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On harmonic functions for trace processes

✍ Scribed by Panki Kim; Renming Song; Zoran Vondraček

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
173 KB
Volume
284
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let X be a standard Markov process with state space E and let F be a closed subset of E. A nonnegative function f on F is extended probabilistically to a function h~f~ on the whole space E. We show that the extension h~f~ is harmonic with respect to X provided that f is harmonic with respect to Y, the trace process on F of the process X. A consequence is that if the Harnack inequality holds for X, it also holds for the trace process Y. Several examples illustrating the usefulness of the result are given.

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