Global Heat Kernel Estimates for Fractional Laplacians in Unbounded Open Sets

TL;DR

This paper establishes precise global heat kernel and Green function estimates for fractional Laplacians in unbounded open sets, including half-space-like and exterior sets, using boundary distance and comparison techniques.

Contribution

It provides explicit, sharp heat kernel and Green function estimates for fractional Laplacians in unbounded C^{1,1} open sets, extending previous results to more general geometries.

Findings

Sharp heat kernel estimates for all t>0 and x,y in the domain.

Green function estimates for Dirichlet fractional Laplacian.

Extension of estimates to censored stable processes in exterior sets.

Abstract

In this paper, we derive global sharp heat kernel estimates for symmetric alpha-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C^{1,1} open sets in R^d: half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p_D(t,x,y) for all t>0 and x, y\in D. Our approach is based on the idea that for x and y in $D$ far from the boundary and t sufficiently large, we can compare p_D(t,x,y) to the heat kernel in a well understood open set: either a half-space or R^d; while for the general case we can reduce them to the above case by pushing $x$ and $y$ inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.