Trace estimates for relativistic stable processes

TL;DR

This paper analyzes the small-time asymptotic behavior of the trace of the semigroup for killed relativistic stable processes in specific open sets, providing detailed expansion formulas and error bounds.

Contribution

It establishes new asymptotic expansions for the trace of relativistic stable processes, including error estimates, in bounded $C^{1,1}$ and Lipschitz open sets, extending prior stable process results.

Findings

Derived asymptotic expansion of the trace as t approaches zero.

Provided error bounds of order $t^{2/\alpha}t^{-d/\alpha}$ and $t^{1/\alpha}t^{-d/\alpha}$.

Identified additional terms in the expansion compared to stable processes.

Abstract

In this paper, we study the asymptotic behavior, as the time $t$ goes to zero, of the trace of the semigroup of a killed relativistic $\alpha$-stable process in bounded $C^{1,1}$ open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of $t$ of the trace with an error bound of order $t^{2/\alpha}t^{-d/\alpha}$ for $C^{1,1}$ open sets and of order $t^{1/\alpha}t^{-d/\alpha}$ for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders $t^{-d/\alpha}$ and $t^{(2-d)/\alpha}$ for $C^{1,1}$ open sets, and, when $\alpha\in (0, 1]$, between the orders $t^{-d/\alpha}$ and $t^{(1-d)/\alpha}$ for Lipschitz open sets.