Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

Heat kernel estimates for stable-like processes on d-sets

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity $$e^{-c_0 (x, y)|x-y|}\, \int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \, \nu (d\alpha)$$
 where ν is a probability measure on $$[\alpha_1, \alpha_2]\subset (0, 2)$$
 , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c
 0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ1 and γ2, where either γ2 ≥ γ1 > 0 or γ1 = γ2 = 0. This example contains mixed symmetric stable processes on $${\mathbb{R}}^n$$
 as well as mixed relativistic symmetric stable processes on $${\mathbb{R}}^n$$
 . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

Heat kernel estimates for jump processes of mixed types on metric measure spaces

We construct a continuous transition density of the semigroup generated by $${\Delta^{\alpha/2} + b(x)\cdot \nabla}$$
 for $${1 < \alpha < 2, d\ge 1}$$
 and b in the Kato class $${\mathcal{K}_d^{\alpha-1}}$$
 on $${\mathbb{R}^d}$$
 . For small time the transition density is comparable with that of the fractional Laplacian.

Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators

We investigate properties of functions which are harmonic with respect to -stable processes on d-sets such as the Sierpi«ski gasket or carpet. We prove the Harnack inequality for such functions. For every process we estimate its transition density and harmonic measure of the ball. We prove continuity of the density of the harmonic measure. We also give some results on the decay rate of harmonic functions on regular subsets of the d-set. In the case of the Sierpi«ski gasket we even obtain the Boundary Harnack Principle.

/pdf/harnack-inequality-for-stable-processes-on-d-sets-542j4fjbm0.pdf

Harnack inequality for stable processes on d-sets

http://prac.im.pwr.wroc.pl/~sztonyk/Prace/Densities_Derivatives.pdf

Estimates of transition densities and their derivatives for jump Lévy processes

We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack’s inequality. We also estimate the corresponding semigroup and the potential kernel.

/pdf/estimates-of-the-potential-kernel-and-harnack-s-inequality-2s4i0f1lgv.pdf

Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian

We derive explicitly the coupling property for the transition semigroup of a Levy process and gradient estimates for the associated semigroup of transition operators This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively Our results can be applied to a large class of Levy processes, including stable Levy processes, layered stable processes, tempered stable processes and relativistic stable processes

Coupling property and gradient estimates of L\'{e}vy processes via the symbol

We derive explicitly the coupling property for the transition semigroup of a Levy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Levy processes, including stable Levy processes, layered stable processes, tempered stable processes and relativistic stable processes.

/pdf/coupling-property-and-gradient-estimates-of-levy-processes-57ponzfdzw.pdf

Paweł Sztonyk

Papers

Harnack inequality for stable processes on d-sets

Estimates of transition densities and their derivatives for jump Lévy processes

Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian

Coupling property and gradient estimates of L\'{e}vy processes via the symbol

Coupling property and gradient estimates of Lévy processes via the symbol