Point-wise estimates for nonlocal heat kernel of convolution type operators

TL;DR

This paper derives sharp point-wise estimates for the heat kernel of nonlocal convolution operators, revealing how its large-time behavior varies across different spatial regimes, with implications for understanding nonlocal diffusion processes.

Contribution

It provides the first comprehensive point-wise bounds for nonlocal heat kernels across all spatial regimes, highlighting differences from classical kernels.

Findings

Heat kernel behaves classically when |x| << t

Distinct behavior observed when |x| ~ t or |x| >> t

Sharp upper bounds established for all regions

Abstract

The paper deals with point-wise estimates for the heat kernel of a nonlocal convolution type operator with a kernel that decays at least exponentially at infinity. It is shown that the large time behaviour of the heat kernel depends essentially on whether $|x|\ll t$, or $|x|\sim t$, or $|x|\gg t$. We obtain sharp point-wise upper bounds for the heat kernel in all these regions. In the first region the nonlocal heat kernel behaves like the classical one, while in the other regions we observe an essential difference.