Estimates of Dirichlet heat kernels for subordinate Brownian motions

TL;DR

This paper provides explicit, sharp two-sided estimates for the transition densities of subordinate Brownian motions in $C^{1,1}$ domains, applicable to a broad class with various scaling orders, enhancing understanding of their probabilistic behavior.

Contribution

It establishes the first explicit two-sided estimates for subordinate Brownian motions in $C^{1,1}$ domains with general scaling orders, extending previous results.

Findings

Sharp two-sided estimates derived for transition densities

Estimates expressed explicitly in terms of dimension, boundary distance, and Laplace exponent

Applicable to subordinate Brownian motions with diverse scaling orders

Abstract

In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and Laplace exponent of the corresponding subordinator only.