Mathematische Zeitschrift

, Volume 271, Issue 3, pp 1211–1239

On stochastic completeness of jump processes


  • Alexander Grigor’yan
    • Department of MathematicsUniversity of Bielefeld
    • Department of MathematicsUniversity of Bielefeld
  • Jun Masamune
    • Department of Mathematics and StatisticsPennsylvania State University

DOI: 10.1007/s00209-011-0911-x

Cite this article as:
Grigor’yan, A., Huang, X. & Masamune, J. Math. Z. (2012) 271: 1211. doi:10.1007/s00209-011-0911-x


We prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces: if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete. We use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure: if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete, where the cubic volume growth is sharp.


Jump processesRandom walksStochastic completenessNon-local Dirichlet formsPhysical Laplacian

Mathematics Subject Classification (2000)

Primary 60J75Secondary 60J2560J2705C81
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© Springer-Verlag 2011