Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

Abstract

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.

Intrinsic Ultracontractivity for Stable Semigroups on Unbounded Open Sets

“…[1-3, 19, 20, 33]. This problem is closely related to estimates on the eigenfunctions of the Dirichlet Laplacian, on the expected lifetime of some conditioned processes, and to the parabolic backward boundary Harnack inequality [8,21,22,24,27], see also [35][36][37]. There are similar connections for the case of stable processes, and this motivates our study.…”

Potential theory of subordinate killed Brownian motion

“…Despite its usefulness, the potential theory of subordinate killed Brownian motions has been studied only sporadically, see e.g. [17,33,16] for stable subordinators, and [34,36] for more general subordinators. In particular, [36] contains versions of HI and BHP (with respect to the subordinate killed Brownian motion in a bounded Lipschitz domain D) which are very weak in the sense that the results are proved only for non-negative functions which are harmonic in all of D. Those results are easy consequences of the fact that there is a one-to-one correspondence between non-negative harmonic functions (in all of D) with respect to W D and those with respect to Y D .…”

On the Potential Theory of One-dimensional Subordinate Brownian Motions with Continuous Components

“…Thus the subordinator S is the sum of a unit drift and an α/2-stable subordinator, while X is the sum of a Brownian motion and a symmetric α-stable process. We will use the fact that S is a special subordinator, that is, the restriction to (0, ∞) of the potential measure of S has a decreasing density with respect to the Lebesgue measure (for more details see [15] or [17] …”