Dirichlet Heat Kernel Estimates for Rotationally Symmetric L\'evy processes

TL;DR

This paper establishes sharp two-sided estimates for the transition densities of a broad class of rotationally symmetric Levy processes, providing explicit bounds in various geometric settings and confirming a conjecture in potential analysis.

Contribution

It provides the first sharp two-sided Dirichlet heat kernel estimates for a wide class of symmetric Levy processes with general Levy exponents.

Findings

Sharp two-sided estimates for transition densities in open sets

Explicit bounds in terms of Levy exponents and boundary distance

Affirmative answer to a conjecture in potential analysis

Abstract

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a \kappa-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by \Psi(\xi)= \phi(|\xi|^2) with \phi being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of \Psi, the distance function to the boundary of D and the jumping kernel of X, which give an affirmative answer to the conjecture posted in [Potential Anal., 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy processes with general Levy exponents. We also derive an explicit lower bound estimate for symmetric Levy processes on R^d in terms of their Levy exponents.