[论文解读] Stochastic Flows on Non-compact Manifolds

Xue-Mei Li|arXiv (Cornell University)|May 31, 2021

Mathematical Dynamics and Fractals参考文献 36被引用 27

一句话总结

本工作研究非紧凑流形上 SDE 的解流的存在性及性质，通过导数流引入强 p-完备性，并推导针对形式的热半群的基于鞅的表示，以及其几何/拓扑意义。

ABSTRACT

I was asked to make my, by now quite old PhD thesis, available on the arxiv, for parts of it was never submitted for publication. The thesis offers a systematic study of stochastic differential equations (SDEs) on non-compact spaces. In particular we solve the open problem on strong completeness. An SDE is strongly complete if its solution can be chosen to depend continuously in space and in time for all time. The question is whether non-explosion, with possibly additional assumptions, implies strong completeness. Strong completeness of an SDE implies that its solution depends continuously on the initial condition, opening up possibility for numerical solutions, and the existence of a perfect Cocycle (a basic assumption on random dynamical systems). This was known only for compact manifolds and for linear state spaces, methods for either are not applicable to a general space. We also obtain existence of the global smooth solution flow of SDEs on $R^n$ (sometimes allowing substantial growth of the derivative of the coefficients, removing the global Lipschitz continuity condition). Non-explosion, the $C_0$-property, and the derivative flow are studied. We showed Bismut-Witten Laplacians are essentially self-adjojnt, paving the way for studying theirs semigroups acting on functions and on differential forms. We relate the Markovian semi-group on differential one forms with the semi-group $P_t$ on functions (inter-twining), find a method for proving path integration formulas for $dP_tf$, path integration formula for semi-group on differential forms, moment bounds for the derivative flows. Relation are obtained between intrinsic topological and geometrical properties of the manifold and that of SDEs. Information on the homotopy and cohomology of the manifolds are obtained from moment stability of the stochastic flows.

研究动机与目标

• 研究非紧凑流形上随机流的存在性及性质。
• 在导数流条件下建立强完备性和强 p-完备性。
• 将扩散过程的行为与底层流形的几何/拓扑特征联系起来。

提出的方法

• 通过将解流与关于初始数据的导数耦合，引入导数流方法。
• 给出关于系数增长的条件，以实现强 p-完备性和非爆炸。
• 推导互相缠绕关系 dP_t f = δP_t(df)，并研究在函数和微分形式上的半群。
• 给出对加权热半群在形式上的基于鞅的积分表示，扩展 Bismut 的公式。
• 使用弱一致覆盖技术分析无穷远处的行为与非爆炸。
• 通过热半群的 Lp 有界性，将矩稳定性、π1(M) 消失与上同调的联系建立起来。

实验结果

研究问题

• RQ1 在系数和导数流满足何种条件时，非紧凑流形上存在全局定义的光滑解流？
• RQ2 强 p-完备性如何与扩散的非爆炸及矩稳定性相关？
• RQ3 导数流增长限制对诸如 π1(M) 和调和形式等拓扑特征有何含义？
• RQ4 是否可以获得作用于非紧凑流形上函数与微分形式的半群的缠绕与梯度公式？
• RQ5 鞅方法如何为超越1-形式的形式热半群提供积分表示？

主要发现

• 在导数流的可积性界限下，可以实现强 1-完备性和强 p-完备性。
• 对于 R^n 的开集上的 SDE，以及在二阶曲面有界的子流形上的布朗运动系统，非爆炸结果。
• 在导数流的条件下，互缠属性 dP_t f = δP_t(df) 成立，能够表示梯度。
• 对微分形式的热半群的 Lp 有界性和压缩性的框架导致同调为零的结果。
• 基于鞅的 d(P_t f) 的积分表示公式将 Bismut 的公式拓展到形式和高阶形式。
• 通过梯度 SDE 推导出的 q-形式（q≥1）的 d(P_t φ) 公式，在某些情形下提供对数热核的梯度。