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Volume and time doubling of graphs and random walks: The strongly recurrent case. (English) Zbl 1021.60037

The author proves upper and lower off-diagonal, sub-Gaussian transition probability estimates for strongly recurrent random walks under necessary and sufficient conditions. Besides the known conditions, volume doubling and the elliptic Harnack inequality, a new property of time doubling is introduced. If \(E(x,R)\) is the expected time until exit from the ball of radius \(R\) centred at \(x\) for a random walk starting from \(x\), this property requires that \(E(x,2R)\leq D_E E(x,R)\) for a constant \(D_E>0\).

MSC:

60G50 Sums of independent random variables; random walks
31C20 Discrete potential theory
35K05 Heat equation
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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