Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor

A succesful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well chosen mean differential equation. Under an attainability condition, convergence to a given attractor of the flow induced by this dynamical system was proved to occur with positive probability (Benaïm, 1999) for a class of Robbins Monro algorithms. Benaïm et al. (2005) generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.

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