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About the $ L^2$ analyticity of Markov operators on graphs


Author: Joseph Feneuil
Journal: Proc. Amer. Math. Soc. 146 (2018), 1793-1805
MSC (2010): Primary 60J10; Secondary 35P05, 47D07
DOI: https://doi.org/10.1090/proc/13825
Published electronically: November 7, 2017
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Abstract: Let $ \Gamma $ be a graph and $ P$ be a reversible random walk on $ \Gamma $. From the $ L^2$ analyticity of the Markov operator $ P$, we deduce that an iterate of odd exponent of $ P$ is `lazy', that is, there exists an integer $ k$ such that the transition probability (for the random walk $ P^{2k+1}$) from a vertex $ x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $ \Gamma $ but only a polynomial control of the volume.


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Additional Information

Joseph Feneuil
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, Minnesota 55455
Email: jfeneuil@umn.edu

DOI: https://doi.org/10.1090/proc/13825
Keywords: Markov chains, graphs, discrete analyticity, lazy random walks
Received by editor(s): September 15, 2015
Received by editor(s) in revised form: July 12, 2016, August 24, 2016, and May 17, 2017
Published electronically: November 7, 2017
Additional Notes: The author was supported by the ANR project “Harmonic Analysis at its Boundaries”, ANR-12-BS01-0013-03.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2017 American Mathematical Society

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