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On harmonic functions for trace processes
Author(s) -
Kim Panki,
Song Renming,
Vondraček Zoran
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910008
Subject(s) - trace (psycholinguistics) , mathematics , harmonic function , extension (predicate logic) , harmonic , function (biology) , space (punctuation) , pure mathematics , state space , harnack's inequality , combinatorics , mathematical analysis , physics , computer science , statistics , philosophy , linguistics , quantum mechanics , evolutionary biology , biology , programming language , operating system
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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