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Mathematische Nachrichten
Volume 284, Issue 14-15 p. 1889-1902
Research Article

On harmonic functions for trace processes

Panki Kim,

Corresponding Author

Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea

Panki Kim, Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea. Phone: +82-2-880-4077, Fax: +82-2-887-4694

Renming Song, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. Phone: +1-217-244-6604, Fax: +1-217-333-9576

Zoran Vondraček, Department of Mathematics, University of Zagreb, Zagreb, Croatia. Phone: +385-1-4605-792, Fax: +385-1-4680-335.

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Renming Song,

Corresponding Author

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Panki Kim, Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea. Phone: +82-2-880-4077, Fax: +82-2-887-4694

Renming Song, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. Phone: +1-217-244-6604, Fax: +1-217-333-9576

Zoran Vondraček, Department of Mathematics, University of Zagreb, Zagreb, Croatia. Phone: +385-1-4605-792, Fax: +385-1-4680-335.

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Zoran Vondraček,

Corresponding Author

Department of Mathematics, University of Zagreb, Zagreb, Croatia

Panki Kim, Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea. Phone: +82-2-880-4077, Fax: +82-2-887-4694

Renming Song, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. Phone: +1-217-244-6604, Fax: +1-217-333-9576

Zoran Vondraček, Department of Mathematics, University of Zagreb, Zagreb, Croatia. Phone: +385-1-4605-792, Fax: +385-1-4680-335.

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First published: 08 July 2011
https://doi.org/10.1002/mana.200910008
Citations: 4

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 hf on the whole space E. We show that the extension hf 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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