Random dynamics and Geometric analysis

I propose a project in mathematics on the theory of stochastic processes and their applications. I am particularly interested in the interactions between certain random dynamics and the geometry of the space on which they are defined. The focus will be on three parallel ambitious research directions:(i) Stochastic partial differential equations on fractals, with a focus on the stochastic heat equation and the parabolic Anderson model;(ii) Diffusion methods in sub-Riemannian geometry with a focus on novel notions of curvature bounds;(iii) Random matrices with a focus on its connections with the theory of flag manifolds.