Heat Kernels and Exit Times for Isotropic Lévy Processes

Heat kernels and exit times for isotropic L´evy processes

Heat kernels and exit times for isotropic Levy´ processes

MichałRyznar

Wrocław University of Technology

Third FCPNLO Workshop Fractional Calculus, Probability and Non-Local Operators BCAM, Bilbao, November 18-20, 2015

The presentation is based mostly on a joint project with K. Bogdan, T. Grzywny. Heat kernels and exit times for isotropic L´evy processes

Definitions

A (Borel) measure on Rd is called isotropic unimodal, if on Rd 0 it is \{ } absolutely continuous with respect to the Lebesgue measure, and has a ﬁnite radial nonincreasing density function. Heat kernels and exit times for isotropic L´evy processes

Definitions

A (Borel) measure on Rd is called isotropic unimodal, if on Rd 0 it is \{ } absolutely continuous with respect to the Lebesgue measure, and has a ﬁnite radial nonincreasing density function. d AL´evy process X =(Xt,t! 0) is a stochastic process with values in R , which has stationary and idependent increments and cad-lag paths:

Xt Xt ,...,Xt Xt are independent for any 0 " t0 " ..." tn. 1 − 0 n − n−1

Xt+h Xt has the same distribution as Xh,t,h>0 − Heat kernels and exit times for isotropic L´evy processes

Definitions

A (Borel) measure on Rd is called isotropic unimodal, if on Rd 0 it is \{ } absolutely continuous with respect to the Lebesgue measure, and has a ﬁnite radial nonincreasing density function. d AL´evy process X =(Xt,t! 0) is a stochastic process with values in R , which has stationary and idependent increments and cad-lag paths:

Xt Xt ,...,Xt Xt are independent for any 0 " t0 " ..." tn. 1 − 0 n − n−1

Xt+h Xt has the same distribution as Xh,t,h>0 − X =(Xt,t! 0) is called unimodal isotropic if all its one-dimensional distributions pt(dx) are isotropic unimodal. Heat kernels and exit times for isotropic L´evy processes

Definitions

A (Borel) measure on Rd is called isotropic unimodal, if on Rd 0 it is \{ } absolutely continuous with respect to the Lebesgue measure, and has a ﬁnite radial nonincreasing density function. d AL´evy process X =(Xt,t! 0) is a stochastic process with values in R , which has stationary and idependent increments and cad-lag paths:

Xt Xt ,...,Xt Xt are independent for any 0 " t0 " ..." tn. 1 − 0 n − n−1

Xt+h Xt has the same distribution as Xh,t,h>0 − X =(Xt,t! 0) is called unimodal isotropic if all its one-dimensional distributions pt(dx) are isotropic unimodal.

We call the transition density pt(x, y)=pt(y x)=p(t, x, y) the heat − kernel. Ttf(x)=f(x + y)pt(y)dy form a semigroup on various function spaces. Heat kernels and exit times for isotropic L´evy processes

Characteristic function

i ξ,Xt i ξ,x tψ(ξ) d E e ⟨ ⟩ = e ⟨ ⟩pt(dx)=e− ,ξR . ∈ !Rd Isotropic unimodal L´evy processes are characterized by L´evy-Kchintchine exponents of the form ([Watanabe 1983])

ψ(ξ)=σ2 ξ 2 + (1 cos ξ,x ) ν( x )dx, (1) | | − ⟨ ⟩ | | !Rd with nonincreasing ν and σ ! 0. N(dx)=ν( x )dx is called the L´evy | | measure of the process.

pt(dx) N(dx)=limt 0 → t Heat kernels and exit times for isotropic L´evy processes

Characteristic function

i ξ,Xt i ξ,x tψ(ξ) d E e ⟨ ⟩ = e ⟨ ⟩pt(dx)=e− ,ξR . ∈ !Rd Isotropic unimodal L´evy processes are characterized by L´evy-Kchintchine exponents of the form ([Watanabe 1983])

ψ(ξ)=σ2 ξ 2 + (1 cos ξ,x ) ν( x )dx, (1) | | − ⟨ ⟩ | | !Rd with nonincreasing ν and σ ! 0. N(dx)=ν( x )dx is called the L´evy | | measure of the process.

pt(dx) N(dx)=limt 0 → t tψ(ξ) 1 If e− L then ∈

1 i ξ,x tψ(x) p (ξ)= e− ⟨ ⟩e− dx " p (0). t (2π)d t !Rd Heat kernels and exit times for isotropic L´evy processes

Examples

d α Isotropic stable processes: ν(x)=c x − − | | d α Truncated α-stable processes: ν(r)=cr− − 1(0,1)(r). Heat kernels and exit times for isotropic L´evy processes

Examples

d α Isotropic stable processes: ν(x)=c x − − | | d α Truncated α-stable processes: ν(r)=cr− − 1(0,1)(r). Subordinate Brownian motions: Let Tt be a subordinator (positive and increasing one-dimensional L´evy λT tϕ(λ) process) without drift; Ee− t = e− ,where(Bernstein function)

∞ λs ϕ(λ)= (1 e− )µ(ds). − !0 Let B(t) be an independent Brownian motion with ψ(ξ)= ξ 2.Then | | B(Tt) has an absolutely continuous L´evy measure N(dx)=ν(x)dx,where

∞ ν(x)= gs(x)µ(ds), !0 |x|2 d/2 and gs(x)=(4πs)− e− 4s is the Gaussian kernel. Heat kernels and exit times for isotropic L´evy processes

Generators

f(x)=σ2∆+p.v. (f(x + y) f(x)) ν( x )dx, f C2(Rd) A − | | ∈ !Rd Heat kernels and exit times for isotropic L´evy processes

Generators

f(x)=σ2∆+p.v. (f(x + y) f(x)) ν( x )dx, f C2(Rd) A − | | ∈ !Rd Fractional Laplacian: α-stable isoperimetric process

α/2 1 ∆ f(x)=p.v. cd,α (f(x + y) f(x)) dx − x d+α !Rd | | Heat kernels and exit times for isotropic L´evy processes

Generators

f(x)=σ2∆+p.v. (f(x + y) f(x)) ν( x )dx, f C2(Rd) A − | | ∈ !Rd Fractional Laplacian: α-stable isoperimetric process

α/2 1 ∆ f(x)=p.v. cd,α (f(x + y) f(x)) dx − x d+α !Rd | | = I (I ∆)α/2: relativistic α-stable isoperimetric process A − −

K(d+α)/2( x ) f(x)=p.v. cd,α (f(x + y) f(x)) | | dx A − x (d+α)/2 !Rd | | Heat kernels and exit times for isotropic L´evy processes

Topics

Heat kernels estimates and weak scaling. Heat kernels and exit times for isotropic L´evy processes

Topics

Heat kernels estimates and weak scaling.

Exit times. Estimates of the mean exit time and surviaval probability for smooth sets. Heat kernels and exit times for isotropic L´evy processes

Topics

Heat kernels estimates and weak scaling.

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

Heat kernels for the killed process on exiting a set. Estimates. Heat kernels and exit times for isotropic L´evy processes

Topics

Heat kernels estimates and weak scaling.

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

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

Hitting times. Heat kernels and exit times for isotropic L´evy processes

Our goal: Estimates for pt(x) in terms of the symbol ψ of the process (or ψ∗) Heat kernels and exit times for isotropic L´evy processes

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 . | | t d pt(x)=Cd for x R . (t2 + x 2)(d+1)/2 ∈ | | Heat kernels and exit times for isotropic L´evy processes

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 . | | t d pt(x)=Cd for x R . (t2 + x 2)(d+1)/2 ∈ | | Below f g means that there is a constant c>0 such that 1 ≈ c− g " f(x) " cg(x).Thisc is called the comparability constant. Heat kernels and exit times for isotropic L´evy processes

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

Example: isotropic α-stable process, ψ(x)= x α, 0 <α<2. | | d/α t d pt(x) min t− , for x R . ≈ { x d+α } ∈ | | Heat kernels and exit times for isotropic L´evy processes

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

Example: isotropic α-stable process, ψ(x)= x α, 0 <α<2. | | d/α t d pt(x) min t− , for x R . ≈ { x d+α } ∈ | |

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

Example: isotropic α-stable process, ψ(x)= x α, 0 <α<2. | | d/α t d pt(x) min t− , for x R . ≈ { x d+α } ∈ | |

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

pt(2r)( B2r Br ) " P(r " Xt " 2r) " pt(r)( B2r Br ) | |−| | | | | |−| | Heat kernels and exit times for isotropic L´evy processes

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

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

d pt(r) " C(d)r− P(r/2 " Xt " r) | | Heat kernels and exit times for isotropic L´evy processes

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

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

d pt(r) " C(d)r− P(r/2 " Xt " r) | |

d pt(r) ! c(d, κ)r− P(r " Xt " κr),κ>1 | | Heat kernels and exit times for isotropic L´evy processes

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

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

d pt(r) " C(d)r− P(r/2 " Xt " r) | |

d pt(r) ! c(d, κ)r− P(r " Xt " κr),κ>1 | | There are many results about heat kernel estimates-usually assumptions are in terms of L´evy measures (Chen, Kaleta, Kumagai, Kim, Knopova, Mimica, Schilling, Song, Sztonyk and many others). Heat kernels and exit times for isotropic L´evy processes

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

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

d pt(r) " C(d)r− P(r/2 " Xt " r) | |

In our approach we derive estimates using the symbol ψ (ψ∗ ) and the assumptions will be formulated in terms of ψ.

Let ψ∗(u):=supx !u ψ(x),foru ! 0. | | Proposition

2 d ψ(x) " ψ∗( x ) " π ψ(x) for x R . (2) | | ∈ Heat kernels and exit times for isotropic L´evy processes

Proposition

For r>0 we have

2e tψ∗(1/r) P( Xt ! r) " (2d + 1) 1 e− . | | e 1 − − ( ) Heat kernels and exit times for isotropic L´evy processes

Proposition

For r>0 we have

2e tψ∗(1/r) P( Xt ! r) " (2d + 1) 1 e− . | | e 1 − − ( ) Recall

d d d pt(r) " C(d)r− P(r/2 " Xt " r) " Cdtψ∗(2/r)r− " 10Cdtψ∗(1/r)r− | | Heat kernels and exit times for isotropic L´evy processes

Proposition

For r>0 we have

2e tψ∗(1/r) P( Xt ! r) " (2d + 1) 1 e− . | | e 1 − − ( ) Recall

d d d pt(r) " C(d)r− P(r/2 " Xt " r) " Cdtψ∗(2/r)r− " 10Cdtψ∗(1/r)r− | |

Corollary

There is C = C(d) such that

d d pt(x) " Ctψ∗(1/ x )/ x for x R 0 . | | | | ∈ \{ } Heat kernels and exit times for isotropic L´evy processes

Proposition

For r>0 we have

2e tψ∗(1/r) P( Xt ! r) " (2d + 1) 1 e− . | | e 1 − − ( ) Recall

d d d pt(r) " C(d)r− P(r/2 " Xt " r) " Cdtψ∗(2/r)r− " 10Cdtψ∗(1/r)r− | |

Corollary

There is C = C(d) such that

d d pt(x) " Ctψ∗(1/ x )/ x for x R 0 . | | | | ∈ \{ } Example: isotropic α-stable process, ψ(x)= x α, 0 <α<2. | | d+α d pt(x) " Ct/ x for x R 0 . | | ∈ \{ } Heat kernels and exit times for isotropic L´evy processes

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 choose the following parametrization of the tails:

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

Consider the Laplace transform of ft:

∞ λρ ft(λ)= e− ft(ρ)dρ,λ ! 0. L !0 Heat kernels and exit times for isotropic L´evy processes

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 choose the following parametrization of the tails:

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

Consider the Laplace transform of ft:

∞ λρ ft(λ)= e− ft(ρ)dρ,λ ! 0. L !0

Lemma

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

1 1 tψ∗(√λ) 1 tψ∗(√λ) C− 1 e− " ft(λ) " C1 1 e− ,λ>0. 1 λ − L λ − ( ) ( ) Heat kernels and exit times for isotropic L´evy processes

Proof.

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

hˆ(x)k(x)dx = h(x)kˆ(x)dx !Rd !Rd Heat kernels and exit times for isotropic L´evy processes

Proof.

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

hˆ(x)k(x)dx = h(x)kˆ(x)dx !Rd !Rd

Proof.

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

hˆ(x)k(x)dx = h(x)kˆ(x)dx !Rd !Rd

By the result of Hoh, 1998’,

2 ψ(su) " ψ∗(su) " 2(s + 1)ψ∗(u), s, u ! 0. (4) Heat kernels and exit times for isotropic L´evy processes

By the result of Hoh, 1998’,

2 ψ(su) " ψ∗(su) " 2(s + 1)ψ∗(u), s, u ! 0. (4)

Note that bt t 1 e− " b(1 e− ),t! 0,b! 1, (5) − − Heat kernels and exit times for isotropic L´evy processes

By the result of Hoh, 1998’,

2 ψ(su) " ψ∗(su) " 2(s + 1)ψ∗(u), s, u ! 0. (4)

Note that bt t 1 e− " b(1 e− ),t! 0,b! 1, (5) − −

tψ(x√λ) 2t( x 2+1)ψ∗(√λ) 2 tψ∗(√λ) 1 e− " 1 e− | | " 2( x + 1) 1 e− − − | | − ( )

d/2 tψ(x√λ) x 2/4 λ ft(λ)=(4π)− 1 e− e−| | dx L Rd − ! ( ) tψ∗(√λ) 2 d/2 x 2/4 " 1 e− 2( x + 1)(4π)− e−| | dx − Rd | | ( ) ! Heat kernels and exit times for isotropic L´evy processes

On the other hand, if x ! 1,then | | 2 2 ψ x√λ ! ψ∗ x √λ /π ! ψ∗ √λ /π | | by (2). Thus, " # " # " #

d/2 tψ(x√λ) x 2/4 λ ft(λ)=(4π)− 1 e− e−| | dx L Rd − ! ( ) d/2 x 2/4 tψ∗(√λ)/π2 ! (4π)− e−| | dx 1 e− Bc − ! 1 ( ) tψ∗(√λ) 2 d/2 x 2/4 ! 1 e− π− (4π)− e−| | dx, − Bc ( ) ! 1 where we use (5). Heat kernels and exit times for isotropic L´evy processes

Upper bound for ft(r)=P( Xt ! √r) ?? | | Heat kernels and exit times for isotropic L´evy processes

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

The upper bound for ft(λ) immediately provides the upper bound for ft(r) L since ft(r) is decreasing. Heat kernels and exit times for isotropic L´evy processes

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

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

1/λ 1 λr tψ∗(√λ) ft(1/λ)(1 e− ) " λ e− ft(r)dr " λ ft(λ) " Cd 1 e− − 0 L − ! ( ) Heat kernels and exit times for isotropic L´evy processes

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

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

1/λ 1 λr tψ∗(√λ) ft(1/λ)(1 e− ) " λ e− ft(r)dr " λ ft(λ) " Cd 1 e− − 0 L − ! ( ) Lower bound?? Heat kernels and exit times for isotropic L´evy processes

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

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

1/λ 1 λr tψ∗(√λ) ft(1/λ)(1 e− ) " λ e− ft(r)dr " λ ft(λ) " Cd 1 e− − 0 L − ! ( ) Lower bound??

The lower bound for ft(λ)+some type of weak scaling condition for the L symbol (WUSC) give the lower bound for ft(r). This argument is still simple yet requires more eﬀort than the upper bound. Heat kernels and exit times for isotropic L´evy processes

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

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

1/λ 1 λr tψ∗(√λ) ft(1/λ)(1 e− ) " λ e− ft(r)dr " λ ft(λ) " Cd 1 e− − 0 L − ! ( ) Lower bound??

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

WUSC implies some type of scaling property for ft(λ). L Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Scaling conditions

Let φ :[0, ) [0, ). ∞ → ∞

φ WLSC(α,θ,c)ifα > 0, θ ! 0 and c (0, 1] exist, such that ∈ ∈ φ(λθ) ! cλ αφ(θ) for λ ! 1,θ>θ. (6) Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Scaling conditions

Let φ :[0, ) [0, ). ∞ → ∞

φ WLSC(α,θ,c)ifα > 0, θ ! 0 and c (0, 1] exist, such that ∈ ∈ φ(λθ) ! cλ αφ(θ) for λ ! 1,θ>θ. (6)

φ WUSC(α, θ, C)ifα<2, θ ! 0 and C ! 1 exist, such that ∈ φ(λθ) " Cλαφ(θ) for λ ! 1,θ>θ. (7) Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Scaling conditions

Let φ :[0, ) [0, ). ∞ → ∞

φ WLSC(α,θ,c)ifα > 0, θ ! 0 and c (0, 1] exist, such that ∈ ∈ φ(λθ) ! cλ αφ(θ) for λ ! 1,θ>θ. (6)

φ WUSC(α, θ, C)ifα<2, θ ! 0 and C ! 1 exist, such that ∈ φ(λθ) " Cλαφ(θ) for λ ! 1,θ>θ. (7)

The scaling conditions are global if we can set θ =0(or θ =0for WUSC). Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Examples: ψ(θ)= θ α + θ β , 0 <α<β<2 has global WLSC(α), WUSC(β): | | | | λα( θ α + θ β ) " λθ α + λθ β " λβ ( θ α + θ β ). | | | | | | | | | | | | This symbol corresponds to the sum of two independent isotropic stable processe with indices α,β, respectively Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

ψ(θ)=(θ2 + 1)α/2 1, 0 <α<2 has global WLSC(α), but only local − WUSC(α). Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

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

ψ(θ) min θ 2, θ α ≈ {| | | | } hence

ψ(λθ) min λθ 2, λθ α ! λα min θ 2, θ α ≈ {| | | | } {| | | | } Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

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

ψ(θ) min θ 2, θ α ≈ {| | | | } hence

ψ(λθ) min λθ 2, λθ α ! λα min θ 2, θ α ≈ {| | | | } {| | | | }

ψ(λθ) min λθ 2, λθ α " λθ α = λα min θ 2, θ α ,θ! 1 ≈ {| | | | } | | {| | | | } Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Lemma

C = C(d) exists such that if ψ WUSC(α, θ, C) and 2 α−2 ∈ a =[(2 α)C] 2−α C 2 ,then − tψ∗(1/r) P( Xt ! r) ! a 1 e− , 0

Lemma

C = C(d) exists such that if ψ WUSC(α, θ, C) and 2 α−2 ∈ a =[(2 α)C] 2−α C 2 ,then − tψ∗(1/r) P( Xt ! r) ! a 1 e− , 0

Lemma

Let g ! 0 be nonincreasing, β>0 and g WUSC( β,θ, C).Thereis L ∈ − b = b(β,C) (0, 1) such that ∈ b b 1 1 g(r) ! e r− g(r− ), 0

We apply the lemma to ft(r)=P( Xt ! √r) and then use the estimates for | | f. First using the lower and the upper bounds for the Laplace transform L together with WUSC(α, θ, C) condition for ψ we prove that f has L WUSC(α/2 1) property. − Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

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

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

d tψ∗(1/ x ) 2 pt(x) " C∗ min ψ−(1/t) , | | if tψ∗(θ) <π− . x d $ | | ' % & If ψ WLSC(α,θ,c) WUSC(α,θ, C),thenc∗ = c∗(d, α,c, α, C), ∈ ∩ r0 = r0(d, α,c, α, C) exist with

d tψ∗(1/ x ) r0 pt(x) ! c∗ min ψ− (1/t) , | | ,tψ∗(θ/r0) < 1 and x < . x d | | θ $ | | ' % &

Here ψ−(r)=inf s : ψ∗(s) ! r is the generalized inverse of the { } nondecreasing function ψ∗. Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

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

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

d tψ∗(1/ x ) r0 pt(x) ! c∗ min ψ− (1/t) , | | ,tψ∗(θ/r0) < 1 and x < . x d | | θ $ | | ' % &

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

1 tψ(x) d pt(0) = e− dx ψ− (1/t) . (2π)d ≈ ! % & Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Corollary

There is C = C(d) such that

d ν(x) " Cψ∗(1/ x ) x − . | | | |

If ψ WLSC(α,θ,c) WUSC(α,θ, C) and x

Corollary

There is C = C(d) such that

d ν(x) " Cψ∗(1/ x ) x − . | | | |

If ψ WLSC(α,θ,c) WUSC(α,θ, C) and x

Corollary

If ψ WLSC(α,0,c) WUSC(α,0,C), then ∈ ∩

d tψ∗(1/ x ) d pt(x) min ψ−(1/t) , | | ,t>0,x R . ≈ x d ∈ $ | | ' % & (α) (β) (α) (β) Example. Xt = Xt + Xt , 0 <β<α<2. Xt ,Xt idependent isotropic stable and ψ(ξ)= ξ α + ξ β . (Chen, Kumgai 2008’) | | | |

d/α d/β t t d pt(x) min (1/t) (1/t) , + ,t>0,x R . ≈ ∧ x d+α x d+β ∈ $ | | | | ' Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

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

d Let Xt be an isotropic unimodal Lévy process in R with transition density p, Lévy-Khintchine exponent ψ and Lévy measure density ν. The following are equivalent: (i) WLSC and WUSC [global WLSC and WUSC] hold for ψ.

(ii) For some r0 (0, )[r0 = ] and a constant c, ∈ ∞ ∞ 1 tψ∗( x − ) 1 pt(x) ! c | | , 0 < x

Theorem (Kulczycki and Ryznar, 2015’, TAMS)

Let X be an isotropic Lévy process in Rd such that ν(r) is nonincreasing, absolutely continuous such that ν′(r)/r is nonincreasing. Then its transition − density pt(x)=pt( x ) satisfies | | d 1 tψ∗(1/r) pt(r) " c(d) ∧ ,t,r>0. dr rd+1 + + If additionally ψ satisfies+ WLSC(+ α,θ ,c), then + + 0

d r d+2 tψ∗(1/r) 2 pt(r) " c(d, α) [ψ−(1/t)] ,tψ∗(θ0) " 1/π . dr c(d+2)/α+1 ∧ rd+2 + + , - + + If+ additionally+ ψ satisﬁes WLSC(α,θ0,c) and WUSC(α,θ0, C), then we have

d d+2 tψ∗(1/r) pt(r) ! c∗r [ψ−(1/t)] ,tψ∗(θ0/r0) " 1,r 0. Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Let X be an isotropic L´evy process in Rd such that ν(r) is nonincreasing, absolutely continuous such that ν′(r)/r is nonincreasing. − Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Let X be an isotropic Lévy process in Rd such that ν(r) is nonincreasing, absolutely continuous such that ν′(r)/r is nonincreasing. − (d+2) Rd+2 Then there exists a Lévy process Xt in with the characteristic exponent ψ(d+2)(ξ)=ψ( ξ ), ξ Rd+2 and the radial, radially nonincreasing | | ∈ transition density p(d+2)(x)=p(d+2)( x ) satisfying t t | | (d+2) 1 d p (r)= − pt(r),r>0. t 2πr dr Heat kernels and exit times for isotropic Lévy processes Consequence of scaling

The estimates of the gradient of pt enabled analysis of gradients of harmonic functions 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 and we derived its estimates from above. Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Asymptotic properties of pt(x)

Theorem (Cygan, Grzywny, Trojan, 2015’, to appear in TAMS )

Let X be an isotropic unimodal L´evy process on Rd with the characteristic exponent ψ which is regularly varying at with index 0 <α<2.Then ∞ p(t, x) lim d 1 = c(α, d). x→0 t x − ψ( x − ) tψ(|x|−1)→0 | | | | Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Asymptotic properties of pt(x)

Theorem (Cygan, Grzywny, Trojan, 2015’, to appear in TAMS )

Actually the converse is true. If the limit above exists then there is 0 <α<2 such that ψ is regularly varying at with index α. ∞ Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Asymptotics of pt(x) when the symbol is slowly varying

Given a function ℓ slowly varying function at inﬁnity, by Πℓ∞ we denote a class of functions f :[x0, ) [0, + ) such that for all λ R ∞ → ∞ ∈ f(λx) f(x) lim − = log λ. (8) x + ℓ(x) → ∞

Theorem

Let X be an isotropic unimodal L´evy process on Rd with the characteristic exponent ψ Π∞.Then ∈ ℓ p(t, x) Γ(d/2) lim d 1 = d/2 . x→0 t x − ℓ( x − ) 4π tψ(|x|−1)→0 | | | | Heat kernels and exit times for isotropic L´evy processes Consequence of scaling

Asymptotics of pt(x) when the symbol is slowly varying

Theorem

Let X be an isotropic unimodal L´evy process on Rd with the characteristic exponent ψ Π∞.Then ∈ ℓ p(t, x) Γ(d/2) lim d 1 = d/2 . x→0 t x − ℓ( x − ) 4π tψ(|x|−1)→0 | | | |

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ } Heat kernels and exit times for isotropic L´evy processes Exit time

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − Heat kernels and exit times for isotropic L´evy processes Exit time

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B . Heat kernels and exit times for isotropic L´evy processes Exit time

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B

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

x P (τD >t)= pD(t, x, y)dy. ! Heat kernels and exit times for isotropic L´evy processes Exit time

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B

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

x P (τD >t)= pD(t, x, y)dy. !

Green function: G (x, y)= ∞ p (t, x, y)dt. D 0 D . Heat kernels and exit times for isotropic L´evy processes Exit time

Mean exit time.

Denote Br = B(0,r). Ex We will be interested in estimating τBr , in particular at points which are close to the boundary. The knowledge of the rate of decay of the mean exit time provides the rate decay of harmonic functions in a smooth domain (with respect to the generator), which continously vanish outside a part of the complement of a domain. This property is known as Boundary Harnack Principle. They also play important role in estimating the Dirichlet heat kernels. Heat kernels and exit times for isotropic L´evy processes Exit time

Mean exit time.

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

Mean exit time.

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

x 2 2 α/2 E τB = Cd,α(r x ) , x " r. r −| | | | Heat kernels and exit times for isotropic L´evy processes Exit time

Symmetric processes

d Let Xt, t ! 0, - a symmetric L´evy process in R , d N; with the characteristic ∈ exponent

ψ(x)= x, Ax + (1 cos x, z ) N(dz),xRd, ⟨ ⟩ − ⟨ ⟩ ∈ !Rd where A - a symmetric and non-negative definite matrix, N - a symmetric Lévy measure (1 z 2)N(dz) < . Rd ∧| | ∞ ". # Heat kernels and exit times for isotropic Lévy processes Exit time

Symmetric processes

d Let Xt, t ! 0, - a symmetric L´evy process in R , d N; with the characteristic ∈ exponent

ψ(x)= x, Ax + (1 cos x, z ) N(dz),xRd, ⟨ ⟩ − ⟨ ⟩ ∈ !Rd where A - a symmetric and non-negative definite matrix, N - a symmetric Lévy measure (1 z 2)N(dz) < . Rd ∧| | ∞ The classical result of Pruitt 1981’ provides constant C = C(d) ". # 1 C− C " E0τ " ,r>0, h(r) B(0,r) h(r) where Tr A z 2 h(r)= + 1 | | N(dz). r2 ∧ r2 !Rd , - Heat kernels and exit times for isotropic Lévy processes Exit time

Symmetric processes

d Let Xt, t ! 0, - a symmetric L´evy process in R , d N; with the characteristic ∈ exponent

Let ψ∗(r)=sup ψ(x): x " r .Then { | | } 1 1 1 ψ∗(r− ) " h(r) " 2dψ∗(r− ) 8(1 + 2d) Heat kernels and exit times for isotropic L´evy processes Exit time

The renewal function

Let X1 be the first coordinate of the process X which is now assumed isotropic and unimodal. 1 1 Let Mt =sup X and let Lt be the local time at 0 for Mt X ,thefirst s!t s − t coordinate of X reflected at the supremum. Heat kernels and exit times for isotropic Lévy processes Exit time

The renewal function

1 We also deﬁne the ascending ladder-height process, Hs = X −1 = M −1 . Ls Ls Heat kernels and exit times for isotropic L´evy processes Exit time

The renewal function

1 We also deﬁne the ascending ladder-height process, Hs = X −1 = M −1 . Ls Ls

The renewal function V of the ascending ladder-height process H is deﬁned as

∞ V (x)= P(Hs " x)ds, x R. ∈ !0 V is subaddative: V (x + y) " V (x)+V (y). Heat kernels and exit times for isotropic L´evy processes Exit time

The renewal function

1 We also deﬁne the ascending ladder-height process, Hs = X −1 = M −1 . Ls Ls

The renewal function V of the ascending ladder-height process H is deﬁned as

∞ V (x)= P(Hs " x)ds, x R. ∈ !0 V is subaddative: V (x + y) " V (x)+V (y). The importance of the function V in the potential theory of d-dimensional L´evy processes was discovered by Vondracek, Chen and Song. Heat kernels and exit times for isotropic L´evy processes Exit time

For isotropic stable process (ψ(ξ)= ξ α) we have V (x)=xα/2,x! 0.There | | are no explicit formulas in general case, but we have very precise estimates of V . Heat kernels and exit times for isotropic L´evy processes Exit time

For isotropic stable process (ψ(ξ)= ξ α) we have V (x)=xα/2,x! 0.There | | are no explicit formulas in general case, but we have very precise estimates of V . Proposition

1 1 V 2(r) ,r! 0 ≈ h(r) ≈ ψ(1/r) Comparability constants are either absolute or depend only on dimension. Heat kernels and exit times for isotropic L´evy processes Exit time

For isotropic stable process (ψ(ξ)= ξ α) we have V (x)=xα/2,x! 0.There | | are no explicit formulas in general case, but we have very precise estimates of V . Proposition

1 1 V 2(r) ,r! 0 ≈ h(r) ≈ ψ(1/r) Comparability constants are either absolute or depend only on dimension.

relativistic α-stable processes, ψ(x)=(x 2 + 1)α/2 1, 0 <α<2, | | − V (r) rα/2 + r ≈ sum of two independent isotropic stable processes,

V (r) rα/2 rβ/2 ≈ ∧ geometric α-stable processes, ψ(x) = log(1 + x α), 0 <α" 2, | | 1 V (r) ≈ log(1 + r α) / − Heat kernels and exit times for isotropic L´evy processes Exit time

One-dimensional case, symmetric process

Theorem (Grzywny, Ryznar 2012’, PMS)

There is an absolute constant C such that for every symmetric L´evy process in R which is not compound Poisson,

x CV (R x )V (R) " E τ( R,R) " 2V (R x )V (R), x

One-dimensional case, symmetric process

Theorem (Grzywny, Ryznar 2012’, PMS)

There is an absolute constant C such that for every symmetric L´evy process in R which is not compound Poisson,

x CV (R x )V (R) " E τ( R,R) " 2V (R x )V (R), x

We can replace the function V by more explicit functions: h or ψ∗.

1 C− x C " E τ( r,r) " , x

Upper bound, One-dimensional case

Let τ = τ(0, ) and x (0,R). ∞ ∈ According to the result of Silverstein 1980’, for any measurable non-negative function f :[0, ) [0, ), we have ∞ ,→ ∞ τ x E f(Xt)dt = V (dy) V (dz)f(x + y z). 0 [0, ) [0,x] − 1! 2 ! ∞ ! Heat kernels and exit times for isotropic L´evy processes Exit time

Upper bound, One-dimensional case

We take f = I[0,R].Then

τ(0,R) τ x x x E τ(0,R) = E f(Xt)dt " E f(Xt)dt) 1!0 2 1!0 2 = V (dy) V (dz)f(x + y z) [0, ) [0,x] − ! ∞ ! " V (dy) V (dz)=V (R)V (x), ![0,R] ![0,x] which completes the proof of the upper bound. Heat kernels and exit times for isotropic L´evy processes Exit time

Exit time properties, one-dimensional case

Theorem (Kwasnicki, Malecki, Ryznar 2013’, AP)

Let τ be the exit time from (0, ). There is an absolute constant C1 such that ∞

x V (x) P (τ>t) ! C1 1 ,x,t>0. (9) ∧ √t , - Heat kernels and exit times for isotropic L´evy processes Exit time

Exit time properties, one-dimensional case

Theorem (Kwasnicki, Malecki, Ryznar 2013’, AP)

Let τ be the exit time from (0, ). There is an absolute constant C1 such that ∞

x V (x) P (τ>t) ! C1 1 ,x,t>0. (9) ∧ √t , -

Lemma

Let 0

Lower bound, one-dimensional case

To prove the lower bound we observe that Exτ P x(τ>t) " P x(τ >t)+P x(τ <τ) " (0,R) + P x(τ <τ). (0,R) (0,R) t (0,R) Heat kernels and exit times for isotropic L´evy processes Exit time

Lower bound, one-dimensional case

To prove the lower bound we observe that Exτ P x(τ>t) " P x(τ >t)+P x(τ <τ) " (0,R) + P x(τ <τ). (0,R) (0,R) t (0,R) Hence from (9), for √t>V(x), and from (10) we obtain

x x x V (x) V (x) E τ(0,R) ! t(P (τ>t) P (τ(0,R) <τ)) ! t C1 − √t − V (R) , - √t = V (x)√t C1 . − V (R) , - Heat kernels and exit times for isotropic L´evy processes Exit time

Lower bound, one-dimensional case

To prove the lower bound we observe that Exτ P x(τ>t) " P x(τ >t)+P x(τ <τ) " (0,R) + P x(τ <τ). (0,R) (0,R) t (0,R) Hence from (9), for √t>V(x), and from (10) we obtain

x x x V (x) V (x) E τ(0,R) ! t(P (τ>t) P (τ(0,R) <τ)) ! t C1 − √t − V (R) , - √t = V (x)√t C1 . − V (R) , - √ C1 " Let t = 2 V (R) then, for 2V (x) C1V (R), we have C2 Exτ ! 1 V (x)V (R). (0,R) 4 Heat kernels and exit times for isotropic L´evy processes Exit time

The upper bound for the mean exit time from a d-dimensional ball follows immediately from the one-dimensional result since a ball can be included in a tangent strip. Heat kernels and exit times for isotropic L´evy processes Exit time

The upper bound for the mean exit time from a d-dimensional ball follows immediately from the one-dimensional result since a ball can be included in a tangent strip. Corollary

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

The upper bound for the mean exit time from a d-dimensional ball follows immediately from the one-dimensional result since a ball can be included in a tangent strip. Corollary

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

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

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

C 1 Ex " τBr , x

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H: Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

1. X is a subordinate Brownian motion governed by a special subordinator. In this case V is concave. Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

1. X is a subordinate Brownian motion governed by a special subordinator. In this case V is concave. A function φ is a special Bernstein function if it is a Bernstein function λ and φ(λ) is also a Bernstein function. Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

2. σ>0. That is the process has a non-trivial Brownian part. Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

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

3. Scale invariant Harnack inequality holds. Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

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

3. Scale invariant Harnack inequality holds. β For instance: d ! 3 and s− ψ(s) is almost increasing on (θ, ) for some ∞ β>0. Heat kernels and exit times for isotropic L´evy processes Exit time

Assumption H

V (z) V (y) " Hr V ′(x)(z y) whenever 0

Each of the following situations imply H:

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

3. Scale invariant Harnack inequality holds. β For instance: d ! 3 and s− ψ(s) is almost increasing on (θ, ) for some ∞ β>0. β γ d ! 1 and s− ψ(s) is almost increasing and s− ψ(s) is almost decreasing on (θ, ) for some β>0 and γ<2. ∞ Heat kernels and exit times for isotropic L´evy processes Exit time

Uniform estimate Heat kernels and exit times for isotropic L´evy processes Exit time

Uniform estimate

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

1 C(d)− Ex C(d) " τBr " , x

Uniform estimate

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

1 C(d)− Ex C(d) " τBr " , x

Futher examples

sum of two independent isotropic stable processes, Heat kernels and exit times for isotropic L´evy processes Exit time

Futher examples

sum of two independent isotropic stable processes, geometric stable processes, ψ(x) = log(1 + x α), 0 <α" 2, | | Heat kernels and exit times for isotropic L´evy processes Exit time

Futher examples

sum of two independent isotropic stable processes, geometric stable processes, ψ(x) = log(1 + x α), 0 <α" 2, | | conjugate to geometric stable processes, ψ(x)= x 2/ log(1 + x α), | | | | α (0, 2), ∈ ψ(x)= x 2/ log(1 + x 2) 1, α =2, | | | | − Heat kernels and exit times for isotropic L´evy processes Exit time

Futher examples

sum of two independent isotropic stable processes, geometric stable processes, ψ(x) = log(1 + x α), 0 <α" 2, | | conjugate to geometric stable processes, ψ(x)= x 2/ log(1 + x α), | | | | α (0, 2), ∈ ψ(x)= x 2/ log(1 + x 2) 1, α =2, | | | | − 1 isotropic stable-like processes, ν(x)= x d+α 1 x <1 and | | {| | } x 2ν(x)dx < .Thenψ(x) x α x 2, 0 <α<2: | | ∞ ≈| | ∧| | x α/2 α/2 . E τB (r x ) +(r x ) r + r , x

Dynkin approximating operator

x x Let tg(x)=E [g(Xτ ) g(x)]/E τ . A B(x,t) − B(x,t)

x E.g., if sD(x)=E τD and 0

Dynkin approximating operator

x x Let tg(x)=E [g(Xτ ) g(x)]/E τ . A B(x,t) − B(x,t)

x E.g., if sD(x)=E τD and 0

Theorem

d Let x0 R , r>0 and g(x)=V (δ (x)).Then, ∈ B(x0,r) CH " " r 0 lim supt 0 tg(x) if 0 <δB(x0,r)(x) 0 , which means that tV (x)=0in a halfspace. { } A Hence using a convenient coordinate system we may estimate t[g V ](x) if A − x =(x1, 0 ...,0). Heat kernels and exit times for isotropic L´evy processes Exit time Ex Proof of the lower bound of τBr .

x d Denote s(x)=E τB and g(x)=V (δB (x)), x R . By the Pruit’s estimate r r ∈ and subaddativity of V

2 2 s(x) V (r) g (x),δB (x) ! r/4. ≈ ≈ r Heat kernels and exit times for isotropic L´evy processes Exit time Ex Proof of the lower bound of τBr .

x d Denote s(x)=E τB and g(x)=V (δB (x)), x R . By the Pruit’s estimate r r ∈ and subaddativity of V

2 2 s(x) V (r) g (x),δB (x) ! r/4. ≈ ≈ r Hence there is C = C(d) so large that

Cs(x) V (r)g(x) ! 0 if δB (x) ! r/4. − r

Let 0 <δBr (x)

x d Denote s(x)=E τB and g(x)=V (δB (x)), x R . By the Pruit’s estimate r r ∈ and subaddativity of V

2 2 s(x) V (r) g (x),δB (x) ! r/4. ≈ ≈ r Hence there is C = C(d) so large that

Cs(x) V (r)g(x) ! 0 if δB (x) ! r/4. − r

Let 0 <δBr (x) 0 is small, then tg(x) " C1Hr/V (r), and |A |

t [(C1Hr + 1)s V (r)g](x)= (C1Hr + 1) V (r) tg(x) " 1. A − − − A − Heat kernels and exit times for isotropic L´evy processes Exit time Ex Proof of the lower bound of τBr .

x d Denote s(x)=E τB and g(x)=V (δB (x)), x R . By the Pruit’s estimate r r ∈ and subaddativity of V

2 2 s(x) V (r) g (x),δB (x) ! r/4. ≈ ≈ r Hence there is C = C(d) so large that

Cs(x) V (r)g(x) ! 0 if δB (x) ! r/4. − r

Let 0 <δBr (x) 0 is small, then tg(x) " C1Hr/V (r), and |A |

t [(C1Hr + 1)s V (r)g](x)= (C1Hr + 1) V (r) tg(x) " 1. A − − − A −

Let c = C (C1Hr + 1) and f = cs V (r)g, a continuous function. By ∨ − maximum principle, f cannot attain global minimum on Br B .Since \ 3r/4 f ! 0 elsewhere, f ! 0 everywhere. Heat kernels and exit times for isotropic L´evy processes Exit time

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ ⊂ ⊂ Heat kernels and exit times for isotropic L´evy processes Exit time

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ ⊂ ⊂

Let 1 (r)= inf ν(Bc(0,ρ)) ,r>0. J 0<ρ!r ψ2(1/ρ) 1 2 Heat kernels and exit times for isotropic L´evy processes Exit time

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ ⊂ ⊂

Let 1 (r)= inf ν(Bc(0,ρ)) ,r>0. J 0<ρ!r ψ2(1/ρ) 1 2

Theorem Let Condition H hold. Then, there exists constant C = C(d) such that for any bounded C1,1 set at scale r>0 1 H ψ2(1/r) " Ex " r Rd τD ψ∗(1/δD(x))ψ∗(1/r) C 2 2 ,x . CHr ( (r)) ψ (1/diamD) ∈ 3 J Heat kernels and exit times for isotropic L´evy processes Exit time

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ ⊂ ⊂

Let 1 (r)= inf ν(Bc(0,ρ)) ,r>0. J 0<ρ!r ψ2(1/ρ) 1 2

x 1 E τD ≈ ψ∗(1/δD(x))ψ∗(1/r) r with the comparability constant3 dependent on d, ψ and diamD . Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C).Then,forr>0 and ∈ ∩ tψ∗(1/r) " 1,

Px Px 1 (τBr >t)= (sup Xs " r) 1 , x

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C).Then,forr>0 and ∈ ∩ tψ∗(1/r) " 1,

Px Px 1 (τBr >t)= (sup Xs " r) 1 , x

The upper bound always holds. Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C).Then,forr>0 and ∈ ∩ tψ∗(1/r) " 1,

Px Px 1 (τBr >t)= (sup Xs " r) 1 , x

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C).Then,forr>0 and ∈ ∩ tψ∗(1/r) " 1,

Px Px 1 (τBr >t)= (sup Xs " r) 1 , x

The upper bound always holds. In some cases we do not need to assume both global scalings. In particular for the relativistic case the estimate holds, however we do not have the upper global scalings. The fact that the relativistic process is SBM helps. The scaling conditions above can be local and then we have a local version of the proposition. Then the constant in the lower bound depends on the radii in such a way that it is universal for all r " r0. Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C) and d>α,thenforr>0, ∈ ∩ t>0 we have

x 1 1 P (τ c >t) 1 ψ∗(1/r) , x >r, B(0,r) ≈ ∧ ψ∗(1/( x r)) √t ∨ | | 1 | |− , -2 where the" comparability# constant depend on d, α,c, α, C. Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C) and d>α,thenforr>0, ∈ ∩ t>0 we have

x 1 1 P (τ c >t) 1 ψ∗(1/r) , x >r, B(0,r) ≈ ∧ ψ∗(1/( x r)) √t ∨ | | 1 | |− , -2 where the" comparability# constant depend on d, α,c, α, C.

The condition d>α implies that the process is transient ( does not hit the balls with pr. 1). Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C) and d>α,thenforr>0, ∈ ∩ t>0 we have

x 1 1 P (τ c >t) 1 ψ∗(1/r) , x >r, B(0,r) ≈ ∧ ψ∗(1/( x r)) √t ∨ | | 1 | |− , -2 where the" comparability# constant depend on d, α,c, α, C.

The condition d>α implies that the process is transient ( does not hit the balls with pr. 1). The upper bound holds without the condition d>α,butitis not optimal e.g. in the classical isotropic stable case. The related result will be discussed later. Heat kernels and exit times for isotropic L´evy processes Survival probability

Applications

Proposition

Suppose that ψ WLSC(α, 0,c) WUSC(α, 0, C) and d>α,thenforr>0, ∈ ∩ t>0 we have

x 1 1 P (τ c >t) 1 ψ∗(1/r) , x >r, B(0,r) ≈ ∧ ψ∗(1/( x r)) √t ∨ | | 1 | |− , -2 where the" comparability# constant depend on d, α,c, α, C.

The condition d>α implies that the process is transient ( does not hit the balls with pr. 1). The upper bound holds without the condition d>α,butitis not optimal e.g. in the classical isotropic stable case. The related result will be discussed later. We conjecture that the bound holds for much larger class. In particular if t = we have the estimate of the probability of not hitting a ball: ∞

x ψ∗(1/r) P (τ c = ) 1 , x >r. B(0,r) ∞ ≈ ∧ ψ (1/( x r)) | | 1 ∗ | |− 2 " # Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ } Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B . Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B

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

x P (τD >t)= pD(t, x, y)dy. ! Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Survival probability. Dirichlet heat kernel.

The ﬁrst exit time from open D Rd: ⊂

τD =inf t>0:Xt / D { ∈ }

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

x pD(t, x, y)=p(t, x, y) E [τD

where p(t, x, y)=pt(y x). − x We have P (Xt B,τD >t)= pD(t, x, y)dy. ∈ B

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

x P (τD >t)= pD(t, x, y)dy. !

Green function: G (x, y)= ∞ p (t, x, y)dt. D 0 D . Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

x If x D,thentheP -distribution of (τD,Xτ ,Xτ ) restricted to ∈ D − D Xτ = Xτ has the density function D − ̸ D c (0, ) D D (s, u, z) pD(s, x, u)ν(z u). ∞ × × ∋ ,→ − " # Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

x If x D,thentheP -distribution of (τD,Xτ ,Xτ ) restricted to ∈ D − D Xτ = Xτ has the density function D − ̸ D c (0, ) D D (s, u, z) pD(s, x, u)ν(z u). ∞ × × ∋ ,→ − " # Integrating against ds, du and/or dz gives marginal distributions. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

x if x D and P (Xτ ∂D)=0,then ∈ D − ∈

x c P (Xτ dz)= GD(x, u)ν(z u)du dz on (D) . D ∈ − ,!D - Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

x if x D and P (Xτ ∂D)=0,then ∈ D − ∈

x c P (Xτ dz)= GD(x, u)ν(z u)du dz on (D) . D ∈ − ,!D - Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Known results about estimates od DHK

Zhang (2002) - smooth domains, BM Varopulos (2003) - Lipschitz domains, BM Chen, Kim, Song (2010), bounded smooth domains, isotropic stable processes Bogdan, Grzywny, Ryznar (2010), Lipschitz domains, isotropic stable processes Chen, Kim, Song (2012-2014) - smooth domains, subclasses of SBM. Kim, Song, Vondraˇcek (2014) - halfspace, subclasses of SBM. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Known results about estimates od DHK

Below f g means that there is a constant c>0 such that 1 ≈ c− g " f(x) " cg(x).Thisc is called the comparability constant. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded domains

Let D be an open bounded set and pt(0) is bounded for every t>0 tψ 1 2 (e− L ). Then the semigroup of integral operators on L (D) with kernels ∈ pD(t, x, y) " pt(0) is compact. The eigenvalues 0 <λ1(D) <λ2(D) " ... and orthonormal basis of eigenfunctions φ1 ! 0,φ2,φ3 ...;

λk(D)t φk(x)=e pD(t, x, z)φk(z)dz. ! Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded domains

λk(D)t φk(x)=e pD(t, x, z)φk(z)dz. ! Expansion: λk(D)t pD(t, x, y)= e− φk(x)φk(y). 4 Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded domains

λk(D)t φk(x)=e pD(t, x, z)φk(z)dz. ! Expansion: λk(D)t pD(t, x, y)= e− φk(x)φk(y). 4 Proposition

Let D be an open bounded set containing a ball of radius r.Thereisc = c(d)

2 d/2 1 r 2 diam D c− " λ (D)V (r) " c . diam D 1 r ( ) ( ) Recall that V (r) 1 . ≈ √ψ(1/r) Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Heat kernel on bounded smooth domains

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ c ⊂ ⊂ δD(x)=dist(x, D ). Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Heat kernel on bounded smooth domains

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ c ⊂ ⊂ δD(x)=dist(x, D ).

Theorem (Bogdan, Grzywny, Ryznar (2014))

Let ψ satisfy WLSC and WUSC.Thereisr0 = r0(d, ψ) > 0 such that for d 1,1 0 0, we have for all x, y Rd, t>0, ∈ x y 2 pD(t, x, y) P (τD >t/2)P (τD >t/2)p t V (r),x,y ≈ ∧ and " # Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Heat kernel on bounded smooth domains

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ c ⊂ ⊂ δD(x)=dist(x, D ).

Theorem (Bogdan, Grzywny, Ryznar (2014))

Let ψ satisfy WLSC and WUSC.Thereisr0 = r0(d, ψ) > 0 such that for d 1,1 0 0, we have for all x, y Rd, t>0, ∈ x y 2 pD(t, x, y) P (τD >t/2)P (τD >t/2)p t V (r),x,y ≈ ∧ and " #

x λ1(D)t V (δD(x)) P (τD >t) e− 1 . ≈ √t V (r) ∧ , ∧ - Recall that V (r) 1 ,whereψ is the symbol of the process. ≈ √ψ(1/r) Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Heat kernel on bounded smooth domains

D is of class C1,1 at scale r if D Rd, D is open, non-empty, r>0 and for ⊂ c every Q ∂D there are balls B(x′,r) D and B(x′′,r) D tangent at Q. ∈ c ⊂ ⊂ δD(x)=dist(x, D ).

Theorem (Bogdan, Grzywny, Ryznar (2014))

Let ψ satisfy WLSC and WUSC.Thereisr0 = r0(d, ψ) > 0 such that for d 1,1 0 0, we have for all x, y Rd, t>0, ∈ x y 2 pD(t, x, y) P (τD >t/2)P (τD >t/2)p t V (r),x,y ≈ ∧ and " #

x λ1(D)t V (δD(x)) P (τD >t) e− 1 . ≈ √t V (r) ∧ , ∧ - Recall that V (r) 1 ,whereψ is the symbol of the process. ≈ √ψ(1/r) Extends results obtained by Chen, Kim, Song in the case of subordinate Brownian motions governed by complete subordinators with weak scaling conditions. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded smooth domains

Let ψ satisfy global WLSC and WUSC. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded smooth domains

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

λ1(Br )t V (r x ) V (r y ) 2 pB(0,r)(t, x, y) e− −| | 1 −| | 1 p(t V (r),x,y), ≈ √t V (r) ∧ √t V (r) ∧ ∧ , ∧ -, ∧ - where comparability constant depends only on d and ψ. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded smooth domains

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

λ1(Br )t V (r x ) V (r y ) 2 pB(0,r)(t, x, y) e− −| | 1 −| | 1 p(t V (r),x,y), ≈ √t V (r) ∧ √t V (r) ∧ ∧ , ∧ -, ∧ - where comparability constant depends only on d and ψ. Moreover c (d) c (d) 1 " λ (B ) " 2 V 2(r) 1 r V 2(r) Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Bounded smooth domains

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

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

Corollary r Let φ1 be the (positive) eigenfunction corresponding to λ1(Br).Thereis c = c(d, ψ) such that

1 V (r x ) r V (r x ) d c− −| | " φ (x) " c −| | ,xR . rd/2V (r) 1 rd/2V (r) ∈

r r λ (B )t IU: p (t, x, y) φ (x)φ (y)e− 1 r ,t B(0,r) ≈ 1 1 →∞ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Upper bound

Lemma

x y pD(t, x, y) " pt/3(0)P (τD >t/3)P (τD >t/3). Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Upper bound

Lemma

x y pD(t, x, y) " pt/3(0)P (τD >t/3)P (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 !!

" pt/3(0) pD(t/3,x,z)dz pD(t/3,w,y)dw.

!x x ! " pt/3(0)P (τD >t/3)P (τD >t/3).

Under some conditions (eg. WLSC) for small t and x close to y we have pt/3(0) pt(x y). x≈ − x Also P (τD >t/3) P (τD >t), t small. The lemma is suﬃcient (almost) for ≈ the sharp upper bound for points which distance is small relatively to time (V 2( x y ) " t). | − | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Lemma

Consider open D1,D3 D such that dist(D1,D3) > 0.Let ⊂ D2 = D (D1 D3). If x D1, y D3 and t>0,then \ ∪ ∈ ∈ x pD(t, x, y) " P (Xτ D2)supp(s, z, y) D1 ∈ s

Lemma

Consider open D1,D3 D such that dist(D1,D3) > 0.Let ⊂ D2 = D (D1 D3). If x D1, y D3 and t>0,then \ ∪ ∈ ∈ x pD(t, x, y) " P (Xτ D2)supp(s, z, y) D1 ∈ s

Let h(r)= min z /r2, 1 ν( z )dz. {| | } | | !Rd Lemma

x x If r>0, D Br and x B ,thenP ( Xτ ! r) " 24 h(r) E τD. ⊂ ∈ r/2 | D | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Let h(r)= min z /r2, 1 ν( z )dz. {| | } | | !Rd Lemma

x x If r>0, D Br and x B ,thenP ( Xτ ! r) " 24 h(r) E τD. ⊂ ∈ r/2 | D |

Apply Dynkin’s formula,

τD x x d E g(Xs)ds = E g(Xτ ) g(x),xR , (11) A D − ∈ !0 Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Let h(r)= min z /r2, 1 ν( z )dz. {| | } | | !Rd Lemma

x x If r>0, D Br and x B ,thenP ( Xτ ! r) " 24 h(r) E τD. ⊂ ∈ r/2 | D |

Apply Dynkin’s formula,

τD x x d E g(Xs)ds = E g(Xτ ) g(x),xR , (11) A D − ∈ !0 to function φ(x)=g( x /r), such that g :[0, ) [ 1, 0], g(t)= 1 for | | ∞ ,→ − − 0 " t " 1/2, g(t)=0for t ! 1, Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Let h(r)= min z /r2, 1 ν( z )dz. {| | } | | !Rd Lemma

x x If r>0, D Br and x B ,thenP ( Xτ ! r) " 24 h(r) E τD. ⊂ ∈ r/2 | D |

Apply Dynkin’s formula,

τD x x d E g(Xs)ds = E g(Xτ ) g(x),xR , (11) A D − ∈ !0 to function φ(x)=g( x /r), such that g :[0, ) [ 1, 0], g(t)= 1 for | | ∞ ,→ − − 0 " t " 1/2, g(t)=0for t ! 1,

in fact g′′ = 16 on (1/2, 3/4) and g′′ = 16 on (3/4, 1), which gives − g′′ = 16 and g′ =4, ∥ ∥∞ ∥ ∥∞ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Let h(r)= min z /r2, 1 ν( z )dz. {| | } | | !Rd Lemma

x x If r>0, D Br and x B ,thenP ( Xτ ! r) " 24 h(r) E τD. ⊂ ∈ r/2 | D |

Apply Dynkin’s formula,

τD x x d E g(Xs)ds = E g(Xτ ) g(x),xR , (11) A D − ∈ !0 to function φ(x)=g( x /r), such that g :[0, ) [ 1, 0], g(t)= 1 for | | ∞ ,→ − − 0 " t " 1/2, g(t)=0for t ! 1,

in fact g′′ = 16 on (1/2, 3/4) and g′′ = 16 on (3/4, 1), which gives − g′′ = 16 and g′ =4, and ∥ ∥∞ ∥ ∥∞ 1 4sup g′(t) + sup g′′(t) = 24. t"0 | | 2 t"0 | | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Case of global WLSC and WUSC and D = BR

Proof of the upper bound. Assume that V 2( x y ) ! t and pick r : V 2(4r)=t,inparticular x y ! 4r. | − | | − | Recall that we have the following consequences of global weak scalings: if V 2( x y ) ! t then | − | t pt( x y ) pt( x y /3). | − | ≈ x y dV 2( x y ) ≈ | − | | − | | − | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Case of global WLSC and WUSC and D = BR

c x0 = Rx/ x ,D1 = B (x0,r) D, D3 = B(x, 2 x y /3) D. | | ∩ | − | ∩ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Case of global WLSC and WUSC and D = BR

c x0 = Rx/ x ,D1 = B (x0,r) D, D3 = B(x, 2 x y /3) D. | | ∩ | − | ∩

Note that z y ! x y /3 if z D2 = D (D1 D3). Radial monotonicity | − | | − | ∈ \ ∪ of pt implies

sup p(s, z, y) " sup ps( x y /3) " Cpt( x y ). s

Case of global WLSC and WUSC and D = BR

c x0 = Rx/ x ,D1 = B (x0,r) D, D3 = B(x, 2 x y /3) D. | | ∩ | − | ∩

Note that z y ! x y /3 if z D2 = D (D1 D3). Radial monotonicity | − | | − | ∈ \ ∪ of pt implies

sup p(s, z, y) " sup ps( x y /3) " Cpt( x y ). s

If u D1,z D3,then z u ! 2 x y /3 x x0 x0 u > x y /3. ∈ ∈ | − | | − | −| − |−| − | | − | Hence, pt( x y )) sup ν(z u) " ν((x y)/3) " C | − | . u D ,z D − − t ∈ 1 ∈ 3 Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

By the previous Lemma and subadditivity of V , Ex Ex x x τD1 τD1 P (Xτ D2) " 24h(r)E τD . (12) D1 ∈ 1 ≈ V 2(r) ≈ t Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

By the previous Lemma and subadditivity of V , Ex Ex x x τD1 τD1 P (Xτ D2) " 24h(r)E τD . (12) D1 ∈ 1 ≈ V 2(r) ≈ t By the estimate of the mean exit time,

x E τD " V (r)V (δD(x)) √tV (δD(x)). (13) 1 ≈ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

By the previous Lemma and subadditivity of V , Ex Ex x x τD1 τD1 P (Xτ D2) " 24h(r)E τD . (12) D1 ∈ 1 ≈ V 2(r) ≈ t By the estimate of the mean exit time,

x E τD " V (r)V (δD(x)) √tV (δD(x)). (13) 1 ≈ Therefore by Lemma 6, (12) and (13),

x pD(t, x, y) " P (Xτ D2)supp(s, z, y) D1 ∈ s

V (δD(x)) " C max , 1 pt( x y ). { √t } | − | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

2 For t ! V ( x y ) we have p (0) pt( x y ) (global scaling) hence | − | t/2 ≈ | − |

x V (δD(x)) pD(t, x, y) " pt/2(0)P (τD ! t/2) " C max , 1 pt( x y ). { √t } | − | Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

2 For t ! V ( x y ) we have p (0) pt( x y ) (global scaling) hence | − | t/2 ≈ | − |

x V (δD(x)) pD(t, x, y) " pt/2(0)P (τD ! t/2) " C max , 1 pt( x y ). { √t } | − | Finally by the semigroup property

pD(t, x, y)= pD(t/2,x,z)pD(t/2,z,y)dz ! V (δD(x)) V (δD(y)) " C max , 1 max , 1 pt/2(x, z)pt/2(z,y)dz { √t } { √t } ! V (δD(x)) V (δD(y)) = C max , 1 max , 1 pt(x, y). { √t } { √t } Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Lower bound

Lemma

Consider open D1,D3 D such that dist(D1,D3) > 0.Let ⊂ D2 = D (D1 D3). If x D1, y D3 and t>0,then \ ∪ ∈ ∈ Px Py pD(t, x, y) ! t (τD1 >t) (τD3 >t)infν( z u ) . u D1,z D3 | − | ∈ ∈ Py The lemma + estimates of (τBr >t) + the semigroup property + estimates of the free semigroup give the lower estimate of the Dirichlet kernel if t is small relative to the size of D. Details are more technical then for the upper bound. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2014))

1,1 Let ψ WLSC(α, 0) WUSC(α, 0) and d>α.LetD be a C at scale R1 and ∈ c ∩ d such that D BR . For all x, y R and t>0 we have ⊂ 2 ∈ x y pD(t, x, y) P (τD >t)P (τD >t)p(t, x, y), ≈ and x V (δD(x)) P (τD >t) 1 ≈ √t V (R1) ∧ , ∧ - with comparability constants C = C(d, ψ,R2/R1). Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2014))

c In particular for D =(BR1 ) comparability constants depend only on d and ψ. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2010))

c Let ψ(ξ)= ξ α, α (0, 2), and D = B(0, 1) . For all x, y Rd and t>0 we | | ∈ ∈ have x y p c (t, x, y) P (τD >t)P (τD >t)p(t, x, y), B(0,r) ≈ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2010))

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

α/2 δ (x) 1 D , if αt) ⎧ ≈ ⎪ ⎨⎪

⎪ ⎩⎪ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2010))

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

α/2 δ (x) 1 D , if αt) 1 1/2 , if α = d =1, ≈ ⎪ ∧ log(1+t ) ⎨⎪

⎪ ⎩⎪ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Exterior domains

Theorem (Bogdan, Grzywny, Ryznar (2010))

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

α/2 δ (x) 1 D , if αt) ⎧ 1 D , if α = d =1, ≈ log(1+t1/2) ⎪ ∧ α/2 ⎪ δα−1(x) δ (x) ⎨ D ∧ D (t1/α δ (x))α/2, if α>d=1, (t1/α δ (x))α−1 D ∨ D ∧ ∨ ⎪ with comparability⎩⎪ constants C = C(d, α). Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Halfspace-like domains

Let ψ satisfy global WLSC and WUSC. Denote Ha = (x1,...,xd):xd >a . { } Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Halfspace-like domains

Let ψ satisfy global WLSC and WUSC. Denote Ha = (x1,...,xd):xd >a . { } Theorem (Bogdan, Grzywny, Ryznar (2014))

1,1 d Let D be C at scale R and Ha D Hb. Then for all x, y R and t>0, ⊂ ⊂ ∈ x y pD(t, x, y) P (τD >t)P (τD >t)p(t, x, y) and ≈ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Halfspace-like domains

Let ψ satisfy global WLSC and WUSC. Denote Ha = (x1,...,xd):xd >a . { } Theorem (Bogdan, Grzywny, Ryznar (2014))

1,1 d Let D be C at scale R and Ha D Hb. Then for all x, y R and t>0, ⊂ ⊂ ∈ x y pD(t, x, y) P (τD >t)P (τD >t)p(t, x, y) and ≈

x V (δD(x)) P (τD >t) 1, ≈ √t ∧ and comparability constants depend only on d, ψ, a b and R. − Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Halfspace-like domains

Let ψ satisfy global WLSC and WUSC. Denote Ha = (x1,...,xd):xd >a . { } Theorem (Bogdan, Grzywny, Ryznar (2014))

1,1 d Let D be C at scale R and Ha D Hb. Then for all x, y R and t>0, ⊂ ⊂ ∈ x y pD(t, x, y) P (τD >t)P (τD >t)p(t, x, y) and ≈

x V (δD(x)) P (τD >t) 1, ≈ √t ∧ and comparability constants depend only on d, ψ, a b and R. − Corollary (Bogdan, Grzywny, Ryznar (2014)) Let D = H. Then for all x, y Rd and t>0, ∈ x y pD(t, x, y) P (τD >t)P (τD >t)p(t, x, y) and ≈

x V (δD(x)) P (τD >t) 1, ≈ √t ∧ and comparability constants depend only on d and ψ. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Examples

Isotropic stable processes Relativistic stable processes Sum of two independent isotropic stable processes Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Examples

Isotropic stable processes Relativistic stable processes Sum of two independent isotropic stable processes d Let Xt ν( x )dx and α1,α2 (0, 2), ν(r)=f(r)r− ,where ∼ | | ∈ 1 α f(r)=(r− log(r +1/r)) 1 , or α f(r)=(r log(r +1/r))− 1 , or α r− 1 if 0 1. $ Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

Examples

One dimensional point reccurent processes

Theorem

Let ψ WLSC(α, 0), α>1. If D =[ R, R]c and x >R, ∈ − | |

x V ( x R) 2 P (τD >t) | |− 1,t" V (R) ≈ √t V (R) ∧ ∧ and x V ( x R)V ( x ) 2 P (τD >t) | |− | | 1,t>V(R). ≈ x tψ 1(1/t) ∧ | | − The comparability constants depend only on the scaling characteristics. Heat kernels and exit times for isotropic L´evy processes Dirichlet heat kernel estimates

References

K. Bogdan, T. Grzywny, and M. Ryznar. Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 2010. K. Bogdan, T. Grzywny, and M. Ryznar. Density and tails of unimodal convolution semigroups. J. Funct. Anal. 2014. K. Bogdan, T. Grzywny, and M. Ryznar. Barriers, exit time and survival probability for unimodal Lévy processes, PTRF 2015. K. Bogdan, T. Grzywny, and M. Ryznar. Dirichlet heat kernel for unimodal Lévy processes. SPA, 2014. Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimates for Dirichlet fractional Laplacian. J. Eur. Math. Soc. 2010 Z.-Q. Chen, P. Kim, R. Song, Dirichlet Heat Kernel Estimates for Rotationally Symmetric Lévy processes. Proc. Lond. Math. Soc. 2014. Heat kernels and exit times for isotropic Lévy processes Dirichlet heat kernel estimates

T. Grzywny. Intrinsic ultracontractivity for Lévy processes. Probab. Math. Statist. 2008. T. Grzywny. On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 2014. T. Grzywny and M. Ryznar. Potential theory of one-dimensional geometric stable processes. Colloq. Math. 2012.