We study two examples of infinite dimensional stochastic processes. Situations and techniques involved are quite varied, however in both cases we achieve a progress in describing their long time behaviour.

The first case concerns interacting particle system of diffusions. We construct rigorously the process using finite dimensional approximation and the notion of martingale solution. The existence of invariant measure for the process is proved. The novelty of the results lies in the fact, that our methods enable us to consider such examples, where the generator of the diffusion is subelliptic.

The other project is related to stochastic partial differential equations and their stability properties. In particular it is shown that Robbins-Monro procedure can be extended to infinite dimensional setting. Thus we achieve results about pathwise convergence of solution. To be able to define corresponding solution, we rely on so-called variational approach to stochastic partial differential equations as pioneered by E. Pardoux, N. Krylov and B. Rozovskii. Our examples covers situations such as p-Laplace operator or Porous medium operator.