Stochastic Dynamics: Finite and Infinite Dimensional

本研究计划的重点是在随机外力影响下发展的物理和生物系统的时间演化。特别令人感兴趣的是物理系统内部的内在驱动力是高度不规则的情况。在物理学、流体力学、流行病学和金融数学中使用的许多数学模型中普遍存在这种不规则性。我们的目标是全面理解这些模型的演变。具体而言，计划是建立物理学和生物学中大量数学模型的存在，这些模型足够丰富，可以广泛应用，同时足够灵活，可以容纳在估计基础模型参数时不可避免的统计误差。本文拟开展的随机动力系统研究主要包括三个方面：1.随机动力系统研究；布朗运动和Borel可测线性有界漂移系数驱动的奇异随机微分方程随机不变流形的存在性和空间规律性。对于这样的奇异边，一个稳定流形定理应该成立。考虑到驱动向量场的极端粗糙度，不变流形的存在性和可微性将是惊人的和反直觉的。结果将通过马利文演算和乘法遍历理论得到。2. 紧黎曼流形上光滑向量场驱动的随机微分方程的Kupka-Smale猜想的证明策略。这种基本结果背后的基本直觉是，为了使双曲向量场族表示可行的物理/生物模型的动力学，必须在所有“合理”向量场的集合中“丰富”，同时对系数中的扰动不敏感。3. 线性随机偏微分方程有限维随机不变子空间密度（相对于Lebesgue测度）的存在性和光滑性。

This research program is focused on the time evolution of physical and biological systems which develop under the influence of random external forces. Of particular interest is the scenario when the intrinsic driving forces within the physical system are highly irregular. Such irregularities are prevalent in many mathematical models used in physics, fluid mechanics, epidemiology and financial mathematics. The goal is to reach a comprehensive understanding of the evolution of such models. In particular, the plan is to establish the existence of a large class of mathematical models in physics and biology which is sufficiently rich to allow for a broad range of applications and at the same time flexible enough to accommodate unavoidable statistical errors in estimating the parameters of the underlying models. The proposed research consists of three major undertakings concerning stochastic dynamical systems:1. Existence and spatial regularity of random invariant manifolds for singular stochastic differential equations (sdes) driven by Brownian motion and Borel measurable linearly bounded drift coefficients. A Stable Manifold Theorem should hold for such singular sdes. Given the extreme roughness of the driving vector fields, the existence and differentiability of the invariant manifolds would be striking and counter-intuitive. The results will be attained via Malliavin calculus and multiplicative ergodic theory. 2. A strategy for a proof of the Kupka-Smale Conjecture for stochastic differential equations driven by smooth vector fields on compact Riemannian manifolds. The basic intuition behind such a fundamental result is that in order for a family of hyperbolic vector fields to represent the dynamics of viable physical/biological models, it is imperative that the family be "rich" within the set of all "reasonable" vector fields, and at the same time be insensitive to perturbations in the coefficients. 3. Existence and smoothness of densities (with respect to Lebesgue measure) for the finite-dimensional random invariant subspaces of linear stochastic partial differential equations.