Heat kernel estimates for Markov processes in bounded sets with jump kernels decaying at the boundary

2025/12/15 by Cho, Soobin, Kim, Panki, Song, Renming +1
#35K08 #60J45 #60J46 #60J50 #60J76 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 60J35 #Probability (math.PR) #Secondary 31C25

paper · doi:10.48550/arxiv.2512.12991

Abstract

In this paper, we study two types of purely discontinuous symmetric Markov processes X in bounded smooth subsets of \mathbb R^d: conservative processes and processes killed either upon approaching the boundary of the set or by a killing potential κ. The jump kernel of X is of the form J(x,y)=\cal B(x,y)|x-y|^-d-α, α∈ (0,2), where the function \cal B(x,y) decays to 0 at the boundary and is described in terms of two O-regularly varying functions and one slowly varying function. Under the conditions, introduced in \citeCKSV24, on \cal B(x,y) and on the killing potential κ, we establish sharp two-sided estimates on the heat kernel of X: in Lipschitz sets when X is conservative, and in C^1,1 open sets for the killed process.

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