Two-sided eigenvalue estimates for subordinate processes in domains

Let X={Xt , t 0} be a symmetric Markov process in a state space E and D an open set of E. Denote by XD the subprocess of X killed upon leaving D. Let S ={St , t 0} be a subordinator with Laplace exponent that is independent of X. The processes X := {XSt , t 0} and (XD) := {XD St , t 0} are called the subordinate processes of X and XD, respectively. Under some mild conditions, we show that, if {− n, n 1} and {− n, n 1} denote the eigenvalues of the generators of the subprocess of X killed upon leaving D and of the process XD respectively, then n ( n) for every n 1. We further show that, when X is a spherically symmetric -stable process in Rd with ∈ (0, 2] and D ⊂ Rd is a bounded domain satisfying the exterior cone condition, there is a constant c = c(D)>0 such that c ( n) n ( n) for every n 1.The above constant c can be taken as 1/2 if D is a bounded convex domain in Rd . In particular, when X is Brownian motion in Rd , S is an /2-subordinator (i.e., ( )= /2) with ∈ (0, 2), and D is a bounded domain in Rd satisfying the exterior cone condition, {− n, n 1} and {− n, n 1} are the eigenvalues for the Dirichlet Laplacian in D and for the generator of the spherically symmetric -stable process killed upon exiting the domain D, respectively. In this case, we have c /2 n n /2 n for every n 1. When D is a bounded convex domain in Rd , we further show that c 1 Inr(D)− 1 c 2 Inr(D)− , where Inr(D) is the inner radius of D and c2 >c1 >0 are two constants depending only on the dimension d. © 2005 Elsevier Inc. All rights reserved