Pointwise estimates for heat kernels of convolution-type operators

Abstract

We study the large‐time behaviour of the fundamental solution of parabolic equations with an elliptic part being non‐local convolution‐type operator. We assume that this operator is a generator of a Markov jump process, and that its convolution kernel decays at least exponentially at infinity. The fundamental solution shows rather different asymptotic behaviour depending on whether | x | ≲ t , or t ≪ | x | ≪ t , or | x | ∼ t , or | x | ≫ t . In each of these regions we obtain sharp pointwise estimates for the fundamental solution.