Associated with random walks on groups, we consider three fundamental quantities: entropy, speed and volume growth (exponential growth rate of the group). The fundamental inequality due to Guivarc'h tells that the entropy does not exceed the speed times the olume growth. Vershik (2000) asked about the genuine equality case. We focus on hyperbolic groups, and characterise the equality case; namely, the equality holds if and only if the harmonic measure and a natural measure--a Patterson-Sullivan measure--on the boundary are equivalent. We also show some open problems related to this question. I start with a history of this problem and mention about recent progresses as well. All the notion will be explained during the talk.