Reachable sets under Kolmogorov dynamics in the probability simplex

Abstract

The finite-time reachability problem - whether two probability vectors can be transformed in a time t through a generator from a given set - is a key question in Markovian dynamics. We address this problem for normalized Kolmogorov generators with tools from differential geometry. The shortest time T connecting an ordered pair of states defines an (asymmetric) quasi-distance in the probability simplex, which distinguishes between outgoing and incoming reachable sets, and captures the inherent irreversibility of Markov dynamics. The Finsler metric corresponding to normalized generators with bounded diagonal entries is characterized, and time-independent evolutions that saturate speed limits are presented. Analytical boundaries of both reachable sets are derived and the ratio of their corresponding volumes assess the bounds obtained. The asymmetry with respect to the uniform probability vector is shown to be related to the Kullback-Leibler relative entropy.