Transition probability estimates for long range random walks

Abstract

Let $(M,d,\mu )$ be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on $M$ symmetric with respect to $\mu$ and whose one-step transition density is comparable to $(V_h(d(x,y))\varphi (d(x,y){)}^{-1}$, where $\varphi$ is a positive continuous regularly varying function with index $\beta \in (0,2)$ and $V_h$ is the homogeneous volume growth function. Extending several existing work by other authors, we prove global upper and lower bounds for $n$-step transition probability density that are sharp up to constants.

31 pages; incorporated referee comments; published in the New York Journal of Mathematics (http://nyjm.albany.edu/j/2015/21-32.html)

References in corpus (3)

• Random walks and isoperimetric profiles under moment conditions
• Anomalous threshold behavior of long range random walks
• Random walks under slowly varying moment conditions on groups of polynomial volume growth

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• Anomalous threshold behavior of long range random walks
• Quasi-ergodicity of compact strong Feller semigroups on $L^2$
• Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces
• Heat kernel estimates for symmetric jump processes with mixed polynomial growths
• Stability results for symmetric jump processes on metric measure spaces with atoms. Long version