Some aspects of large time behavior of the heat kernel: an overview with perspectives

TL;DR

This paper reviews various developments concerning the large time behavior of the positive minimal heat kernel associated with second-order parabolic operators on domains and manifolds, emphasizing general results and settings.

Contribution

It provides an overview of recent advances in understanding the asymptotic behavior of heat kernels for non-symmetric operators on complex geometric structures.

Findings

Summarizes key results on heat kernel decay rates.

Highlights the role of geometric properties in heat kernel behavior.

Discusses generalizations to non-symmetric operators and manifolds.

Abstract

We discuss a variety of developments in the study of large time behavior of the positive minimal heat kernel of a time independent (not necessarily symmetric) second-order parabolic operator defined on a domain M in $R^d$, or more generally, on a noncompact Riemannian manifold M. Our attention is mainly focused on general results in general settings.