Two-sided eigenvalue estimates for subordinate processes in domains

Let X = {X_t, t ≥ 0} be a symmetric Markov process in a state space E and D an open set of E. Denote by X^D the subprocess of X killed upon leaving D. Let S = {S_t, t ≥ 0} be a subordinator with Laplace exponent φ that is independent of X. The processes X^φ := {X_St, t ≥ 0} and (X^D)^φ := {X^D_St, t ≥} are called the subordinate processes of X and X^D, respectively. Under some mild conditions, we show that, if {-μ_n, n ≥ 1} and {-λ_n, n ≥ 1} denote the eigenvalues of the generators of the subprocess of X^φ killed upon leaving D and of the process X^D respectively, then μ_n ≤ φ (λ_n) f or every n ≥ 1. We further show that, when X is a spherically symmetric α-stable process in R^d with α ∈ (0,2)] and D ⊂ R^d is a bounded domain satisfying the exterior cone condition, there is a constant c = c (D) > 0 such that c φ (λ_n) ≤ μ_n ≤ φ (λ_n) for every n ≥ 1. The above constant c can be taken as 1/2 if D is a bounded convex domain in R^d. In particular, when X is Brownian motion in R^d, S is an α/2-subordinator (i.e., φ(λ) = λ^α/2) with α ∈ (0, 2), and D is a bounded domain in R^d satisfying the exterior cone condition, {-λ_n, n ≥ 1} and {-μ_n, 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 have c λ_n^α/2 ≤ μ_n ≤ λ_n^α/2 for every n ≥ 1. When D is a bounded convex domain in R^d, we further show that c₁^α Inr(D)^-α ≤ μ₁ ≤ c₂^α Inr(D)^-α, where Inr (D) is the inner radius of D and c₂ > C₁ > 0 are two constants depending only on the dimension d.