Positive Harmonic Functions of Transformed Random Walks
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In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a discrete group under the change of probability measure by a randomized stopping time. We show that this space remains unchanged after applying a bounded randomized stopping time.
KeywordsRandom walk Harmonic functions Martin boundary
Mathematics Subject Classification (2010)60J50 60B15 31C05
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The authors wish to use the opportunity to thank Vadim Kaimanovich and Wolfgang Woess for numerous valuable suggestions on an earlier version of this paper. We also thank Sébastian Gouzël and Iddo Ben-Ari for fruitful discussions. During the completion of this project, the second author was partially supported by the DFG grant DI506/14-1.
- 1.Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential theory—surveys and problems (Prague, 1987), vol. 1344 of Lecture Notes in Math, pp 1–23. Springer, Berlin (1988)Google Scholar
- 8.Forghani, B., Kaimanovich, V.A.: Boundary preserving transformations of random walks in preparation (2016)Google Scholar
- 9.Furstenberg, H.: Random Walks and Discrete Subgroups of Lie Groups. In: Advan. Probab. Related Topic, vol. 1, pp 1–63. Dekker, New York (1971)Google Scholar
- 15.Sawyer, S.A.: Martin boundaries and random walks. In: Harmonic functions on trees and buildings (New York, 1995), vol. 206 of Contemp. Math. Amer. Math. Soc.Providence, RI, pp 17–44 (1997)Google Scholar
- 16.Solan, E., Tsirelson, B.V.N.: Random stopping times in stopping problems and stopping games. arXiv:1211.5802 (2012)