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 φ that is independent of X. The processes Xφ≔{XSt,t⩾0} and (XD)φ≔{XStD,t⩾0} are called the subordinate processes of X and XD, respectively. Under some mild conditions, we show that, if {-μn,n⩾1} and {-λn,n⩾1} denote the eigenvalues of the generators of the subprocess of Xφ killed upon leaving D and of the process XD respectively, thenμn⩽φ(λn)for every n⩾1.We further show that, when X is a spherically symmetric α-stable process in Rd with α∈(0,2] and D⊂Rd is a bounded domain satisfying the exterior cone condition, there is a constant c=c(D)>0 such thatcφ(λn)⩽μn⩽φ(λn)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 α/2-subordinator (i.e., φ(λ)=λα/2) with α∈(0,2), and D is a bounded domain in Rd satisfying the exterior cone condition, {-λn,n⩾1} and {-μn,n⩾1} are the eigenvalues for the Dirichlet Laplacian in D and for the generator of the spherically symmetric α-stable process killed upon exiting the domain D, respectively. In this case, we havecλnα/2⩽μn⩽λnα/2for every n⩾1.When D is a bounded convex domain in Rd, we further show thatc1αInr(D)-α⩽μ1⩽c2αInr(D)-α,where Inr(D) is the inner radius of D and c2>c1>0 are two constants depending only on the dimension d.