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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: 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. 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: 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. 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 } ∈ | | d/α 1 d t tψ(1/ x ) t− = ψ− (1/t) , = | | = ctν(x) x d+α x d | | | | " # 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. 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 } ∈ | | d/α 1 d t tψ(1/ x ) t− = ψ− (1/t) , = | | = ctν(x) x d+α x d | | | | " # 1 d tψ(1/ x ) pt(x) min ψ− (1/t) , | | ≈ x d $ | | ' % & Heat kernels and exit times for isotropic L´evy processes Recall pt(x) is radially decreasing.
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