Carne-Varopoulos bounds for centered random walks (Q2497168)

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.