The proposed research concentrates on probabilistic methods connecting and unifying certain topics of Pure and Applied Mathematics. Geometric aspects of diffusion processes and stochastic differential equations, both in finite and infinite dimension, are a common central theme in all parts of the project. Methods of Stochastic Analysis and the Stochastic Calculus of Variations (Malliavin Calculus) are combined with tools from Geometric Analysis to study evolutions of random dynamical systems. New probabilistic tools are developed and their efficiency is tested on eminent problems. The topics range from stochastic approaches to finite time blow-up in geometric evolution equations, like in the deformation of Riemannian metrics under Ricci flow, to phenomena of path transitions as they appear in molecular modelling under rough energy landscapes. Another topic of interest is the analysis of degenerate high dimensional systems in financial engineering, as required in multi-asset markets with degenerate cross-volatilities for a fast and stable numerical computation of option price sensitivities in the hedging and risk management of financial claims.