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Long range estimates for Markov chains. (English) Zbl 0583.60063
The author considers a reversible Markov chain \(\{Z_ n\}\) with a countable state space. The main result of the paper is the following ”a priori estimate” \[ P\{Z_ n=j| Z_ 0=i\}\leq Cn^{3/4}\exp (-d^ 2(i,j)/Cn) \] where d(i,j) is the minimal length of a path connecting i and j, and having non-zero probability. As an application of this estimate the author gives criteria for existence of bounded harmonic functions on some finitely generated groups.
Reviewer: Y.Kifer

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F10 Large deviations
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F05 Central limit and other weak theorems
60J60 Diffusion processes