Continuity of asymptotic characteristics for random walks on hyperbolic groups (Q366113)

The asymptotic entropy of a random walk on a countable group \(G\) determined by a probability measure \(\mu\) is the limit \[ h(\mu )=\lim\limits_n\frac{H(\mu^{*n})}n \] of normalized entropies of \(n\)-fold convolutions of the measure \(\mu\). The authors study conditions under which \(h(\mu )\) is continuous in \(\mu\). In their words, the approach ``is based on using conditional random walks and amounts to checking uniformity in the strip criterion for the identification of the Poisson boundary.'' Applications to word hyperbolic groups are described. In a remark added after the manuscript was accepted, the authors mention some recent works establishing the Lipschitz continuity and differentiability of the function \(h(\mu )\); see, in particular [\textit{F. Ledrappier}, Ann. Probab. 41, No. 5, 3582--3605 (2013; Zbl 1283.60010)].