ERGODIC PROPERTIES OF WEAK ASYMPTOTIC PSEUDOTRAJECTORIES FOR SET-VALUED DYNAMICAL SYSTEMS

Abstract

A successful method to describe the asymptotic behavior of various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes is to relate them to an appropriately chosen limit semiflow. Benaïm and Schreiber (2000) define a general class of such stochastic processes, which they call weak asymptotic pseudotrajectories and study their ergodic behavior. In particular, they prove that the weak* limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic semiflow. Continuing a program started by Benaïm, Hofbauer and Sorin (2005), we generalize the ergodic results mentioned above to weak asymptotic pseudotrajectories relative to set-valued dynamical systems.