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Stochastic Completeness of Graphs: Bounded Laplacians, Intrinsic Metrics, Volume Growth and Curvature

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Abstract

The goal of this article is to survey various results concerning stochastic completeness of graphs. In particular, we present a variety of formulations of stochastic completeness and discuss how a discrepancy between uniqueness class and volume growth criteria in the continuous and discrete settings was ultimately resolved via the use of intrinsic metrics. Along the way, we discuss some equivalent notions of boundedness in the sense of geometry and of analysis. We also discuss various curvature criteria for stochastic completeness and discuss how weakly spherically symmetric graphs establish the sharpness of results.

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Acknowledgements

I would like to thank Józef Dodziuk for suggesting such a fruitful area of study and for sustaining me over the years. I am also happy to acknowledge the inspiration and support offered by Isaac Chavel and Leon Karp. Furthermore, I would like to thank Alexander Grigor\('\)yan for encouragement and support, early in my career up until the present moment. And also many thanks to my coauthors who contributed to this story and whom I also consider to be good friends. In alphabetical order: Sebastian Haeseler, Bobo Hua, Xueping Huang, Matthias Keller, Daniel Lenz, Jun Masamune, Florentin Münch and Marcel Schmidt. Finally, I would like to thank Isaac Pesenson for the invitation to contribute this article and the referees for helpful comments.

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  1. Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Radosław K. Wojciechowski

  2. York College of the City University of New York, 94-20 Guy R. Brewer Blvd., Jamaica, NY, 11451, USA

    Radosław K. Wojciechowski

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The author gratefully acknowledges financial support from PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York, and the Collaboration Grant for Mathematicians, funded by the Simons Foundation.

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Wojciechowski, R.K. Stochastic Completeness of Graphs: Bounded Laplacians, Intrinsic Metrics, Volume Growth and Curvature. J Fourier Anal Appl 27, 30 (2021). https://doi.org/10.1007/s00041-021-09821-6

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