Abstract
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.
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Grigor’yan, A., Huang, X. & Masamune, J. On stochastic completeness of jump processes. Math. Z. 271, 1211–1239 (2012). https://doi.org/10.1007/s00209-011-0911-x
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DOI: https://doi.org/10.1007/s00209-011-0911-x