Ergodic Properties of weak Asymptotic Pseudotrajectories for Set-valued Dynamical Systems

TL;DR

This paper extends the ergodic analysis of weak asymptotic pseudotrajectories to set-valued dynamical systems, linking stochastic processes to deterministic semiflows and exploring their invariant measures.

Contribution

It generalizes the ergodic properties of weak asymptotic pseudotrajectories from single-valued to set-valued dynamical systems, broadening the theoretical framework.

Findings

Weak* limit points are almost surely invariant for the deterministic semiflow.

Extension of ergodic properties to set-valued dynamical systems.

Provides a unified approach for stochastic and deterministic asymptotic behaviors.

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 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.