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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 2 / 42
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) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 3 / 42
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) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 3 / 42
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) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 3 / 42
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) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 3 / 42
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) =
∫Hc
P(x , z)h(z) dz .
x 7→ P(x , z) is not harmonic in H.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 4 / 42
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) =
∫Hc
P(x , z)h(z) dz .
x 7→ P(x , z) is not harmonic in H.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 4 / 42
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) =
∫Hc
P(x , z)h(z) dz .
x 7→ P(x , z) is not harmonic in H.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 4 / 42
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) =
∫Hc
P(x , z)h(z) dz .
x 7→ P(x , z) is not harmonic in H.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 4 / 42
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) =
∫Hc
P(x , z)h(z) dz .
x 7→ P(x , z) is not harmonic in H.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 4 / 42
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 ∪ {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 5 / 42
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 ∪ {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 5 / 42
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 ∪ {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 5 / 42
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 ∪ {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 5 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 6 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 6 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 6 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 6 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 7 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 7 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 7 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 8 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 8 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 8 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 8 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 8 / 42
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)
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 9 / 42
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)
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 9 / 42
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)
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 9 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 10 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 10 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 10 / 42
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 10 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 11 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 11 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 11 / 42
Description of the class of processes - subordinate BM
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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 12 / 42
Description of the class of processes - subordinate BM
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
a3λδ4φ(t) ≤ φ(λt) ≤ a4λ
δ3φ(t), λ ≤ 1, t ≤ 1 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 13 / 42
Description of the class of processes - subordinate BM
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
a3λδ4φ(t) ≤ φ(λt) ≤ a4λ
δ3φ(t), λ ≤ 1, t ≤ 1 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 13 / 42
Description of the class of processes - subordinate BM
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
a3λδ4φ(t) ≤ φ(λt) ≤ a4λ
δ3φ(t), λ ≤ 1, t ≤ 1 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 13 / 42
Description of the class of processes - subordinate BM
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
a3λδ4φ(t) ≤ φ(λt) ≤ a4λ
δ3φ(t), λ ≤ 1, t ≤ 1 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 13 / 42
Description of the class of processes - subordinate BM
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 14 / 42
Description of the class of processes - subordinate BM
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 14 / 42
Description of the class of processes - subordinate BM
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 14 / 42
Description of the class of processes - subordinate BM
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,∞) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 15 / 42
Description of the class of processes - subordinate BM
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,∞) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 15 / 42
Description of the class of processes - subordinate BM
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 16 / 42
Description of the class of processes - subordinate BM
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 16 / 42
Description of the class of processes - subordinate BM
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 16 / 42
Description of the class of processes - subordinate BM
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 16 / 42
Description of the class of processes - subordinate BM
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 17 / 42
Description of the class of processes - subordinate BM
Green function and Levy measure
Theorem: Assume (H1) and (H2).(a) Then
J(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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 17 / 42
Description of the class of processes - subordinate BM
Green function and Levy measure
Theorem: Assume (H1) and (H2).(a) Then
J(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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 17 / 42
Description of the class of processes - subordinate BM
Green function and Levy measure
Theorem: Assume (H1) and (H2).(a) Then
J(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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 17 / 42
Boundary Harnack principle
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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 18 / 42
Boundary Harnack principle
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
u(x)
v(x)≤ c
u(y)
v(y)for all x , y ∈ D ∩ B(z0, r/2) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 19 / 42
Boundary Harnack principle
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
u(x)
v(x)≤ c
u(y)
v(y)for all x , y ∈ D ∩ B(z0, r/2) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 19 / 42
Boundary Harnack principle
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 20 / 42
Boundary Harnack principle
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)).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 21 / 42
Boundary Harnack principle
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)).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 21 / 42
Boundary Harnack principle
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)).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 21 / 42
Boundary Harnack principle
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) .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 22 / 42
Boundary Harnack principle
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
u(x)
(φ(δD(x)−2))−1/2≤ c
u(y)
(φ(δD(y)−2))−1/2for all x , y ∈ D ∩ B(Q,
r
2).
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 23 / 42
Boundary Harnack principle
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
u(x)
(φ(δD(x)−2))−1/2≤ c
u(y)
(φ(δD(y)−2))−1/2for all x , y ∈ D ∩ B(Q,
r
2).
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 23 / 42
Boundary Harnack principle at infinity
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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 24 / 42
Boundary Harnack principle at infinity
Boundary Harnack principle at infinity
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
c,
KU(x , z) =
∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U
c.
If u regular harmonic in U, then u(x) =∫U
c KU(x , z)u(z) dz .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 25 / 42
Boundary Harnack principle at infinity
Boundary Harnack principle at infinity
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
c,
KU(x , z) =
∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U
c.
If u regular harmonic in U, then u(x) =∫U
c KU(x , z)u(z) dz .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 25 / 42
Boundary Harnack principle at infinity
Boundary Harnack principle at infinity
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
c,
KU(x , z) =
∫UGU(x , y)j(|y − z |) dy , x ∈ U, z ∈ U
c.
If u regular harmonic in U, then u(x) =∫U
c KU(x , z)u(z) dz .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 25 / 42
Boundary Harnack principle at infinity
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
1
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 26 / 42
Boundary Harnack principle at infinity
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
1
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 26 / 42
Boundary Harnack principle at infinity
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
1
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 26 / 42
Boundary Harnack principle at infinity
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
1
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 26 / 42
Boundary Harnack principle at infinity
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
1
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 .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 26 / 42
Boundary Harnack principle at infinity
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
u(y)
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) =∞.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 27 / 42
Boundary Harnack principle at infinity
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
u(y)
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) =∞.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 27 / 42
Boundary Harnack principle at infinity
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
u(y)
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) =∞.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 27 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 28 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 28 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 28 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 28 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 29 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 29 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 29 / 42
Boundary Harnack principle at infinity
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 29 / 42
Boundary Harnack principle at infinity
Ingredients of the proof – continuation
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
Px
(XτU ∈ B(0, r)
)≤ CrdKU(x , 0) , x ∈ U ∩ B(0, ar)c .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 30 / 42
Boundary Harnack principle at infinity
Ingredients of the proof – continuation
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
Px
(XτU ∈ B(0, r)
)≤ CrdKU(x , 0) , x ∈ U ∩ B(0, ar)c .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 30 / 42
Boundary Harnack principle at infinity
Ingredients of the proof – continuation
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
).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 31 / 42
Martin boundary of unbounded sets
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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 32 / 42
Martin boundary of unbounded sets
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 33 / 42
Martin boundary of unbounded sets
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 33 / 42
Martin boundary of unbounded sets
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).
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 33 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 34 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 34 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 34 / 42
Martin boundary of unbounded sets
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
u(x)
v(x),
and we have ∣∣∣∣u(x)
v(x)− g
∣∣∣∣ ≤ C
(|x |r
)−ν, x ∈ D ∩ B(0, r)c .
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 35 / 42
Martin boundary of unbounded sets
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 36 / 42
Martin boundary of unbounded sets
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 {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 37 / 42
Martin boundary of unbounded sets
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 {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 37 / 42
Martin boundary of unbounded sets
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 {∞}.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 37 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 38 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 38 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 38 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 38 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 39 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 39 / 42
Martin boundary of unbounded sets
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 39 / 42
Minimal thinness
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
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 40 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 41 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 41 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 41 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 41 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 42 / 42
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.
Zoran Vondracek (University of Zagreb) Martin boundary for SBM Wroc law, 15-19.7.2013. 42 / 42