Stochastic Analysis on Manifolds

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The proposed research falls into the following broad areas of stochastic analysis: stochastic dynamical systems on manifold, the geometry of stochastic flows and Hormander type second order differential operators, analysis on path spaces, and the interplay between the dynamical behavior of stochastic dynamical systems and the geometry and topology of the underlying spaces. The investigator shall concentrate on (1) L^2 theory for differential forms on path and loop spaces; (2) linear connections associated to Hormander type second order elliptic operators; (3) stochastic flows and in particular stochastic differential equations associated to Hamiltonian systems. The study of lattice models with spin variable in a manifold is also included, as is the study of bounded and L^2 harmonic forms on finite dimensional manifolds. The investigator will study these problems using techniques from stochastic differential equations. This work would lead to a better understanding of random perturbation of dynamical systems and geometric analysis of infinite dimensional spaces. The proposed project concentrates in the areas of stochastic dynamical systems and analysis on path spaces. Path spaces are the spaces of trajectories of particles or other physical objects. Methods from stochastic analysis, especially the techniques arising from the study of stochastic differential equations, shall be employed. Stochastic differential equations are themselves very important objects to study as they arise naturally from physics, engineering, biology, and finance, to model the influence of white noise. The study of such equations on a variety of geometric objects such as spheres in connection with the curvature of such spaces arises naturally from the mathematics and also in applications, for example in the computation of recovering data by filtering out the white noise in the transmission.