀x ∈ G ,

h(x) = Exh(XτH) , ∀x ∈ H .

h(x) =

P(x , z)h(z) dz .

Motivation

h(x) = Exh(XτG ) , ∀x ∈ G ,

h(x) = Exh(XτH) , ∀x ∈ H .

h(x) =

P(x , z)h(z) dz .

Motivation

Representation of singular α/2-harmonic functions

Suppose h : Rd → [0,∞) is α/2-harmonic and h ≡ 0 on Hc . Then h iscalled singular harmonic. This corresponds to harmonic functions of thekilled process XH.

Representation of harmonic functions for XH. Let

M(x , z) :=xα/2d

|x − z |d(1 + |z |2)d/2 , M(x ,∞) := x

α/2d .

If h : H→ [0,∞) is harmonic wrt XH (singular α/2-harmonic), then thereexists a unique measure ν on ∂H ∪ {∞} such that

h(x) =

∫∂H∪{∞}

M(x , z) ν(dz) = cxα/2d +

∫∂H

xα/2d

|x − z |d(1 + |z |2)d/2 ν(dz) .

Martin boundary of H with respect to X : ∂H ∪ {∞}.

Motivation

Suppose h : Rd → [0,∞) is α/2-harmonic and h ≡ 0 on Hc . Then h iscalled singular harmonic. This corresponds to harmonic functions of thekilled process XH.Representation of harmonic functions for XH. Let

M(x , z) :=xα/2d

|x − z |d(1 + |z |2)d/2 , M(x ,∞) := x

α/2d .

h(x) =

∫∂H∪{∞}

∫∂H

xα/2d

|x − z |d(1 + |z |2)d/2 ν(dz) .

Motivation

M(x , z) :=xα/2d

|x − z |d(1 + |z |2)d/2 , M(x ,∞) := x

α/2d .

h(x) =

∫∂H∪{∞}

∫∂H

xα/2d

|x − z |d(1 + |z |2)d/2 ν(dz) .

Motivation

M(x , z) :=xα/2d

|x − z |d(1 + |z |2)d/2 , M(x ,∞) := x

α/2d .

h(x) =

∫∂H∪{∞}

∫∂H

xα/2d

|x − z |d(1 + |z |2)d/2 ν(dz) .

Motivation

Martin boundary

Let X = (Xt ,Px) be rotationally invariant Levy process in Rd , D ⊂ Rd

open, XD the killed process, GD(x , y) the Green function of XD .

Fix x0 ∈ D and define MD(x , y) := GD(x ,y)GD(x0,y) , x , y ∈ D.

D has a Martin boundary ∂MD with respect to XD satisfying the followingproperties:

(1) D ∪ ∂MD is compact metric space;

(2) D is open and dense in D ∪ ∂MD, and its relative topology coincideswith its original topology;

(3) MD(x , · ) can be uniquely extended to ∂MD in such a way that,MD(x , y) converges to MD(x , z) as y → z ∈ ∂MD, the functionx → MD(x , z) is excessive with respect to XD , the function(x , z)→ MD(x , z) is jointly continuous on D × ∂MD andMD(·, z1) 6= MD(·, z2) if z1 6= z2.

Motivation

Martin boundary

Motivation

Martin boundary

Motivation

Martin boundary

Motivation

Minimal Martin boundary

A harmonic function h : D → [0,∞) is minimal (with respect to XD), ifg ≤ h, g harmonic, implies that g = ch.

The minimal Martin boundary of XD is defined as

∂mD = {z ∈ ∂MD : MD(·, z) is minimal harmonic with respect to XD}.

A function h : D → [0,∞) is harmonic if and only if there exists a finitemeasure ν on ∂mD such that

h(x) =

∫∂mD

MD(x , z) ν(dz) , Martin integral representation.

Motivation

A harmonic function h : D → [0,∞) is minimal (with respect to XD), ifg ≤ h, g harmonic, implies that g = ch.The minimal Martin boundary of XD is defined as

h(x) =

∫∂mD

Motivation

A harmonic function h : D → [0,∞) is minimal (with respect to XD), ifg ≤ h, g harmonic, implies that g = ch.The minimal Martin boundary of XD is defined as

h(x) =

∫∂mD

Motivation

Some history

The notion of Martin boundary goes back to Robert S. Martin (1941) forthe case classical harmonic functions (i.e. X is Brownian motion).

General theory of Martin boundary for strong Markov processes (induality) developed by Kunita and Watanabe (1965).

X is Brownian motion, D bounded Lipschitz domain D: Hunt andWheeden (1970) proved that the (minimal) Martin boundary can beidentified with the Euclidean boundary.

X rotationally invariant α-stable process, 0 < α < 2. Identification of the(minimal) Martin boundary with the Euclidean boundary:

(1) Bounded Lipschitz domain: Chen and Song (1998) and Bogdan(1999);

(2) Bounded κ-fat open set: Song and Wu (1999).

Certain subordinate BM, D bounded κ-fat open set: Kim, Song, V.(2009).

Motivation

Some history

The notion of Martin boundary goes back to Robert S. Martin (1941) forthe case classical harmonic functions (i.e. X is Brownian motion).General theory of Martin boundary for strong Markov processes (induality) developed by Kunita and Watanabe (1965).

Motivation

Some history

Motivation

Some history

X is Brownian motion, D bounded Lipschitz domain D: Hunt andWheeden (1970) proved that the (minimal) Martin boundary can beidentified with the Euclidean boundary.X rotationally invariant α-stable process, 0 < α < 2. Identification of the(minimal) Martin boundary with the Euclidean boundary:

Motivation

Some history

X is Brownian motion, D bounded Lipschitz domain D: Hunt andWheeden (1970) proved that the (minimal) Martin boundary can beidentified with the Euclidean boundary.X rotationally invariant α-stable process, 0 < α < 2. Identification of the(minimal) Martin boundary with the Euclidean boundary:

Motivation

Martin boundary for unbounded sets?

In all mentioned results D is bounded. The reason: Proofs depend on theboundary Harnack principle for non-negative harmonic functions whichimplies the existence of the limit limy→z∈∂D MD(x , y).

Known results for unbounded sets. Complete description of the Martinboundary only for Brownian motion and isotropic stable processes. Explicitformulae for the Martin kernel in case of the half-space H.

In case of unbounded open D, inversion through the sphere implies theexistence of MD(x ,∞) := lim|y |→∞, y∈D MD(x , y):Bogdan, Kulczycki, Kwasnicki (2008)

Motivation

Finite part and infinite part of Martin boundary

Partial results for some subordinate Brownian motion – description of thefinite part of the Martin boundary.

A point z ∈ ∂MD is called a finite Martin boundary point if there exists abounded sequence (yn)n≥1 converging to z in the Martin topology.

A point z is called an infinite Martin boundary point if every sequence(yn)n≥1 converging to z in the Martin topology is unbounded.

In case X is a subordinate Brownian motion satisfying certain condition,the finite part of the Martin boundary of H can be identified with theEuclidean boundary ∂H, Kim, Song, V. (2011).

Motivation

Goal of the talk

Describe under what conditions on the process X and the unbounded openset D one can identify the (minimal) Martin boundary of D with∂D ∪ {∞}.

Two types of assumptions for the process: small time-small scale, andlarge time-large scale.

Assumptions on D: κ-fat at each boundary point, and κ-fat at infinity.

Motivation

Goal of the talk

Two types of assumptions for the process: small time-small scale, andlarge time-large scale.

Assumptions on D: κ-fat at each boundary point, and κ-fat at infinity.

Motivation

Goal of the talk

Two types of assumptions for the process: small time-small scale, andlarge time-large scale.Assumptions on D: κ-fat at each boundary point, and κ-fat at infinity.

Description of the class of processes - subordinate BM

1 Motivation

6 Minimal thinness

Subordinators

S = (St)t≥0 a subordinator with the Laplace exponent φ:

E[e−λSt ] = e−tφ(λ) , φ(t) =

∫(0,∞)

(1− e−λt)µ(dt)

Assumptions on φ: φ is CBF – µ(dt) = µ(t) dt where µ is CM.Consequence: the renewal measure has a CM density u. WLOG φ(1) = 1.

Upper and lower scaling conditions at infinity and at zero:(H1): There exist constants 0 < δ1 ≤ δ2 < 1 and a1, a2 > 0 such that

a1λδ1φ(t) ≤ φ(λt) ≤ a2λ

δ2φ(t), λ ≥ 1, t ≥ 1 .

(H2): There exist constants 0 < δ3 ≤ δ4 < 1 and a3, a4 > 0 such that

δ3φ(t), λ ≤ 1, t ≤ 1 .

Subordinators

E[e−λSt ] = e−tφ(λ) , φ(t) =

∫(0,∞)

Assumptions on φ: φ is CBF – µ(dt) = µ(t) dt where µ is CM.Consequence: the renewal measure has a CM density u. WLOG φ(1) = 1.

Upper and lower scaling conditions at infinity and at zero:(H1): There exist constants 0 < δ1 ≤ δ2 < 1 and a1, a2 > 0 such that

δ2φ(t), λ ≥ 1, t ≥ 1 .

δ3φ(t), λ ≤ 1, t ≤ 1 .

Subordinators

E[e−λSt ] = e−tφ(λ) , φ(t) =

∫(0,∞)

Assumptions on φ: φ is CBF – µ(dt) = µ(t) dt where µ is CM.Consequence: the renewal measure has a CM density u. WLOG φ(1) = 1.Upper and lower scaling conditions at infinity and at zero:(H1): There exist constants 0 < δ1 ≤ δ2 < 1 and a1, a2 > 0 such that

δ2φ(t), λ ≥ 1, t ≥ 1 .

δ3φ(t), λ ≤ 1, t ≤ 1 .

Subordinators

E[e−λSt ] = e−tφ(λ) , φ(t) =

∫(0,∞)

Assumptions on φ: φ is CBF – µ(dt) = µ(t) dt where µ is CM.Consequence: the renewal measure has a CM density u. WLOG φ(1) = 1.Upper and lower scaling conditions at infinity and at zero:(H1): There exist constants 0 < δ1 ≤ δ2 < 1 and a1, a2 > 0 such that

δ2φ(t), λ ≥ 1, t ≥ 1 .

δ3φ(t), λ ≤ 1, t ≤ 1 .

Examples

If 0 < α < 2 and ˜ slowly varying at infinity, then

φ(λ) � λα/2 ˜(λ) , λ→∞ ,

implies (H1). Assumption on the behavior of the subordinator (henceSBM) for small time, small space.

If 0 < β < 2 and ` slowly varying at infinity, then

φ(λ) � λβ/2`(λ) , λ→ 0 ,

implies (H2). Assumption on the behavior of the subordinator (henceSBM) for large time, large space.

(H1) (resp. (H2)) is equivalent to φ is an O-regularly varying functions at∞ (resp. at 0) with Matuszewska indices in (0, 1).

Examples

φ(λ) � λα/2 ˜(λ) , λ→∞ ,

φ(λ) � λβ/2`(λ) , λ→ 0 ,

Examples

φ(λ) � λα/2 ˜(λ) , λ→∞ ,

φ(λ) � λβ/2`(λ) , λ→ 0 ,

Properties of the potential and the Levy density

There exists a constant C = C (φ) > 0 such that

u(t) ≤ Ct−1φ(t−1)−1 , µ(t) ≤ Ct−1φ(t−1) , ∀ t ∈ (0,∞) .

(H1): u(t) ≥ C−1t−1φ(t−1)−1 , µ(t) ≥ C−1t−1φ(t−1) , ∀ t ∈ (0, 1] ,

(H2): u(t) ≥ C−1t−1φ(t−1)−1 , µ(t) ≥ C−1t−1φ(t−1) , ∀ t ∈ [1,∞)

We write

u(t) � t−1φ(t−1)−1 , µ(t) � t−1φ(t−1) , t ∈ (0,∞) .

Properties of the potential and the Levy density

There exists a constant C = C (φ) > 0 such that

u(t) ≤ Ct−1φ(t−1)−1 , µ(t) ≤ Ct−1φ(t−1) , ∀ t ∈ (0,∞) .

(H1): u(t) ≥ C−1t−1φ(t−1)−1 , µ(t) ≥ C−1t−1φ(t−1) , ∀ t ∈ (0, 1] ,

(H2): u(t) ≥ C−1t−1φ(t−1)−1 , µ(t) ≥ C−1t−1φ(t−1) , ∀ t ∈ [1,∞)

We write

u(t) � t−1φ(t−1)−1 , µ(t) � t−1φ(t−1) , t ∈ (0,∞) .

Subordinate Brownian motion

W = (Wt ,Px) d-dimensional Brownian motion, S = (St) and independentsubordinator with the Laplace exponent φ satisfying (H1), (H2) and CBF .

The SBM is the process X = (Xt)t≥0 defined as Xt := WSt .

X is a Levy process with characteristic exponent Φ(x) = φ(|x |2),infinitesimal generator A = φ(−∆), and Levy measure with densityJ(x) = j(|x |) where

j(r) =

∫ ∞0

(4πt)−d/2e−r2/4t µ(t) dt , r > 0 .

Assume X is transient; then X has the Green functionG (x , y) = G (x − y) = g(|x − y |) where

g(r) =

∫ ∞0

(4πt)−d/2e−r2/4t u(t) dt , r > 0 .

W = (Wt ,Px) d-dimensional Brownian motion, S = (St) and independentsubordinator with the Laplace exponent φ satisfying (H1), (H2) and CBF .The SBM is the process X = (Xt)t≥0 defined as Xt := WSt .

j(r) =

∫ ∞0

(4πt)−d/2e−r2/4t µ(t) dt , r > 0 .

g(r) =

∫ ∞0

(4πt)−d/2e−r2/4t u(t) dt , r > 0 .

j(r) =

∫ ∞0

(4πt)−d/2e−r2/4t µ(t) dt , r > 0 .

g(r) =

∫ ∞0

(4πt)−d/2e−r2/4t u(t) dt , r > 0 .

j(r) =

∫ ∞0

(4πt)−d/2e−r2/4t µ(t) dt , r > 0 .

g(r) =

∫ ∞0

(4πt)−d/2e−r2/4t u(t) dt , r > 0 .

Green function and Levy measure

Theorem: Assume (H1) and (H2).

(a) ThenJ(x) � |x |−dφ(|x |−2) , x 6= 0 .

(b) If d > 2(δ2 ∨ δ4), then X is transient and

G (x) � |x |−dφ(|x |−2)−1 , x 6= 0 .

Corollary: (Doubling property) J(2x) � J(x), G (2x) � G (x), x 6= 0.

Theorem: Assume (H1) and (H2).(a) Then

J(x) � |x |−dφ(|x |−2) , x 6= 0 .

G (x) � |x |−dφ(|x |−2)−1 , x 6= 0 .

J(x) � |x |−dφ(|x |−2) , x 6= 0 .

G (x) � |x |−dφ(|x |−2)−1 , x 6= 0 .

J(x) � |x |−dφ(|x |−2) , x 6= 0 .

G (x) � |x |−dφ(|x |−2)−1 , x 6= 0 .

Boundary Harnack principle

1 Motivation

6 Minimal thinness

Uniform BHP

Recall that u : Rd → [0,∞) is regular harmonic in open D ⊂ Rd withrespect to X if

u(x) = Ex [u(XτD ) : τD <∞] , for all x ∈ D .

Theorem: There exists a constant c = c(φ, d) > 0 such that for everyz0 ∈ Rd , every open set D ⊂ Rd , every r > 0 and for any nonnegativefunctions u, v in Rd which are regular harmonic in D ∩ B(z0, r) withrespect to X and vanish in Dc ∩ B(z0, r), we have

v(x)≤ c

v(y)for all x , y ∈ D ∩ B(z0, r/2) .

Uniform BHP

Recall that u : Rd → [0,∞) is regular harmonic in open D ⊂ Rd withrespect to X if

u(x) = Ex [u(XτD ) : τD <∞] , for all x ∈ D .

Theorem: There exists a constant c = c(φ, d) > 0 such that for everyz0 ∈ Rd , every open set D ⊂ Rd , every r > 0 and for any nonnegativefunctions u, v in Rd which are regular harmonic in D ∩ B(z0, r) withrespect to X and vanish in Dc ∩ B(z0, r), we have

v(x)≤ c

v(y)for all x , y ∈ D ∩ B(z0, r/2) .

Lemma: (Approximate factorization) For every z0 ∈ Rd , every open setU ⊂ B(z0, r) and for any nonnegative function u in Rd which is regularharmonic in U with respect to X and vanishes a.e. in Uc ∩ B(z0, r),

u(x) � Ex [τU ]

∫B(z0,r/2)c

j(|z − z0|)u(z)dz , x ∈ U ∩ B(z0, r/2) .

For all r ∈ (0, 1] under (H1) (Kim, Song, V. (2011)), for all r ∈ (0,∞)under (H1) and (H2) (Kim, Song, V (2012)).

u(x) � Ex [τU ]

∫B(z0,r/2)c

j(|z − z0|)u(z)dz , x ∈ U ∩ B(z0, r/2) .

u(x) � Ex [τU ]

∫B(z0,r/2)c

j(|z − z0|)u(z)dz , x ∈ U ∩ B(z0, r/2) .

Take z0 = 0. Then the above reads:

u(x) �∫UGU(x , y) dy

∫B(0,r/2)c

j(|y |)u(y)dy , x ∈ U ∩ B(0, r/2) .

Global and uniform BHP in smooth sets with explicit decayrate

Theorem: Assume (H1) and (H2). There exists c = c(φ) > 0 such thatfor every open set D satisfying the interior and exterior ball conditionswith radius R > 0, every r ∈ (0,R], every Q ∈ ∂D and every nonnegativefunction u in Rd which is harmonic in D ∩ B(Q, r) with respect to X andvanishes continuously on Dc ∩ B(Q, r), we have

(φ(δD(x)−2))−1/2≤ c

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

Global: it holds for all R > 0 with the comparison constant not dependingon R.Uniform: it holds for all balls with radii r ≤ R and the comparisonconstant depends neither on D nor on r .

Global and uniform BHP in smooth sets with explicit decayrate

Theorem: Assume (H1) and (H2). There exists c = c(φ) > 0 such thatfor every open set D satisfying the interior and exterior ball conditionswith radius R > 0, every r ∈ (0,R], every Q ∈ ∂D and every nonnegativefunction u in Rd which is harmonic in D ∩ B(Q, r) with respect to X andvanishes continuously on Dc ∩ B(Q, r), we have

(φ(δD(x)−2))−1/2≤ c

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

Global: it holds for all R > 0 with the comparison constant not dependingon R.Uniform: it holds for all balls with radii r ≤ R and the comparisonconstant depends neither on D nor on r .

Boundary Harnack principle at infinity

1 Motivation

6 Minimal thinness

In case of rotationally invariant α-stable process, M. Kwasnicki (2009)used the inversion through the sphere B(0,

√r) to obtain a BHP at infinity.

Recall that the Poisson kernel KU(x , z) is the exit density from an openset U: Px(XτU ∈ B) =

∫B KU(x , z) dy , B ⊂ U

KU(x , z) =

∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U

If u regular harmonic in U, then u(x) =∫U

c KU(x , z)u(z) dz .

∫B KU(x , z) dy , B ⊂ U

KU(x , z) =

∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U

∫B KU(x , z) dy , B ⊂ U

KU(x , z) =

∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U

BHP at infinity – continuation

Theorem: There exists C = C (φ) > 1 such that for all r ≥ 1, for all opensets U ⊂ B(0, r)c and all nonnegative functions u on Rd that are regularharmonic in U and vanish on B(0, r)c \ U, it holds that

Go back

C≤ u(x)

KU(x , 0)∫B(0,2r) u(z) dz

≤ C , for all x ∈ U ∩ B(0, 2r)c .

u(x) �∫UGU(x , y)j(|y |) dy

∫B(0,2r)

u(z) dz , x ∈ U ∩ B(0, 2r)c .

Go back

C≤ u(x)

KU(x , 0)∫B(0,2r) u(z) dz

≤ C , for all x ∈ U ∩ B(0, 2r)c .

∫B(0,2r)

u(z) dz , x ∈ U ∩ B(0, 2r)c .

Go back

C≤ u(x)

KU(x , 0)∫B(0,2r) u(z) dz

≤ C , for all x ∈ U ∩ B(0, 2r)c .

∫B(0,2r)

u(z) dz , x ∈ U ∩ B(0, 2r)c .

Go back

C≤ u(x)

KU(x , 0)∫B(0,2r) u(z) dz

≤ C , for all x ∈ U ∩ B(0, 2r)c .

∫B(0,2r)

u(z) dz , x ∈ U ∩ B(0, 2r)c .

Theorem: There exists C = C (φ) > 1 such that for all r ≥ 1, for all opensets U ⊂ B(0, r)c and all nonnegative functions u on Rd that are regularharmonic in U and vanish on B(0, r)c \ U, it holds that Go back

C≤ u(x)

KU(x , 0)∫B(0,2r) u(z) dz

≤ C , for all x ∈ U ∩ B(0, 2r)c .

∫B(0,2r)

u(z) dz , x ∈ U ∩ B(0, 2r)c .

Corollaries

Corollary: There exists C = C (φ) > 1 such that for all r ≥ 1, for all opensets U ⊂ B(0, r)c and all nonnegative functions u and v on Rd that areregular harmonic in U and vanish on B(0, r)c \ U, it holds that

C−1 u(y)

v(y)≤ u(x)

v(x)≤ C

v(y), for all x , y ∈ U ∩ B(0, 2r)c .

Corollary: Let r ≥ 1 and U ⊂ B(0, r)c . If u is a non-negative function onRd which is regular harmonic in U and vanishes on B(0, r)c \ U, then

lim|x |→∞

u(x) = 0 .

Not true if regular harmonic is replaced by harmonic: w(x) := (x+d )α/2 is

harmonic in the upper half-space H ⊂ B((0,−1), 1)c , vanishes onB((0,−1), 1)c \H, but limxd→∞ w(x) =∞.

Corollaries

C−1 u(y)

v(y)≤ u(x)

v(x)≤ C

v(y), for all x , y ∈ U ∩ B(0, 2r)c .

lim|x |→∞

u(x) = 0 .

Corollaries

C−1 u(y)

v(y)≤ u(x)

v(x)≤ C

v(y), for all x , y ∈ U ∩ B(0, 2r)c .

lim|x |→∞

u(x) = 0 .

Ingredients of the proof

Upper bound on the Green function B(0, r)c , r ≥ 1: Let 1 < p < q < 4and b > 0. There exist a constant C = C (φ, p, q, b) > 0 such that for allr ≥ 1, all x ∈ A(0, pr , qr) and all y ∈ A(0, r , 2qr) such that br < |x − y | itholds that

GB(0,r)c (x , y) ≤ cφ(r−2)1/2

φ(δB(0,r)c (y)−2)1/2g(r)

≤ Cφ(r−2)−1/2φ(δB(0,r)c (y)−2)−1/2r−d .

The proof uses the global, uniformBHP with explicit decay rate.

GB(0,r)c (x , y) ≤ cφ(r−2)1/2

φ(δB(0,r)c (y)−2)1/2g(r)

≤ Cφ(r−2)−1/2φ(δB(0,r)c (y)−2)−1/2r−d .

GB(0,r)c (x , y) ≤ cφ(r−2)1/2

φ(δB(0,r)c (y)−2)1/2g(r)

≤ Cφ(r−2)−1/2φ(δB(0,r)c (y)−2)−1/2r−d .

GB(0,r)c (x , y) ≤ cφ(r−2)1/2

φ(δB(0,r)c (y)−2)1/2g(r)

≤ Cφ(r−2)−1/2φ(δB(0,r)c (y)−2)−1/2r−d .

Ingredients of the proof – continuation

Upper bound for the Poisson kernel of B(0, r)c , r ≥ 1: Let 1 < p < q < 4.There exists C = C (φ, p, q) > 1 such that for all r ≥ 1, allx ∈ A(0, pr , qr) and z ∈ B(0, r) it holds that

KB(0,r)c (x , z) ≤ Cr−dφ(r−2)−1/2φ((r − |z |)−2)1/2 .

For α-stable process

KB(0,r)c (x , z)

= c(α, d)(|x |2 − r2)α/2

(r2 − |z |2)α/2|x − z |−d

� r−drα/2

(r − |z |)α/2.

KB(0,r)c (x , z) ≤ Cr−dφ(r−2)−1/2φ((r − |z |)−2)1/2 .

KB(0,r)c (x , z)

= c(α, d)(|x |2 − r2)α/2

(r2 − |z |2)α/2|x − z |−d

� r−drα/2

(r − |z |)α/2.

KB(0,r)c (x , z) ≤ Cr−dφ(r−2)−1/2φ((r − |z |)−2)1/2 .

KB(0,r)c (x , z)

= c(α, d)(|x |2 − r2)α/2

(r2 − |z |2)α/2|x − z |−d

� r−drα/2

(r − |z |)α/2.

KB(0,r)c (x , z) ≤ Cr−dφ(r−2)−1/2φ((r − |z |)−2)1/2 .

KB(0,r)c (x , z)

= c(α, d)(|x |2 − r2)α/2

(r2 − |z |2)α/2|x − z |−d

� r−drα/2

(r − |z |)α/2.

Exit probability estimate: For every a ∈ (1,∞), there exists a positiveconstant C = C (φ, a) > 0 such that for any r ∈ (0,∞) and any open setU ⊂ B(0, r)c we have

(XτU ∈ B(0, r)

)≤ CrdKU(x , 0) , x ∈ U ∩ B(0, ar)c .

Exit probability estimate: For every a ∈ (1,∞), there exists a positiveconstant C = C (φ, a) > 0 such that for any r ∈ (0,∞) and any open setU ⊂ B(0, r)c we have

(XτU ∈ B(0, r)

)≤ CrdKU(x , 0) , x ∈ U ∩ B(0, ar)c .

Regularization of the Poisson kernel in the spirit of Bogdan, Kulczycki andKwasnicki (2008) leading to

KU(x , z) � KU(x , 0)

(∫U∩B(0,2r)

KU(y , z) dy + 1

Martin boundary of unbounded sets

1 Motivation

6 Minimal thinness

κ-fat sets

Let κ ∈ (0, 1/2]. An open set D is said to be κ-fat open at Q ∈ ∂D, ifthere exists R > 0 such that for each r ∈ (0,R) there exists a point Ar (Q)satisfying B(Ar (Q), κr) ⊂ D ∩ B(Q, r).

If D is κ-fat at each boundary point Q ∈ ∂D with the same R > 0, D iscalled κ-fat with characteristics (R, κ).

κ-fat sets

Let κ ∈ (0, 1/2]. An open set D is said to be κ-fat open at Q ∈ ∂D, ifthere exists R > 0 such that for each r ∈ (0,R) there exists a point Ar (Q)satisfying B(Ar (Q), κr) ⊂ D ∩ B(Q, r).If D is κ-fat at each boundary point Q ∈ ∂D with the same R > 0, D iscalled κ-fat with characteristics (R, κ).

κ-fat sets

Let κ ∈ (0, 1/2]. An open set D is said to be κ-fat open at Q ∈ ∂D, ifthere exists R > 0 such that for each r ∈ (0,R) there exists a point Ar (Q)satisfying B(Ar (Q), κr) ⊂ D ∩ B(Q, r).If D is κ-fat at each boundary point Q ∈ ∂D with the same R > 0, D iscalled κ-fat with characteristics (R, κ).

κ-fat sets at infinity

An open set D in Rd is κ-fat at infinity if there exists R > 0 such that forevery r ∈ [R,∞) there exists Ar ∈ Rd such that B(Ar , κr) ⊂ D ∩ B(0, r)c

and |Ar | < κ−1r . The pair (R, κ) will be called the characteristics of theκ-fat open set D at infinity.

All half-space-like open sets, allexterior open sets and all infinitecones are κ-fat at infinity.

Oscillation reduction

Lemma: Let D ⊂ Rd be an open set which is κ-fat at infinity withcharacteristics (R, κ). There exist C = C (φ, d) > 0 and ν = ν(d , φ) > 0such that for all r ≥ 1 and all non-negative functions u and v on Rd whichare regular harmonic in D ∩ B(0, r/2)c , vanish in Dc ∩ B(0, r/2)c andsatisfy u(Ar ) = v(Ar ), there exists the limit

g = lim|x |→∞, x∈D

and we have ∣∣∣∣u(x)

v(x)− g

∣∣∣∣ ≤ C

(|x |r

)−ν, x ∈ D ∩ B(0, r)c .

Martin kernel at infinity

Fix x0 ∈ D ∩ B(0,R)c and recall that

MD(x , y) =GD(x , y)

GD(x0, y), x , y ∈ D ∩ B(0,R)c .

For r > (2|x | ∨ R), both functions y 7→ GD(x , y) and y 7→ GD(x0, y) areregular harmonic in D ∩ B(0, r/2)c and vanish on Dc ∩ B(0, r/2)c .

Theorem: (Kim, Song, V 2012) For each x ∈ D there exists the limit

MD(x ,∞) := limy∈D, |y |→∞

MD(x , y) .

This implies that every infinite Martin boundary point can be mapped to{∞}. Since Martin kernels for different Martin boundary points aredifferent, the infinite part of the Martin boundary is exactly {∞}.

GD(x0, y), x , y ∈ D ∩ B(0,R)c .

MD(x ,∞) := limy∈D, |y |→∞

MD(x , y) .

GD(x0, y), x , y ∈ D ∩ B(0,R)c .

MD(x ,∞) := limy∈D, |y |→∞

MD(x , y) .

Harmonicity and minimality

Lemma: For each x ∈ D and ρ ∈ (0, 13δD(x)],

MD(x ,∞) = Ex [MD(XτB(x,ρ),∞)] .

Theorem: The function MD(·,∞) is minimal harmonic in D with respectto X .

By use of the lemma above, one shows that MD(·,∞) is harmonic (exittime from a relatively compact D1 ⊂ D is an increasing limit of exit timesform balls with radii comparable to the distance to the boundary).Minimality: h positive harmonic, h ≤ MD(·,∞). Write

h(x) =

∫∂fMD

MD(x ,w)µ(dw) + MD(x , ∂∞)µ({∂∞}) ,

and show that µ(∂fMD) = 0.

MD(x ,∞) = Ex [MD(XτB(x,ρ),∞)] .

Theorem: The function MD(·,∞) is minimal harmonic in D with respectto X .

By use of the lemma above, one shows that MD(·,∞) is harmonic (exittime from a relatively compact D1 ⊂ D is an increasing limit of exit timesform balls with radii comparable to the distance to the boundary).Minimality: h positive harmonic, h ≤ MD(·,∞). Write

h(x) =

∫∂fMD

MD(x ,w)µ(dw) + MD(x , ∂∞)µ({∂∞}) ,

MD(x ,∞) = Ex [MD(XτB(x,ρ),∞)] .

Theorem: The function MD(·,∞) is minimal harmonic in D with respectto X .By use of the lemma above, one shows that MD(·,∞) is harmonic (exittime from a relatively compact D1 ⊂ D is an increasing limit of exit timesform balls with radii comparable to the distance to the boundary).

Minimality: h positive harmonic, h ≤ MD(·,∞). Write

h(x) =

∫∂fMD

MD(x ,w)µ(dw) + MD(x , ∂∞)µ({∂∞}) ,

MD(x ,∞) = Ex [MD(XτB(x,ρ),∞)] .

Theorem: The function MD(·,∞) is minimal harmonic in D with respectto X .By use of the lemma above, one shows that MD(·,∞) is harmonic (exittime from a relatively compact D1 ⊂ D is an increasing limit of exit timesform balls with radii comparable to the distance to the boundary).Minimality: h positive harmonic, h ≤ MD(·,∞). Write

h(x) =

∫∂fMD

MD(x ,w)µ(dw) + MD(x , ∂∞)µ({∂∞}) ,

Finite part of Martin boundary: If D is κ-fat open set, then the finite partof the Martin boundary can be identified with the Euclidean boundary.

Define w(x) = w(x , xd) := V (xd) where V is the renewal functional of thelast component of X .

Corollary: The Martin boundary and the minimal Martin boundary of thehalf space H with respect to X can be identified with ∂H ∪ {∞} andMH(x ,∞) = w(x)/w(x0) for x ∈ H.

Minimal thinness

1 Motivation

6 Minimal thinness

Minimal thinness

Let A ⊂ D, TA = inf{t > 0 : XDt ∈ A} the hitting time to A,

PAf (x) := Ex [f (XDTA

)] the hitting operator to A.

Analytically, PAf = RAf – the balayage of f onto A.

Let z ∈ ∂mD. Then A ⊂ D is minimally thin at z if PAMD(·, z) 6= M(·, z).

Equivalently, there exists x ∈ D such that the MD(·, z)-conditionedprocess XD,z will not hit A with positive probability.

Minimal thinness

Let z ∈ ∂mD. Then A ⊂ D is minimally thin at z if PAMD(·, z) 6= M(·, z).Equivalently, there exists x ∈ D such that the MD(·, z)-conditionedprocess XD,z will not hit A with positive probability.

Minimal thinness

Minimal thinness in the halfspace

Theorem: Assume that X is SBM satisfying (H1) and z ∈ ∂H. If A ⊂ H isminimally thin in H at z , then∫

A∩B(z,1)|x − z |−d dx <∞ .

Conversely, suppose that A is a union of Whitney cubes. If A is notminimally thin at z , then the above integral is infinite.

Theorem: Assume that X is SBM satisfying (H1) and (H2). If A ⊂ H isminimally thin in H at ∞, then∫

A∩B(0,1)c|x − z |−d dx <∞ .

Conversely, suppose that A is a union of Whitney cubes. If A is notminimally thin at ∞, then the above integral is infinite.

Minimal thinness

Minimal thinness in the halfspace

Theorem: Assume that X is SBM satisfying (H1) and z ∈ ∂H. If A ⊂ H isminimally thin in H at z , then∫

A∩B(z,1)|x − z |−d dx <∞ .

Conversely, suppose that A is a union of Whitney cubes. If A is notminimally thin at z , then the above integral is infinite.

Theorem: Assume that X is SBM satisfying (H1) and (H2). If A ⊂ H isminimally thin in H at ∞, then∫

A∩B(0,1)c|x − z |−d dx <∞ .

Conversely, suppose that A is a union of Whitney cubes. If A is notminimally thin at ∞, then the above integral is infinite.

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