On transformations of Markov chains and Poisson boundary
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- by Iddo Ben-Ari and Behrang Forghani PDF
- Trans. Amer. Math. Soc. 373 (2020), 2207-2227 Request permission
Abstract:
A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions under some additional conditions. Our work was motivated by and is analogous to the well-studied case when the Markov chain is a random walk on a discrete group.References
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Additional Information
- Iddo Ben-Ari
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 751387
- Email: iddo.ben-ari@uconn.edu
- Behrang Forghani
- Affiliation: Department of Mathematics, Bowdoin College, Brunswick, Maine 04011
- MR Author ID: 1132817
- Email: bforghan@bowdoin.edu
- Received by editor(s): February 6, 2017
- Received by editor(s) in revised form: November 6, 2018, May 22, 2019, and August 16, 2019
- Published electronically: November 15, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 2207-2227
- MSC (2010): Primary 31C05, 31C35, 60J50
- DOI: https://doi.org/10.1090/tran/7975
- MathSciNet review: 4068296