Manifold-valued stochastic processes and Geometry of infinite dimensional path manifolds – ProbaGeo

Since K. Itô introduced the parallel translations along Brownian paths on Riemannian manifolds at ICM (Stockholm, 1962), interactions between Probability theory and Differential geometry have been become a fertile and wide field of research. In the present program, we plan to develop stochastic analysis on manifold (including Lorentz manifold and singular space) valued stochastic processes and the geometry of some infinite dimensional curved spaces. More precisely, we shall concentrate our attention to the themes: A. Towards Green-Wu's conjectures; B. Martingale approach to some singular spaces; C. Long time behavior of relativistic diffusions; D. Geometry of infinite dimensional path manifolds, including the group of diffeomorphisms of the circle, Riemannian path spaces, loop groups and Wasserstein spaces, these infinite dimensional curved spaces constitute interesting and important examples where Probability and Geometry are merged. These seemingly different aspects are deep connected to each others: path spaces provide a setup for Bismut's integration by parts formula and useful informations on the solvability of Dirichlet problem at infinity on Cartan-Hadamard manifolds since the lower bound of sectional curvatures has been replaced by the lower bound of Ricci curvature. The aim of this project is to merge and synergize research from a French group of mathematicians to deal with challenging problems arisen in the above themes, which require the effort of all of us.