Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow 8 and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely 8-invariant measures; (2) given any semiflow 8, there exists a weak asymptotic pseudotrajectory X of 8 such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for 8; and (3) if X is an asymptotic pseudotrajectory for a semiflow 8, then conditions on 8 that ensure convergence of the empirical measures are derived.