Trace estimates for relativistic 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 $α$-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 $t^{2/α}t^{-d/α}$ for $C^{1,1}$ open sets and of order $t^{1/α}t^{-d/α}$ for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders $t^{-d/α}$ and $t^{(2-d)/α}$ for $C^{1,1}$ open sets, and, when $α\in (0, 1]$, between the orders $t^{-d/α}$ and $t^{(1-d)/α}$ for Lipschitz open sets.