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Carne-Varopoulos bounds for centered random walks. (English) Zbl 1099.60049

Let \(X=(X_{t})_{t\geq 0}\) be a discrete parameter Markov chain taking on values in a discrete set \(V\). First, not assuming any algebraic structure of \(V\), the author introduces a “centering condition” that generalizes the classical reversibility condition. The former is defined in terms of a splitting into oriented cycles of the weighted oriented graph endowed in \(V\) by the transition probabilities of \(X\). A main result is an extension of the Carne-Varopoulos inequality to not necessarily reversible Markov chains (Theorem 2.8). Next, \(V\) is assumed to be a discrete group and \(X\) to be a random walk. Relationships between different notions of centering are investigated. While Carne-Varopoulos bounds can be used to bound the rate of escape of a random walk from its starting state, in the case of random walks on a group it is shown that the rate of escape vanishes if and only if the Poisson boundary is trivial (Proposition 3.11). This generalizes known results for symmetric random walks.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G50 Sums of independent random variables; random walks
60J50 Boundary theory for Markov processes
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References:

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