  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 it to
an appropriately chosen limit semiflow. Bena\"im 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. Bena\"im, Hofbauer and Sorin (2005) generalised this approach to
set-valued dynamical systems. We pursue the analogy by extending to these
settings the ergodic properties of weak asymptotic pseudotrajectories.
