On estimates of transition density for subordinate Brownian motions with Gaussian components in $C^{1,1}$-open sets

TL;DR

This paper derives precise two-sided bounds for the transition density of subordinate Brownian motions with Gaussian components in Euclidean space and $C^{1,1}$-domains, leading to sharp Green function estimates.

Contribution

It provides the first sharp two-sided bounds for the transition density of subordinate Brownian motions with Gaussian components in $C^{1,1}$-open sets.

Findings

Sharp two-sided bounds for transition density in ${f R}^d$

Sharp Green function estimates in $C^{1,1}$-domains

Applicable to processes with scaling order up to 2

Abstract

We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates.