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.


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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