Tuesday, November 10, 2009
10:30 AM in REC 307
Krzysztof Bogdan
Visiting Professor of Mathematics, Purdue University, Permanent Position at Wroclaw University of Technology, Poland

Heat Kernel Estimates for the Fractional Laplacian

Abstract

I will report a joint work with Tomasz Grzywny and Michał Ryznar from WUT, Poland, on approximate factorization of the heat kernel of the Dirichlet fractional Laplacian in Lipschitz domains (the paper is on arXiv).

Theorem. If $D$ is a Lipschitz domain then for $0<tleq 1$ and all MATH,

MATH

Here $p(t,x,y)$ is the heat kernel of the fractional Laplacian on $QTR{bf}{R}^d$, $p_D(t,x,y)$ is the heat kernel of the fractional Laplacian with Dirichlet condition on $D^c$, and

MATH

is the survival probability of the corresponding isotropic $alpha$-stable Lévy process.