Symmetric Markov chains on \Bbb Z^d with unbounded range

2005/08/30 by Richard F. Bass, Takashi Kumagai, Bass, Richard F. +1
Mathematics · #60J25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60J25

paper · pdf · doi:10.48550/arxiv.math/0508619

arxiv created 2005/08/30 · openalex publication_date 2005/08/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider symmetric Markov chains on \Bbb Z^d where we do \bf not assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on \Bbb R^d corresponding to an elliptic operator in divergence form.

Citations

• Transition Probabilities for Symmetric Jump Processes

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