α-Continuity Properties of Stable Processes

Abstract

Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be the symmetric $α$-stable process killed upon exiting $D$. Each element of the set $\{λ_i^α\}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$, regarded as a function of $α\in(0,2)$, is right continuous. In addition, if $D$ is Lipschitz and bounded, then each $ λ_i^α$ is continuous in $α$ and the set of associated eigenfunctions is precompact. We also prove that if $D$ is a domain of finite Lebesgue measure, then for all $0<α<β\leq 2$ and $i\geq 1$, \[λ_i^α\leq [ λ^β_i]^{α/β}.\] Previously, this bound had been known only for $β=2$ and $α$ rational.