## Abstract

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 c_{1}^{α} Inr(D)^{-α} ≤ μ_{1} ≤ c_{2}^{α} Inr(D)^{-α}, where Inr (D) is the inner radius of D and c_{2} > C_{1} > 0 are two constants depending only on the dimension d.

Original language | English (US) |
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Pages (from-to) | 90-113 |

Number of pages | 24 |

Journal | Journal of Functional Analysis |

Volume | 226 |

Issue number | 1 |

DOIs | |

State | 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

## ASJC Scopus subject areas

- Analysis