Stochastic Analysis on Manifolds

小行星9803574 本研究的目的是研究随机过程的性质 连同底层空间的拓扑和几何性质，以及 扩散过程理论的应用。有限维和无限维 流形被认为是（特别是道路和循环空间的黎曼 歧管）。 调查将集中于以下问题： 随机微分方程的（光滑）解流， 动力系统（包括矩估计，几乎必然估计， 递归/瞬态，和稳定流形理论）和积分的部分公式， 各种情况。将特别注意的动力学和几何 受白色噪声扰动的哈密顿系统。倒向随机的研究 时空驱动的微分方程和随机微分方程 还将包括鞅。 研究人员将继续她在随机动力系统的研究， 相关话题。的渐近性、分析性质和几何性质 随机微分方程的解将与几何学一起研究 以及底层空间的拓扑结构。随机微分的特殊例子 应研究包括受白色噪声扰动的哈密顿系统在内的方程。

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.