Empiric stochastic stability of physical and pseudo-physical measures

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Date(s) - 12/09/2017
11 h 00 min - 12 h 00 min


We define the empiric stochastic stability of an invariant probability measure by adapting to the finite-time scenario, the classical definition of stochastic stability. We prove that, for any continuous system, an invariant measure is empirically stochastically stable if and only if it is physical. We define also the empiric stochastic stability of a weak*-compact set of invariant measures, instead of a single measure.
Even if the system has no physical measure, we show that it still has minimal empirically stochastically stable sets of measures.
Besides, 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 empirically stochastically measures or sets of measures satisfy Pesin Entropy Formula.


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