Intrinsic ultracontractivity for isotropic stable processes

Intrinsic ultracontractivity for isotropic stable processes

in unbounded domains

Mateusz Kwa´snicki

mateusz.kwasnicki@pwr.wroc.pl Institute of Mathematics and Computer Science

Wroc law University of Technology

Based on:

MK, Intrinsic ultracontractivity for stable semigroups on unbounded open sets, preprint;

K. Bogdan, T. Kulczycki, MK, Estimates and structure of α-harmonic functions, Probab. Theory Rel. Fields 140 (2008).

Basic definitions d = 1, 2, 3, ...

D ⊆ R^d — open (unbounded)

(X_t) — isotropic α-stable process in R^d, α ∈ (0, 2) E⁰e^{i ξ Xt} = e^{−t |ξ|α}

E^xf (X_t) = Z

p_t(y − x ) f (y ) dy p_t(y − x )  C min



t^αd, t

|y − x|^{d +α}



ν_x(y ) = A_{d ,−α}

|y − x|^{d +α} νx(E ) =

Z

E

νx(y ) dy

Definition

For p : R₊ → R₊, definehorn-shaped region (HSR):

D_p=n

(x₁, ˜x ) ∈ R^d : x₁ > 0 , |˜x | < p(x₁)o .

Definition

HSR Dp is nondegenerateif for some κ > 0,

p(u) > 0 =⇒ ∃x : (u, 0, ..., 0) ∈ B(x, κ p(u)) ⊆ D_p.

Killed process

τ_D — first exit time from D, τ_D = inf {t ≥ 0 : Xt ∈ D}/

(P_t^D) — semigroup of (X_t) killed upon exiting D, P_t^Df (x ) = E^x



f (Xt) ; t < τ_D



= Z

D

p^D_t (x , y ) f (y ) dy f ∈ L²(D) p^D_t positive, continuous and bounded on D × D νx(D^c) — killing intensity

Theorem (MK)

P_t^D are compact ⇐⇒ lim

|x|→∞E^xτ_D = 0 . Remark: This holds true also for the Brownian motion

Theorem (MK)

P_t^D are compact ⇐= lim

|x|→∞νx(D^c) = ∞ .

Example

For nondegenerate HSR D = D_p, P_t^D are compact ⇐⇒ lim

u→∞p(u) = 0 .

From now on we assume thatP_t^D are compact There exists a complete orthonormal sequence (ϕ_n):

P_t^Dϕn= e^−λntϕn 0 < λ1 < λ2≤ λ₃ ≤ ... → ∞

Theorem (MK)

ϕ1(x )  C (D) E^xτD

(1 + |x |)^{d +α}.

Remark: E^xτ_D islocal:

E^xτ_D  C (D) E^xτ_{D∩B(x ,1)}

From now on we assume thatP_t^D are compact There exists a complete orthonormal sequence (ϕ_n):

P_t^Dϕn= e^−λntϕn 0 < λ1 < λ2≤ λ₃ ≤ ... → ∞

Theorem (MK)

ϕ1(x )  C (D) E^xτD

(1 + |x |)^{d +α}.

Proposition (MK)

For a HSR D = D_p with sufficiently smooth p, E^xτ_D  C (D)

f (x₁)²− |˜x |²^α₂ .

Definition

(P_t^D) is said to be intrinsically ultracontractive (IU) if p^D_t (x , y ) ≤ C (D, t) ϕ₁(x ) ϕ₁(y ) .

(P_t^D) is IU if

D is bounded with smooth boundary (Chen,Song, 1997) D is bounded (Kulczycki, 1998)

If (P_t^D) is IU, then sup

x ,y ∈D

p_t^D(x , y )

e^−λ1tϕ₁(x ) ϕ₁(y )− 1    

 C (D) e^{−(λ2−λ1) t} (t ≥ 1)

Theorem (MK)

The following are equivalent:

(a) (P_t^D) is IU;

(b) p^D_t (x , y ) ≤ C (D, t)

(1 + |x |)^{d +α}(1 + |y |)^{d +α}.

(c) sup

x

P^x(τ_{D\B(0,r )}> t) ≤ C (D, t) (1 + r )^{d +α}.

Corollary

D₁ ⊆ D₂, (P_t^D2) is IU =⇒ (P_t^D1) is IU .

Corollary

(P_t^D) is IU =⇒ P_t^D are Hilbert-Schmidt operators .

Theorem (MK)

The following are equivalent:

(a) (P_t^D) is IU;

(b) p^D_t (x , y ) ≤ C (D, t)

(1 + |x |)^{d +α}(1 + |y |)^{d +α}; (c) sup

x

P^x(τ_{D\B(0,r )}> t) ≤ C (D, t) (1 + r )^{d +α}.

Remark: Condition (c) is local at infinity

Theorem (MK)

The following are equivalent:

(a) (P_t^D) is IU;

(b) p^D_t (x , y ) ≤ C (D, t)

(1 + |x |)^{d +α}(1 + |y |)^{d +α}; (c) sup

x

P^x(τ_{D\B(0,r )}> t) ≤ C (D, t) (1 + r )^{d +α}.

Theorem (MK)

(P_t^D) is IU ⇐= lim

|x|→∞

ν_x(D^c) log |x | = ∞ , (P_t^D) is IU =⇒ lim

|x|→∞dist(x , ∂D) (log |x |)^α1 = 0 .

Corollary

For a nondegenerate HSR D = D_p, (P_t^D) is IU ⇐⇒ lim

u→∞(log u)^α1p(u) = 0 .

Example

If p(u) ∼ u^−q (q > 0), then (P_t^D) is IU.

If p(u) ∼ (log u)^−q, then

(P_t^D) is IU ⇐⇒ q > 1 α.

Example If D =

∞

[

n=1

B(x_n, r_n), (disjoint balls, x_n= (u_n, 0, ..., 0)), then (P_t^D) is IU ⇐⇒ lim

n→∞r_n^αlog un= 0 .

Example

Let k_n= 2ⁿ, q_n = (n + 1)^−d−1 and D =

(

x ∈ R^d : |x | ∈

∞

[

n=0 kn−1

[

m=0



n +m + qn

kn

, n + m + 1 kn

)

;

then (P_t^D) is IU, but |D^c| < ∞.

Boundary Harnack inequality (K. Bogdan, T. Kulczycki, MK) If

f , g ≥ 0 and f = g = 0 on B(0, 2) \ D, f (x ) = E^xf (X (τ_D)), g (x ) = E^xg (X (τ_D));

then f (x )

g (x ) ≤ C f (y )

g (y ) for x , y ∈ B(0, 1) ∩ D.

B(0, 2) D

B(0, 1) 0 x y

Boundary Harnack inequality (K. Bogdan, T. Kulczycki, MK) If

f , g ≥ 0 and f = g = 0 on B(0, 2) \ D, f (x ) = E^xf (X (τ_D)), g (x ) = E^xg (X (τ_D));

then f (x )

g (x ) ≤ C f (y )

g (y ) for x , y ∈ B(0, 1) ∩ D.

Remarks:

C does not depend on D

No smoothness assumptions on ∂D History:

D Lipschitz, C = C (D) (Bogdan, 1997) D arbitrary, C = C (D) (Song,Wu, 1999)

Boundary Harnack inequality (K. Bogdan, T. Kulczycki, MK) If

f , g ≥ 0 and f = g = 0 on B(0, 2) \ D, f (x ) = E^xf (X (τ_D)), g (x ) = E^xg (X (τ_D));

then f (x )

g (x ) ≤ C f (y )

g (y ) for x , y ∈ B(0, 1) ∩ D.

Key lemma (K. Bogdan, T. Kulczycki, MK) Under the same hypotheses,

f (x )  C



E^xτ_D∩B(0,2)

 Z

D\B(0,1)

f (y ) ν0(y ) dy

!

for x ∈ B(0, 1) ∩ D.

Definitions

For x , y ∈ D, z ∈ D^c, w ∈ ∂D, GD(x , y ) =

Z ∞ 0

p_t^D(x , y ) dt —Green function PD(x , z) =

Z

D

GD(x , y ) νy(z) dy — ‘Poisson kernel’

M_D(x , w ) = lim

y →w

G_D(x , y )

G_D(x₀, y ) — Martin kernel

Theorem (K. Bogdan, T. Kulczycki, MK) (i) M_D(x , w ) exists for all w ∈ ∂D;

(i) M_D(·, w ) is α-harmonic in D ⇐⇒ w ∈ ∂_MD, where

∂MD =



w ∈ ∂D : Z

D

νy(w ) E^yτDdy = ∞

 .

∂MD =



w ∈ ∂D : Z

D

νw(y ) E^yτDdy = ∞

 .

Example

For p : (0, 1) → [0, ∞) nondecreasing, definethorn:

D_p=n

(x₁, ˜x ) ∈ R^d : 0 < x₁ < 1 , |˜x | < p(x₁)o . Then

0 ∈ ∂MDf ⇐⇒

Z 1 0

(p(u))^{d +α−1}

u^{d +α} du = ∞ .

For p(u) ∼ u| log u|^−q: 0 ∈ ∂_MD ⇐⇒ q ≤ 1

d + α − 1

Theorem (K. Bogdan, T. Kulczycki, MK) If f ≥ 0 is α-harmonic in D, then

f (x ) = Z

D^c

P_D(x , z) f (z) dz + Z

∂MD

M_D(x , w ) µ(dw ) . The representation is unique.

Remarks:

|∂_MD| = 0

PD(x , ·) is density of P^x-distribution of X (τD) on D^c\ ∂_MD (Ikeda, Watanabe: on D^c\ ∂D)

A different point of view

A function f ≥ 0 isα-harmonicin D with outer chargeλ if f (x ) =

Z

D^c

P_D(x , z) λ(dz) + Z

∂MD

M_D(x , w ) µ(dw )

= P_Dλ(x ) + M_Dµ(x ) . (λ — a measure on D^c\ ∂_MD)

Boundary Harnack inequality If

f = P_Dλ + M_Dµ, g = P_Dλ⁰+ M_Dµ⁰,

λ(B(0, 2)) = λ⁰(B(0, 2)) = µ(B(0, 2)) = µ⁰(B(0, 2)) = 0, then

f (x )

g (x ) ≤ C f (y )

g (y ) for x , y ∈ B(0, 1) ∩ D.