Heat kernel estimates for subordinate Brownian motions

TL;DR

This paper derives sharp two-sided estimates for transition probabilities of subordinate Brownian motions with Laplace exponents that vary regularly at infinity, extending understanding of their potential theory and scaling behaviors.

Contribution

It provides new sharp estimates for transition probabilities of subordinate Brownian motions with Laplace exponents that vary regularly at infinity, including non-standard scaling cases.

Findings

Established sharp two-sided transition probability estimates.

Applied estimates to Green function (potential) calculations.

Proved equivalence between lower scaling condition and near diagonal upper estimates.

Abstract

In this article we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, e.g. \[ \phi(\lambda)=\frac{\lambda}{\log(1+\lambda)}-1 \quad \text{ or }\quad \phi(\lambda)=\frac{\lambda}{\log(1+\lambda^{\beta/2})},\ \beta\in (0,2)\, \] that correspond to subordinate Brownian motions with scaling order that is not necessarily strictly between 0 and 2. These estimates are applied to estimate Green function (potential) of subordinate Brownian motion. We also prove the equivalence of the lower scaling condition of the Laplace exponent and the near diagonal upper estimate of the transition estimate.