Stochastic Analysis on Manifolds

本研究的目的是研究随机过程的性质及其底层空间的拓扑和几何性质，以及扩散过程理论的应用。有限维流形和无限维流形都要考虑（特别是黎曼流形的路径空间和环路空间）。研究将集中在以下问题：随机微分方程的（光滑）解流的存在性，随机动力系统的渐近性（包括矩估计，几乎确定估计，递归/瞬态和稳定流形理论）以及各种情况下的分部积分公式。将特别注意受白噪声干扰的哈密顿系统的动力学和几何。倒向随机微分方程和时空鞅驱动的随机微分方程的研究也将包括在内。研究方向为随机动力系统及相关课题。研究随机微分方程解的渐近性、解析性质和几何性质，以及底层空间的几何和拓扑。包括受白噪声干扰的哈密顿系统在内的随机微分方程的特殊例子将被研究。

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.