Upper heat kernel estimates for nonlocal operators via Aronson's method

TL;DR

This paper extends Aronson's method to establish upper heat kernel estimates for nonlocal operators with bounded jumping kernels, providing a unified approach in Euclidean and metric measure spaces.

Contribution

The paper introduces a novel extension of Aronson's technique to nonlocal operators with coercive energy forms and bounded jumping kernels, broadening the scope of heat kernel estimates.

Findings

Established upper heat kernel bounds for nonlocal operators.

Extended Aronson's method to Euclidean and metric measure spaces.

Provided detailed proofs and potential for further generalizations.

Abstract

In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson's proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed.