9803574 Li The objectives of this research are to study the properties of stochastic processes together with the topological and geometrical properties of the underlying spaces, and applications of the theory of diffusion processes. Both finite and infinite dimensional manifolds are to be considered (in particular path and loop spaces of Riemannian manifolds). Investigations will concentrate on the following problems: the existence of a (smooth) solution flow to stochastic differential equations, asymptotics of stochastic dynamical systems (including moment estimates, almost sure estimates, recurrence/transience, and stable manifold theory) and integration by parts formulae in various situations. Special attention will be paid to the dynamics and geometry of Hamiltonian systems perturbed by white noise. The study of backward stochastic differential equations and stochastic differential equations driven by space-time martingales will also be included. The investigator shall continue her research in stochastic dynamical systems and related topics. The asymptotics, analytical properties and geometrical properties of solutions of stochastic differential equations are to be studied together with the geometry and topology of the underlying spaces. Particular examples of stochastic differential equations including Hamiltonian systems perturbed by white noise shall be investigated.
