An Abramov formula for stationary spaces of discrete groups

Let (G,mu) be a discrete group equipped with a generating probability measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G, starting from the identity, returns to Gamma with probability one. Let theta be the hitting measure, or the distribution of the position in which the random walk first hits Gamma. We prove that the Furstenberg entropy of a (G,mu)-stationary space, with respect to the induced action of (Gamma,theta), is equal to the Furstenberg entropy with respect to the action of (G,mu), times the index of Gamma in G. The index is shown to be equal to the expected return time to Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu), we prove that the random walk entropy of (Gamma,theta) is equal to the random walk entropy of (G,mu), times the index of Gamma in G.