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

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