Empiric stochastic stability of physical and pseudo-physical measures

Abstract

We define the empiric stochastic stability of an invariant measure in the finite-time scenario, the classical definition of stochastic stability. We prove that an invariant measure of a continuous system is empirically stochastically stable if and only if it is physical. We also define the empiric stochastic stability of a weak*-compact set of invariant measures instead of a single measure. Even when the system has not physical measures it still has minimal empirically stochastically stable sets of measures. We prove that such sets are necessarily composed by pseudo-physical measures. Finally, we apply the results to the one-dimensional C1-expanding case to conclude that the measures of empirically stochastically sets satisfy Pesin Entropy Formula.