# Two-sided eigenvalue estimates for subordinate processes in domains

Zhen Qing Chen, Renming Song

Research output: Contribution to journalArticlepeer-review

## Abstract

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 := {XDSt, t ≥} 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, then μn ≤ φ (λn) f or 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 that c φ (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 have c λnα/2 ≤ μn ≤ λnα/2 for every n ≥ 1. When D is a bounded convex domain in Rd, we further show that c1α Inr(D) ≤ μ1 ≤ c2α Inr(D), where Inr (D) is the inner radius of D and c2 > C1 > 0 are two constants depending only on the dimension d.

Original language English (US) 90-113 24 Journal of Functional Analysis 226 1 https://doi.org/10.1016/j.jfa.2005.05.004 Published - Sep 1 2005

## Keywords

• Bernstein function
• Borel right process
• Brownian motion
• Complete Bernstein function
• Dirichlet form
• Eigenvalues
• Lévy process
• Resolvent
• Semigroup
• Spherically symmetric stable process
• Subordination
• Subordinator

• Analysis