"The proposed research and training project aims at giving a major progress in the development of the interaction between classical and random dynamical systems, a key subject for the European Research Area with many applications. In particular, we want to deepen understanding of the following topics: (i) Ergodic properties of random billiards. We will analyse a broad class of billiard geometries, study skew-product representations for the billiard flow, and investigate central limit theorems. (ii) Absolute continuity of laws of transformed Brownian motions. We would like to give necessary and sufficient conditions to guarantee that, when transformed according to a family of diffeomorphisms, the laws of a Brownian motion on the path space are absolutely continuous with respect to the original laws. Reduction techniques and Lie-group theoretical arguments are expected to be used in this research. (iii) Bifurcation of random dynamical systems. Several notions of bifurcation available for random systems will be compared. Random dynamical systems with bounded noise will receive special attention and will be studied with control theory techniques. Under bounded noise perturbations, we want to characterise invariant measures and their support as well as domains of attraction. In order to achieve these research objectives, the researcher will be specifically trained by the dynamical systems group (DynamIC) of Imperial College London in ergodic theory, bifurcation theory for autonomous and non-autonomous dynamical systems, and analysis on infinite dimensional spaces."