Mathematical Sciences: Stochastic Analysis on Manifolds

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9626142 Li, Xue-Mei ABSTRACT The aim of this proposal is to study stochastic flows on manifolds, including their existence, geometric properties and applications to the study of partial differential equations, non-linear equations, Hamiltonian systems perturbed by white noise and the geometry and topology of the underlying space. One of the objects to study, which will be a useful tool in many directions, is the affine connection associated to a stochastic differential equation. This connection provides the geometrical structure to obtain good estimates for the derivative flow of stochastic differential equations. Those structures arising from degenerate systems are particularly investigated: there are problems of definition and of the existence of singularities. Although the main work is on finite dimensional systems infinite dimensional ones are also considered. Bismut type formulae and integration by parts formulae on path spaces connected to the above are also examined. On a more directly geometrical level the concepts of spectral positivity for curvatures are considered together with its consequences for the geometry of the manifold. Stochastic flows are typically solutions of stochastic differential equations (equations perturbed by white noise). Properties of stochastic flows including the stability and invariance are particularly interesting when considered together with the geometry of the underlying spaces. The techniques described in the proposal are used to study nonlinear equations including the blow ups and regularities of the solutions. The investigator also examines properties of local martingales, which finds its applications in mathematical finance.