Continuity of eigenvalues of subordinate processes in domains

Abstract

Let X={Xt,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S(n)={S(n)t, t ≥ 0} be a subordinator with Laplace exponent ϕn and S={St,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S(n). In this paper we consider the subordinate processes Open image in new window and Open image in new window and their subprocesses Open image in new window and Xϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of Open image in new window and Xϕ,D are all discrete, with Open image in new window being the eigenvalues of the generator of Open image in new window and Open image in new window being the eigenvalues of the generator of Xϕ,D. We show that, if limn→∞ϕn(λ)=ϕ(λ) for every λ>0, then