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.