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

A successful 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, Benaïm proved that convergence to a given attractor of the flow induced by this dynamical system occurs with positive probability for a class of Robbins Monro algorithms. Benaïm, Hofbauer, and Sorin 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.