Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature

2020/09/29 by Wojciechowski, Radosław K. · 2 citations
#58J35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Primary 39A12 #Probability (math.PR) #Secondary 05C63

paper · doi:10.48550/arxiv.2010.02009

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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• Form uniqueness for graphs with weakly spherically symmetric ends
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