Let mu a probability measure on a countable group G. The Avez entropy of mu provides a way of quantifying the randomness of the random walk on G associated with mu. We build a new framework to compute asymptotic quantities associated with the mu-random walk on G, using constructions that arise from harmonic analysis on groups. We introduce the notion of emph{convolution entropy} and show that, under mild assumptions on mu, it coincides with the Avez entropy of mu when G has the rapid decay property. Subsequently, we apply our results to stationary dynamical systems consisting of an action of a group with the rapid decay property on a probability space, and give several characterizations for when the Avez entropy coincides with the Furstenberg entropy of the stationary space. This leads to a characterization of Zimmer amenability for stationary dynamical systems whenever the acting group has the rapid decay property.

This is based on joint work with B. Anderson-Sackaney, T. de Laat and E. Samei.