bcc.impan.plbcc.impan.pl/16NOPDE/uploads/talks/Michal-Ryznar-talk.pdf · Heat kernels of unimodal L´evy processes with weak scaling conditions Definitions A (Borel) measure on Rdis

Heat kernels of unimodal Levy processes with weak scaling conditions

Heat kernels of unimodal Levy processes with weakscaling conditions

Michał Ryznar

Wrocław University of Science and Technology, Poland

The 3rd Conference on Nonlocal Operators and Partial DifferentialEquations

Będlewo, June 27, 2016.

The presentation is based mostly on a joint project with K. Bogdan, T.Grzywny.


Definitions

A (Borel) measure on Rd is called isotropic unimodal, if on Rd \ {0} it isabsolutely continuous with respect to the Lebesgue measure, and has afinite radial nonincreasing density function.


Definitions


A Levy process X = (Xt, t > 0) is a stochastic process with values in Rd,

which has stationary and idependent increments and cad-lag paths:

Xt1 −Xt0 , . . . , Xtn −Xtn−1 are independent for any 0 6 t0 6 . . . 6 tn.

Xt+h −Xt has the same distribution as Xh, t, h > 0


Definitions






X = (Xt, t > 0) is called unimodal isotropic if all its one-dimensionaldistributions pt(dx) are isotropic unimodal.


Definitions






X = (Xt, t > 0) is called unimodal isotropic if all its one-dimensionaldistributions pt(dx) are isotropic unimodal.

We call the transition density pt(x, y) = pt(y − x) = p(t, x, y) the heatkernel.Ttf(x) = f(x+ y)pt(y)dy form a semigroup on various function spaces.


Characteristic function

E ei〈ξ,Xt〉 =

∫

Rd

ei〈ξ,x〉pt(dx) = e−tψ(ξ), ξ ∈ R

d.

Isotropic unimodal Levy processes are characterized by Levy-Kchintchineexponents of the form ([Watanabe 1983])

ψ(ξ) = σ2|ξ|2 +∫

Rd

(1− cos 〈ξ, x〉) ν(|x|)dx, (1)

with nonincreasing ν and σ > 0. N(dx) = ν(|x|)dx is called the Levymeasure of the process.

N(dx) = limt→0

pt(dx)

t


Characteristic function

E ei〈ξ,Xt〉 =

∫

Rd

ei〈ξ,x〉pt(dx) = e−tψ(ξ), ξ ∈ R

d.

Isotropic unimodal Levy processes are characterized by Levy-Kchintchineexponents of the form ([Watanabe 1983])

ψ(ξ) = σ2|ξ|2 +∫

Rd

(1− cos 〈ξ, x〉) ν(|x|)dx, (1)

with nonincreasing ν and σ > 0. N(dx) = ν(|x|)dx is called the Levymeasure of the process.

N(dx) = limt→0

pt(dx)

t

If e−tψ(ξ) ∈ L1 then

pt(ξ) =1

(2π)d

∫

Rd

e−i〈ξ,x〉e−tψ(x)dx 6 pt(0).


Examples

Isotropic stable processes: ν(x) = c|x|−d−α

Truncated α-stable processes: ν(r) = c r−d−α1(0,1)(r).


Examples

Isotropic stable processes: ν(x) = c|x|−d−α

Truncated α-stable processes: ν(r) = c r−d−α1(0,1)(r).

Subordinate Brownian motions:Let Tt be a subordinator (positive and increasing one-dimensional Levyprocess) without drift; Ee−λTt = e−tϕ(λ), where (Bernstein function)

ϕ(λ) =

∫ ∞

0

(1− e−λs)µ(ds).

Let B(t) be an independent Brownian motion with ψ(ξ) = |ξ|2. ThenB(Tt) has an absolutely continuous Levy measure N(dx) = ν(x)dx, where

ν(x) =

∫ ∞

0

gs(x)µ(ds),

and gs(x) = (4π s)−d/2e−

|x|2

4s is the Gaussian kernel.


Generators

Af(x) = σ2∆+ p.v.∫

Rd

(f(x+ y)− f(x)) ν(|x|)dx, f ∈ C2(Rd)


Generators

Af(x) = σ2∆+ p.v.∫

Rd

(f(x+ y)− f(x)) ν(|x|)dx, f ∈ C2(Rd)

Fractional Laplacian: α-stable isoperimetric process

∆α/2f(x) = p.v. cd,α

∫

Rd

(f(x+ y)− f(x)) 1

|x|d+α dx


Generators

Af(x) = σ2∆+ p.v.∫

Rd

(f(x+ y)− f(x)) ν(|x|)dx, f ∈ C2(Rd)

Fractional Laplacian: α-stable isoperimetric process

∆α/2f(x) = p.v. cd,α

∫

Rd

(f(x+ y)− f(x)) 1

|x|d+α dx

A = I − (I −∆)α/2: relativistic α-stable isoperimetric process

Af(x) = p.v. cd,α

∫

Rd

(f(x+ y)− f(x)) K(d+α)/2(|x|)|x|(d+α)/2 dx


Topics

Heat kernels estimates and weak scaling.


Topics


Exit times. Estimates of the mean exit time and surviaval probability forsmooth sets.


Topics


Exit times. Estimates of the mean exit time and surviaval probability forsmooth sets.

Heat kernels for the killed process on exiting a set. Estimates.


Our goal:Estimates for pt(x) in terms of the symbol ψ of the process (or ψ

∗)



∗)

Example: Cauchy, isotropic 1-stable process, ψ(x) = |x|.

pt(x) = Cdt

(t2 + |x|2)(d+1)/2 for x ∈ Rd.



∗)


pt(x) = Cdt

(t2 + |x|2)(d+1)/2 for x ∈ Rd.

Below f ≈ g means that there is a constant c > 0 such thatc−1g 6 f(x) 6 cg(x). This c is called the comparability constant.



∗)


pt(x) = Cdt

(t2 + |x|2)(d+1)/2 for x ∈ Rd.


Example: isotropic α-stable process, ψ(x) = |x|α, 0 < α < 2.

pt(x) ≈ min{t−d/α,t

|x|d+α } for x ∈ Rd.



∗)


pt(x) = Cdt

(t2 + |x|2)(d+1)/2 for x ∈ Rd.





t−d/α =(

ψ−1(1/t))d,

t

|x|d+α =tψ(1/|x|)|x|d = ctν(x)



∗)


pt(x) = Cdt

(t2 + |x|2)(d+1)/2 for x ∈ Rd.





t−d/α =(

ψ−1(1/t))d,

t

|x|d+α =tψ(1/|x|)|x|d = ctν(x)

pt(x) ≈ min{

[

ψ−1(1/t)]d,tψ(1/|x|)|x|d

}


Recall pt(x) is radially decreasing. Easy observation:

pt(2r)(|B2r| − |Br|) 6 P(r 6 |Xt| 6 2r) 6 pt(r)(|B2r| − |Br|)




pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r)




pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r)

pt(r) > c(d, κ)r−dP(r 6 |Xt| 6 κr), κ > 1




pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r)


There are many results about heat kernel estimates-usually assumptions are interms of Levy measures (Chen, Kaleta, Kumagai, Kim, Knopova, Mimica,Schilling, Song, Sztonyk and many others).




pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r)


There are many results about heat kernel estimates-usually assumptions are interms of Levy measures (Chen, Kaleta, Kumagai, Kim, Knopova, Mimica,Schilling, Song, Sztonyk and many others).

In our approach we derive estimates using the symbol ψ (ψ∗ ) and theassumptions will be formulated in terms of ψ.Let ψ∗(u) := sup|x|6u ψ(x), for u > 0.

Proposition

ψ(x) 6 ψ∗(|x|) 6 π2 ψ(x) for x ∈ Rd. (2)


Proposition

For r > 0 we have

P(|Xt| > r) 62e

e− 1(2d+ 1)(

1− e−tψ∗(1/r))

.


Proposition

For r > 0 we have

P(|Xt| > r) 62e

e− 1(2d+ 1)(

1− e−tψ∗(1/r))

.

Recall

pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r) 6 Cdtψ∗(2/r)r−d 6 10Cdtψ

∗(1/r)r−d


Proposition

For r > 0 we have

P(|Xt| > r) 62e

e− 1(2d+ 1)(

1− e−tψ∗(1/r))

.

Recall


∗(1/r)r−d

Corollary

There is C = C(d) such that

pt(x) 6 Ctψ∗(1/|x|)/|x|d for x ∈ Rd \ {0}.


Proposition

For r > 0 we have

P(|Xt| > r) 62e

e− 1(2d+ 1)(

1− e−tψ∗(1/r))

.

Recall


∗(1/r)r−d

Corollary


pt(x) 6 Ctψ∗(1/|x|)/|x|d for x ∈ Rd \ {0}.


pt(x) 6 Ct/|x|d+α for x ∈ Rd \ {0}.


The method of proof of the estimates of P(|Xt| > r).

For reasons which shall become clear in the proof of the next result, we choosethe following parametrization of the tails:

ft(ρ) = P(|Xt| >√ρ) = P(|Xt|2 > ρ), ρ > 0, t > 0. (3)

Consider the Laplace transform of ft:

Lft(λ) =∫ ∞

0

e−λρft(ρ)dρ, λ > 0.


The method of proof of the estimates of P(|Xt| > r).

For reasons which shall become clear in the proof of the next result, we choosethe following parametrization of the tails:

ft(ρ) = P(|Xt| >√ρ) = P(|Xt|2 > ρ), ρ > 0, t > 0. (3)

Consider the Laplace transform of ft:

Lft(λ) =∫ ∞

0

e−λρft(ρ)dρ, λ > 0.

Lemma

There is a constant C1 = C1(d) such that

C−111

λ

(

1− e−tψ∗(√λ))

6 Lft(λ) 6 C11

λ

(

1− e−tψ∗(√λ))

, λ > 0.


Proof.

By Fubini’s theorem, for integrable functions h, k:∫

Rd

h(x)k(x)dx =

∫

Rd

h(x)k(x)dx


Proof.


Rd

h(x)k(x)dx =

∫

Rd

h(x)k(x)dx

Ee−λ|Xt|2

=

∫

Rd

e−λ|x|2

pt(x)dx = (4π)−d/2∫

Rd

e−tψ(x√λ)e−|x|

2/4dx.


Proof.


Rd

h(x)k(x)dx =

∫

Rd

h(x)k(x)dx

Ee−λ|Xt|2

=

∫

Rd

e−λ|x|2

pt(x)dx = (4π)−d/2∫

Rd

e−tψ(x√λ)e−|x|

2/4dx.

λLft(λ) = E(1− e−λ|Xt|2

)

= 1−∫

Rd

e−λ|x|2

pt(x)dx = (4π)−d/2∫

Rd

(

1− e−tψ(x√λ))

e−|x|2/4dx.


By the result of Hoh, 1998’,

ψ(su) 6 ψ∗(su) 6 2(s2 + 1)ψ∗(u), s, u > 0. (4)



ψ(su) 6 ψ∗(su) 6 2(s2 + 1)ψ∗(u), s, u > 0. (4)

Note that1− e−bt 6 b(1− e−t), t > 0, b > 1, (5)



ψ(su) 6 ψ∗(su) 6 2(s2 + 1)ψ∗(u), s, u > 0. (4)

Note that1− e−bt 6 b(1− e−t), t > 0, b > 1, (5)

1− e−tψ(x√λ)

6 1− e−2t(|x|2+1)ψ∗(

√λ)

6 2(|x|2 + 1)(

1− e−tψ∗(√λ))

λLft(λ) = (4π)−d/2∫

Rd

(

1− e−tψ(x√λ))

e−|x|2/4dx

6

(

1− e−tψ∗(√λ))

∫

Rd

2(|x|2 + 1)(4π)−d/2e−|x|2/4dx


On the other hand, if |x| > 1, then

ψ(

x√λ)

> ψ∗(

|x|√λ)

/π2 > ψ∗(√

λ)

/π2

by (2). Thus,

λLft(λ) = (4π)−d/2∫

Rd

(

1− e−tψ(x√λ))

e−|x|2/4dx

> (4π)−d/2∫

Bc1

e−|x|2/4dx

(

1− e−tψ∗(√λ)/π2)

>

(

1− e−tψ∗(√λ))

π−2(4π)−d/2∫

Bc1

e−|x|2/4dx,

where we use (5).


Upper bound for ft(r) = P(|Xt| >√r) ??



The upper bound for Lft(λ) immediately provides the upper bound for ft(r)since ft(r) is decreasing.




ft(1/λ)(1− e−1) 6 λ

∫ 1/λ

0

e−λrft(r)dr 6 λLft(λ) 6 Cd

(

1− e−tψ∗(√λ))




ft(1/λ)(1− e−1) 6 λ

∫ 1/λ

0


(

1− e−tψ∗(√λ))

Lower bound??




ft(1/λ)(1− e−1) 6 λ

∫ 1/λ

0


(

1− e−tψ∗(√λ))

Lower bound??

The lower bound for Lft(λ) + some type of weak scaling condition for thesymbol (WUSC) give the lower bound for ft(r). This argument is still simpleyet requires more effort than the upper bound.




ft(1/λ)(1− e−1) 6 λ

∫ 1/λ

0


(

1− e−tψ∗(√λ))

Lower bound??

The lower bound for Lft(λ) + some type of weak scaling condition for thesymbol (WUSC) give the lower bound for ft(r). This argument is still simpleyet requires more effort than the upper bound.

WUSC implies some type of scaling property for Lft(λ).


Improvement of the general upper bound

Let

K(r) =1

r2

∫

|x|<r|x|2ν(dx) r > 0.

Proposition

For r > 0 we haveP(r/2 6 |Xt| 6 r) 6 CdtK(r).

pt(r) 6 C(d)r−dP(r/2 6 |Xt| 6 r) 6 CdtK(r)

rd

We always haveK(r) 6 cdψ

∗(1/r)

Example: ψ(x) = log(1 + |x|2). Then ν(x) ≈ 1|x|d , |x| < 1.

Hence K(r) ≈ 1, r < 1 while limr→0 ψ∗(1/r) =∞


Consequence of scaling

Scaling conditions

Let φ : [0,∞)→ [0,∞).

φ ∈WLSC(α, θ, c) if α > 0, θ > 0 and c ∈ (0, 1] exist, such that

φ(λθ) > cλαφ(θ) for λ > 1, θ > θ,



Scaling conditions

Let φ : [0,∞)→ [0,∞).



equivalently

infy>x>θ

φ(y)

φ(x)

(

x

y

)α

> 0,



Scaling conditions

Let φ : [0,∞)→ [0,∞).



equivalently

infy>x>θ

φ(y)

φ(x)

(

x

y

)α

> 0,

equivalently

φ(x)x−α

is comparable to a non-decreasing function on [θ,∞).



Scaling conditions

Let φ : [0,∞)→ [0,∞).



equivalently

infy>x>θ

φ(y)

φ(x)

(

x

y

)α

> 0,

equivalently

φ(x)x−α

is comparable to a non-decreasing function on [θ,∞).The scaling condition WLSC is global if we can set θ = 0



φ ∈WUSC(α, θ, C) if α < 2, θ > 0 and C > 1 exist, such that

φ(λθ) 6 Cλαφ(θ) for λ > 1, θ > θ,





equivalently

supy>x>θ

φ(y)

φ(x)

(

x

y

)α

<∞,





equivalently

supy>x>θ

φ(y)

φ(x)

(

x

y

)α

<∞,

equivalently

φ(x)x−α

is comparable to a non-decreasing function on [θ,∞).





equivalently

supy>x>θ

φ(y)

φ(x)

(

x

y

)α

<∞,

equivalently

φ(x)x−α

is comparable to a non-decreasing function on [θ,∞).The scaling condition WUSC is global if we can set θ = 0



Examples:ψ(θ) = θα + θβ , 0 < α < β < 2 has global WLSC(α), WUSC(β):

ψ(θ)θ−α = 1 + θβ−α is increasing

This symbol corresponds to the sum of two independent isotropic stableprocesse with indices α, β, respectively






ψ(θ) = (θ2 + 1)α/2 − 1, 0 < α < 2 has global WLSC(α), but only localWUSC(α).






ψ(θ) = (θ2 + 1)α/2 − 1, 0 < α < 2 has global WLSC(α), but only localWUSC(α).

ψ(θ)θ−α = (θ−2 + 1)α/2 − θ−α is increasing

and

ψ(θ)θ−α ≈ 1, θ > 1



Lemma

C = C(d) exists such that if ψ∈ WUSC(α, θ, C) anda = [(2− α)C]

2

2−αCα−22 , then

P(|Xt| > r) > a(

1− e−tψ∗(1/r))

, 0 < r <√a/

θ.



Lemma

C = C(d) exists such that if ψ∈ WUSC(α, θ, C) anda = [(2− α)C]

2

2−αCα−22 , then

P(|Xt| > r) > a(

1− e−tψ∗(1/r))

, 0 < r <√a/

θ.

The Lemma above follows from the following result due to Zhale 2009’

Lemma

Let g > 0 be nonincreasing, β > 0 and Lg ∈WUSC(−β, θ, C). There isb = b(β,C) ∈ (0, 1) such that

g(r) >b

2ebr−1Lg(r−1), 0 < r < b/θ.

We apply the lemma to ft(r) = P(|Xt| >√r) and then use the estimates for

Lf . First using the lower and the upper bounds for the Laplace transformtogether with WUSC(α, θ, C) condition for ψ we prove that Lf hasWUSC(α/2− 1) property.



Theorem (Bogdan, Grzywny and Ryznar, 2014’, JFA)

If ψ ∈WLSC(α, θ, c), then there is C∗ = C∗(d, α, c) such that

pt(x) 6 C∗min

{

[

ψ−(1/t)]d,tψ∗(1/|x|)|x|d

}

if tψ∗(θ) < π−2.

If ψ ∈WLSC(α, θ, c)∩WUSC(α, θ, C), then c∗ = c∗(d, α, c, α, C),r0 = r0(d, α, c, α, C) exist with

pt(x) > c∗min

{

[

ψ− (1/t)]d,tψ∗(1/|x|)|x|d

}

, tψ∗(θ/r0) < 1 and |x| <r0θ.

Here ψ−(r) = inf{s : ψ∗(s) > r} is the generalized inverse of thenondecreasing function ψ∗.




If ψ ∈WLSC(α, θ, c), then there is C∗ = C∗(d, α, c) such that

pt(x) 6 C∗min

{

[

ψ−(1/t)]d,tψ∗(1/|x|)|x|d

}

if tψ∗(θ) < π−2.

If ψ ∈WLSC(α, θ, c)∩WUSC(α, θ, C), then c∗ = c∗(d, α, c, α, C),r0 = r0(d, α, c, α, C) exist with

pt(x) > c∗min

{

[

ψ− (1/t)]d,tψ∗(1/|x|)|x|d

}

, tψ∗(θ/r0) < 1 and |x| <r0θ.

Here ψ−(r) = inf{s : ψ∗(s) > r} is the generalized inverse of thenondecreasing function ψ∗.If WLSC holds, then (at least for small t):

pt(0) =1

(2π)d

∫

e−tψ(x)dx ≈[

ψ− (1/t)]d.



Corollary


ν(x) 6 Cψ∗(1/|x|)|x|−d.

If ψ ∈WLSC(α, θ, c)∩WUSC(α, θ, C) and |x| < r0/θ, then

ν(x) > c∗ψ∗(1/|x|)|x|−d.



Corollary


ν(x) 6 Cψ∗(1/|x|)|x|−d.

If ψ ∈WLSC(α, θ, c)∩WUSC(α, θ, C) and |x| < r0/θ, then

ν(x) > c∗ψ∗(1/|x|)|x|−d.

Corollary

If ψ ∈ WLSC(α,0,c)∩WUSC(α,0,C), then

pt(x) ≈ min{

[

ψ−(1/t)]d,tψ∗(1/|x|)|x|d

}

, t > 0, x ∈ Rd.

Example. Xt = X(α)t +X

(β)t , 0 < β < α < 2. X

(α)t , X

(β)t idependent

isotropic stable and ψ(ξ) = |ξ|α + |ξ|β . (Chen, Kumgai 2008’)

pt(x) ≈ min{

(1/t)d/α ∧ (1/t)d/β , t

|x|d+α +t

|x|d+β

}

, t > 0, x ∈ Rd.




Let Xt be an isotropic unimodal Levy process in Rd with transition density p,

Levy-Khintchine exponent ψ and Levy measure density ν. The following areequivalent:

(i) WLSC and WUSC [global WLSC and WUSC ] hold for ψ.

(ii) For some r0 ∈ (0,∞) [r0 =∞] and a constant c,

pt(x) > ctψ∗(|x|−1)|x|d , 0 < |x| < r0, 0 < tψ∗(|x|−1) < 1.

(iii) For some r0 ∈ (0,∞) [r0 =∞] and a constant c,

ν(x) > cψ∗(|x|−1)|x|d , 0 < |x| < r0.



Theorem (Kulczycki and Ryznar, 2016’, TAMS)

Let X be an isotropic Levy process in Rd such that ν(r) is nonincreasing,absolutely continuous such that −ν′(r)/r is nonincreasing. Then its transitiondensity pt(x) = pt(|x|) satisfies

∣

∣

∣

d

drpt(r)∣

∣

∣6 c(d)

1 ∧ tψ∗(1/r)rd+1

, t, r > 0.

If additionally ψ satisfies WLSC(α, θ0, c), then

∣

∣

∣

d

drpt(r)

∣

∣

∣6 c(d, α)

r

c(d+2)/α+1

(

[ψ−(1/t)]d+2 ∧ tψ∗(1/r)

rd+2

)

, tψ∗(θ0) 6 1/π2.

If additionally ψ satisfies WLSC(α, θ0, c) and WUSC(α, θ0, C), then we have

∣

∣

∣

d

drpt(r)∣

∣

∣> c∗r

(

[ψ−(1/t)]d+2 ∧ tψ∗(1/r)

rd+2

)

, tψ∗(θ0/r0) 6 1, r < r0/θ0,

where c∗ = c∗(d, α, α, c, C), r0 = r0(d, α, α, c, C). Note that if the scalingconditions are global, that is θ0 = 0, then the last two estimates hold for allt, r > 0.



Let X be an isotropic Levy process in Rd such that ν(r) is nonincreasing,absolutely continuous such that −ν′(r)/r is nonincreasing.




Then there exists a Levy process X(d+2)t in Rd+2 with the characteristic

exponent ψ(d+2)(ξ) = ψ(|ξ|), ξ ∈ Rd+2 and the radial, radially nonincreasing

transition density p(d+2)t (x) = p

(d+2)t (|x|) satisfying

p(d+2)t (r) =

−12πr

d

drpt(r), r > 0.




Then there exists a Levy process X(d+2)t in Rd+2 with the characteristic

exponent ψ(d+2)(ξ) = ψ(|ξ|), ξ ∈ Rd+2 and the radial, radially nonincreasing

transition density p(d+2)t (x) = p

(d+2)t (|x|) satisfying

p(d+2)t (r) =

−12πr

d

drpt(r), r > 0.

The estimates of the gradient of pt enabled analysis of gradients of harmonicfunctions with respect to the process X. In a joint paper with T. Kulczycki(2015’) we showed that the gradient of a positive harmonic function exists andwe derived its estimates from above.



Symbols which do not have weak scaling properties

Slowly varying: e.g.

ψ(x) = log(1 + |x|α)β , 0 < α 6 2, 0 < β 6 1

Various asymptotics and estimates were recently obtained by T. Grzywny,B. Trojan and M.R (2016) under some additional assumptions on thesymbol or its derivative. One of the main difficulties here is the fact thatvery often p(t, 0) =∞.





ψ(x) = log(1 + |x|α)β , 0 < α 6 2, 0 < β 6 1

Various asymptotics and estimates were recently obtained by T. Grzywny,B. Trojan and M.R (2016) under some additional assumptions on thesymbol or its derivative. One of the main difficulties here is the fact thatvery often p(t, 0) =∞.2-regularly varying: e.g.

ψ(x) =|x|2

log(1 + |x|α) , 0 < α 6 2

This case seems to be even more difficult than the one above. There areworks by A. Mimica (2015), A. Mimica ad P. Kim (2016) where somecases were treated.





ψ(x) = log(1 + |x|α)β , 0 < α 6 2, 0 < β 6 1

Various asymptotics and estimates were recently obtained by T. Grzywny,B. Trojan and M.R (2016) under some additional assumptions on thesymbol or its derivative. One of the main difficulties here is the fact thatvery often p(t, 0) =∞.2-regularly varying: e.g.

ψ(x) =|x|2

log(1 + |x|α) , 0 < α 6 2

This case seems to be even more difficult than the one above. There areworks by A. Mimica (2015), A. Mimica ad P. Kim (2016) where somecases were treated.

Both cases are far from being explored to that extent as regularly varyingsymbols with indices strictly between 0 and 2.


Exit time

Survival probability. Dirichlet heat kernel.

The first exit time from open D ⊂ Rd:

τD = inf{t > 0 : Xt /∈ D}


Exit time



τD = inf{t > 0 : Xt /∈ D}

Transition density of the process killed on leaving D (Dirichlet heat kernel):

pD(t, x, y) = p(t, x, y)− Ex[τD < t; p(t− τD, XτD , y)],

where p(t, x, y) = pt(y − x).


Exit time



τD = inf{t > 0 : Xt /∈ D}




We have Px(Xt ∈ B, τD > t) =∫

BpD(t, x, y)dy.


Exit time



τD = inf{t > 0 : Xt /∈ D}





BpD(t, x, y)dy.

In particular, pD yields the probability of surviving time t:

Px(τD > t) =

∫

pD(t, x, y)dy.


Exit time



τD = inf{t > 0 : Xt /∈ D}





BpD(t, x, y)dy.

In particular, pD yields the probability of surviving time t:

Px(τD > t) =

∫

pD(t, x, y)dy.

Green function: GD(x, y) =∫∞0pD(t, x, y)dt.


Exit time

Known results about estimates of Dirichlet HeatKernels

Zhang (2002) - smooth domains, BM

Varopulos (2003) - Lipschitz domains, BM

Chen, Kim, Song (2010), bounded smooth domains, isotropic stableprocesses

Bogdan, Grzywny, Ryznar (2010), Lipschitz domains, isotropic stableprocesses

Chen, Kim, Song (2012-2014) - smooth domains, subclasses of SBM andLevy processes comparable with SBM

Kim, Song, Vondracek (2014) - halfspace, subclasses of SBM


Exit time

Known results about estimates of Dirichlet HeatKernels

Zhang (2002) - smooth domains, BM

Varopulos (2003) - Lipschitz domains, BM

Chen, Kim, Song (2010), bounded smooth domains, isotropic stableprocesses

Bogdan, Grzywny, Ryznar (2010), Lipschitz domains, isotropic stableprocesses

Chen, Kim, Song (2012-2014) - smooth domains, subclasses of SBM andLevy processes comparable with SBM

Kim, Song, Vondracek (2014) - halfspace, subclasses of SBM



Exit time

Upper bound

Lemma

pD(t, x, y) 6 pt/3(0)Px(τD > t/3)Py(τD > t/3).


Exit time

Upper bound

Lemma

pD(t, x, y) 6 pt/3(0)Px(τD > t/3)Py(τD > t/3).

Proof. By the semigroup property

pD(t, x, y) =

∫ ∫

pD(t/3, x, z)pD(t/3, z, w)pD(t/3, w, y)dzdw

6 pt/3(0)

∫

pD(t/3, x, z)dz

∫

pD(t/3, w, y)dw.

6 pt/3(0)Px(τD > t/3)Px(τD > t/3).

Under some conditions (eg. WLSC) for small t and x close to y we havept/3(0) ≈ pt(x− y).Also Px(τD > t/3) ≈ P

x(τD > t), t small. The lemma is sufficient (almost) forthe sharp upper bound for points which distance is small relatively to time(V 2(|x− y|) 6 t).


Exit time

Lemma

Consider open D1, D3 ⊂ D such that dist(D1, D3) > 0. LetD2 = D \ (D1 ∪D3). If x ∈ D1, y ∈ D3 and t > 0, then

pD(t, x, y) 6 Px(XτD1 ∈ D2) sup

s<t, z∈D2p(s, z, y)

+ (t ∧ ExτD1) sup

u∈D1, z∈D3ν(|z − u|) ,


Exit time

Lemma

Consider open D1, D3 ⊂ D such that dist(D1, D3) > 0. LetD2 = D \ (D1 ∪D3). If x ∈ D1, y ∈ D3 and t > 0, then

pD(t, x, y) 6 Px(XτD1 ∈ D2) sup

s<t, z∈D2p(s, z, y)

+ (t ∧ ExτD1) sup

u∈D1, z∈D3ν(|z − u|) ,

The lemma above is used for points which are relatively far away from eachother t 6 V 2(|x− y|) and close to the boundary. Its application requires goodcontrol of the exit distribution and mean exit time. Recall that in the previouslecture we studied the estimates in the case of a ball.


Exit time

Mean exit time.

Denote Br = B(0, r).We will be interested in estimating ExτBr , in particular at points which areclose to the boundary. The knowledge of the rate of decay of the mean exittime provides the rate decay of harmonic functions in a smooth domain (withrespect to the generator), which continuously vanish outside a part of thecomplement of a domain. This property is known as Boundary HarnackPrinciple. They also play important role in estimating the Dirichlet heat kernels.


Exit time

Mean exit time.


In the case of a Brownian motion we know that |Bt|2 − d · t is a martingale , sostopping at τBr we have

r2 − d · ExτBr = Ex(

|BτBr |2 − d · τBr

)

= Ex|B0|2 = |x|2.

ExτBr =

1

d(r2 − |x|2), |x| 6 r.


Exit time

Mean exit time.


In the case of a Brownian motion we know that |Bt|2 − d · t is a martingale , sostopping at τBr we have

r2 − d · ExτBr = Ex(

|BτBr |2 − d · τBr

)

= Ex|B0|2 = |x|2.

ExτBr =

1

d(r2 − |x|2), |x| 6 r.

In the case of ψ(ξ) = |x|α, 0 < α < 2 (isotropic α-stable ) Getoor 1990’proved that

ExτBr = Cd,α(r

2 − |x|2)α/2, |x| 6 r.


Exit time

Symmetric processes

Let Xt, t > 0, - a symmetric Levy process in Rd, d ∈ N; with the characteristic

exponent

ψ(x) = 〈x,Ax〉+∫

Rd

(1− cos 〈x, z〉)N(dz), x ∈ Rd,

where A - a symmetric and non-negative definite matrix, N - a symmetric Levymeasure

(∫

Rd(1 ∧ |z|2)N(dz) <∞

)

.


Exit time

Symmetric processes


exponent

ψ(x) = 〈x,Ax〉+∫

Rd

(1− cos 〈x, z〉)N(dz), x ∈ Rd,


(∫

Rd(1 ∧ |z|2)N(dz) <∞

)

.

The classical result of Pruitt 1981’ provides a constant C = C(d)

C−1

h(r)6 E

0τB(0,r) 6C

h(r), r > 0,

where

h(r) =TrA

r2+

∫

Rd

(

1 ∧ |z|2

r2

)

N(dz).


Exit time

Symmetric processes


exponent

ψ(x) = 〈x,Ax〉+∫

Rd

(1− cos 〈x, z〉)N(dz), x ∈ Rd,


(∫

Rd(1 ∧ |z|2)N(dz) <∞

)

.

The classical result of Pruitt 1981’ provides a constant C = C(d)

C−1

h(r)6 E

0τB(0,r) 6C

h(r), r > 0,

where

h(r) =TrA

r2+

∫

Rd

(

1 ∧ |z|2

r2

)

N(dz).

Let ψ∗(r) = sup{ψ(x) : |x| 6 r}. Then

1

8(1 + 2d)ψ∗(r−1) 6 h(r) 6 2dψ∗(r−1)


Exit time

One-dimensional case, symmetric process

Theorem (Grzywny, Ryznar, 2012’, PMS)

There is an absolute constant C such that for every symmetric Levy process inR which is not compound Poisson,

C−1√

ψ∗( 1r−|x| )ψ

∗( 1r)6 E

xτ(−r,r) 6C

√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.


Exit time

One-dimensional case, symmetric process

Theorem (Grzywny, Ryznar, 2012’, PMS)

There is an absolute constant C such that for every symmetric Levy process inR which is not compound Poisson,

C−1√

ψ∗( 1r−|x| )ψ

∗( 1r)6 E

xτ(−r,r) 6C

√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.

The upper bound is a consequence of the formula of the Green function for ahalf-line.Let V (r) = 1√

ψ∗(1/r), r > 0. Then the above estimate can be rewritten as

Exτ(−r,r) ≈ V (r − |x|)V (r), |x| < r,


Exit time

d-dimensional case and unimodal isoperimetric process

The upper bound for a d-dimensional ball follows immediately from theone-dimensional result since a ball can be included in a tangent strip.


Exit time



Corollary

There is absolute constant C such that for all r > 0 and x ∈ Rd we have

ExτBr 6

C√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.

The lower bound is much more difficult to obtain and requires precise estimatesof the genarator of the process on suitable test functions.


Exit time



Corollary

There is absolute constant C such that for all r > 0 and x ∈ Rd we have

ExτBr 6

C√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.

The lower bound is much more difficult to obtain and requires precise estimatesof the genarator of the process on suitable test functions.

Theorem (Bogdan, Grzywny, Ryznar, 2015’, PTRF)

If Condition H holds, then there is C = C(d) such that for r > 0,

C

Hr

1√

ψ∗( 1r−|x| )ψ

∗( 1r)6 E

xτBr , |x| < r.

Here Hr > 1 is increasing in r and in many situations we may show that it isbounded uniformly: H∞ <∞.


Exit time

Assumption H

Each of the following situations imply H:


Exit time

Assumption H


1. X is a subordinate Brownian motion governed by a special subordinator.In this case Hr = 1.


Exit time

Assumption H


1. X is a subordinate Brownian motion governed by a special subordinator.In this case Hr = 1.A function φ is a special Bernstein function if it is a Bernstein functionand λ

φ(λ)is also a Bernstein function.


Exit time

Assumption H




2. σ > 0. That is the process has a non-trivial Brownian part.


Exit time

Assumption H





3. Scale invariant Harnack inequality holds.


Exit time

Assumption H





3. Scale invariant Harnack inequality holds.For instance: d > 3 and s−βψ(s) is almost increasing on (θ,∞) for someβ > 0.


Exit time

Assumption H





3. Scale invariant Harnack inequality holds.For instance: d > 3 and s−βψ(s) is almost increasing on (θ,∞) for someβ > 0.d > 1 and s−βψ(s) is almost increasing and s−γψ(s) is almost decreasingon (θ,∞) for some β > 0 and γ < 2.


Exit time

Uniform estimate


Exit time

Uniform estimate

If X is a subordinate Brownian motion governed by a special subordinator then

C(d)−1√

ψ∗( 1r−|x| )ψ

∗( 1r)6 E

xτBr 6C(d)

√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.

ψ(x) = φ(|x|2), where φ(r) and r/φ(r) are Bernstein functions.


Exit time

Uniform estimate

If X is a subordinate Brownian motion governed by a special subordinator then

C(d)−1√

ψ∗( 1r−|x| )ψ

∗( 1r)6 E

xτBr 6C(d)

√

ψ∗( 1r−|x| )ψ

∗( 1r), |x| < r.

ψ(x) = φ(|x|2), where φ(r) and r/φ(r) are Bernstein functions.

(at least for d > 3)If ψ has global WLSC condition then the above two sided estimate is true (atleast for d > 3) but the constant on the left-hand side possibly dependent onthe process.For example for truncated isotropic stable processes, ν(x) = 1

|x|d+α 1{|x|<1},

ψ(x) ≈ |x|α ∧ |x|2, 0 < α < 2:

ExτBr ≈

[

(r − |x|)α/2 + (r − |x|)] [

rα/2 + r]

, |x| < r.

The same estimate for relativistic α-stable processes,ψ(x) = (|x|2 + 1)α/2 − 1, 0 < α < 2.


Exit time

D is of class C1,1 at scale r if D ⊂ Rd, D is open, non-empty, r > 0 and for

every Q ∈ ∂D there are balls B(x′, r) ⊂ D and B(x′′, r) ⊂ Dc tangent at Q.

Theorem

Let ψ ∈WLSC ∩WUSC. Open and bound D is C1,1 at scale r. Then forx ∈ R

d,

ExτD ≈

1√

ψ∗(1/δD(x))ψ∗(1/r)

The comparability constant depends on r, d, ψ and diamD.


Exit time

D is of class C1,1 at scale r if D ⊂ Rd, D is open, non-empty, r > 0 and for

every Q ∈ ∂D there are balls B(x′, r) ⊂ D and B(x′′, r) ⊂ Dc tangent at Q.

Theorem

Let ψ ∈WLSC ∩WUSC. Open and bound D is C1,1 at scale r. Then forx ∈ R

d,

ExτD ≈

1√

ψ∗(1/δD(x))ψ∗(1/r)

The comparability constant depends on r, d, ψ and diamD.

Under certain conditions (global weak scaling)

ExτD ≈

1√

ψ∗(1/δD(x))ψ∗(1/r)

with the comparability constant dependent on d, ψ and rdiamD.


Exit time

Heat kernel on bounded smooth domains

Theorem (Bogdan, Grzywny, Ryznar, 2014, SPA)

Let ψ satisfy WLSC and WUSC. There is r0 = r0(d, ψ) > 0 such that for0 < r < r0 and bounded D ⊂ R

d of class C1,1 with localization radius r andν(diamD) > 0, we have for all x, y ∈ R

d, t > 0,

pD(t, x, y) ≈ Px(τD > t/2)Py(τD > t/2)p

(

t ∧ V 2(r), x, y)

and


Exit time





d, t > 0,


(

t ∧ V 2(r), x, y)

and

Px (τD > t) ≈ e−λ1(D)t

(

V (δD(x))√t ∧ V (r)

∧ 1)

.


Exit time





d, t > 0,


(

t ∧ V 2(r), x, y)

and


(

V (δD(x))√t ∧ V (r)

∧ 1)

.

Here λ1(D) denotes the smallest eigenvalue of the compact semigroupgenerated by pD(t, x, y).

Recall that V (r) = 1√ψ∗(1/r)

, where ψ is the symbol of the process.


Exit time





d, t > 0,


(

t ∧ V 2(r), x, y)

and


(

V (δD(x))√t ∧ V (r)

∧ 1)

.

Here λ1(D) denotes the smallest eigenvalue of the compact semigroupgenerated by pD(t, x, y).

Recall that V (r) = 1√ψ∗(1/r)

, where ψ is the symbol of the process.

Extends results obtained by Chen, Kim, Song in the case of subordinateBrownian motions governed by complete subordinators with weak scalingconditions.


Exit time

Bounded smooth domains

Let ψ satisfy global WLSC and WUSC.


Exit time


Let ψ satisfy global WLSC and WUSC. For every r > 0:

pB(0,r)(t, x, y) ≈ e−λ1(Br)t(

V (r − |x|)√t ∧ V (r)

∧ 1)(

V (r − |y|)√t ∧ V (r)

∧ 1)

p(t∧V 2(r), x, y),

where the comparability constant depends only on d and ψ.


Exit time



pB(0,r)(t, x, y) ≈ e−λ1(Br)t(

V (r − |x|)√t ∧ V (r)

∧ 1)(

V (r − |y|)√t ∧ V (r)

∧ 1)

p(t∧V 2(r), x, y),

where the comparability constant depends only on d and ψ. Moreover

c1(d)

V 2(r)6 λ1(Br) 6

c2(d)

V 2(r)


Exit time



pB(0,r)(t, x, y) ≈ e−λ1(Br)t(

V (r − |x|)√t ∧ V (r)

∧ 1)(

V (r − |y|)√t ∧ V (r)

∧ 1)

p(t∧V 2(r), x, y),

where the comparability constant depends only on d and ψ. Moreover

c1(d)

V 2(r)6 λ1(Br) 6

c2(d)

V 2(r)

A consequence of the above estimate and intrinsic ultracontractivity of PDt(Grzywny 2008’) is:

Corollary

Let φr1 be the (positive) eigenfunction corresponding to λ1(Br). There isc = c(d, ψ) such that

c−1V (r − |x|)rd/2V (r)

6 φr1(x) 6 cV (r − |x|)rd/2V (r)

, x ∈ Rd.

IU: pB(0,r)(t, x, y) ≈ φr1(x)φr1(y)e−λ1(Br)t, t→∞


Exit time

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar, 2014’, SPA)

Let ψ ∈WLSC(α, 0)∩WUSC(α, 0) and d > α. Let D be a C1,1 at scale R1 andsuch that Dc ⊂ BR2 . For all x, y ∈ R

d and t > 0 we have

pD(t, x, y) ≈ Px(τD > t)Py(τD > t)p(t, x, y),

and

Px(τD > t) ≈

(

V (δD(x))√t ∧ V (R1)

∧ 1)

with comparability constants C = C(d, ψ,R2/R1).


Exit time

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar, 2014’, SPA)

Let ψ ∈WLSC(α, 0)∩WUSC(α, 0) and d > α. Let D be a C1,1 at scale R1 andsuch that Dc ⊂ BR2 . For all x, y ∈ R

d and t > 0 we have

pD(t, x, y) ≈ Px(τD > t)Py(τD > t)p(t, x, y),

and

Px(τD > t) ≈

(

V (δD(x))√t ∧ V (R1)

∧ 1)

with comparability constants C = C(d, ψ,R2/R1).

In particular for D = (BR1)c comparability constants depend only on d and ψ.


Exit time

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar, 2010’, AoP)

Let ψ(ξ) = |ξ|α, α ∈ (0, 2), and D = B(0, 1)c. For all x, y ∈ Rd and t > 0 we

havepB(0,r)

c(t, x, y) ≈ Px(τD > t)Py(τD > t)p(t, x, y),


Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,


Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,

1 ∧ log(1+δ1/2

D(x)

log(1+t1/2), if α = d = 1,


Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,

1 ∧ log(1+δ1/2

D(x)

log(1+t1/2), if α = d = 1,

δα−1D

(x)∧δα/2D(x)

(t1/α∨δD(x))α−1∧ (t1/α ∨ δD(x))α/2, if α > d = 1,

with comparability constants C = C(d, α).


Exit time

Exterior domains



havepB(0,r)



Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,


Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,

1 ∧ log(1+δ1/2

D(x)

log(1+t1/2), if α = d = 1,


Exit time

Exterior domains



havepB(0,r)


and

Px(τD > t) ≈

1 ∧ δα/2

D(x)

1∧t1/2 , if α < d,

1 ∧ log(1+δ1/2

D(x)

log(1+t1/2), if α = d = 1,

δα−1D

(x)∧δα/2D(x)

(t1/α∨δD(x))α−1∧ (t1/α ∨ δD(x))α/2, if α > d = 1,

with comparability constants C = C(d, α).

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