Dynamical Systems Seminar

We address the strong stochastic stability of a broad class of discrete-time dynamical systems - non-uniformly expanding maps - when some random noise is introduced in the deterministic dynamics. A weaker form of stochastic stability for this systems was established by J. F. Alves and V. Araújo (2003) in the sense of convergence of the physical stationary probability measure to the SRB probability measure in the weak* topology. We present a strategy to improve this result in order to obtain the strong stochastic stability, i.e., the convergence of the density of the physical measure to the density of the SRB measure in the L¹-norm, and in a more general framework of random perturbations. We illustrate our main result for two examples of non-uniformly expanding maps: the first is related to an open class of local diffeomorphisms introduced by J. F. Alves, C. Bonatti and M. Viana (2000) and the second to Viana maps - an example with critical set introduced by M. Viana (1997). This is a joint work with J. F. Alves.