Collaborative Research: Invariant Manifolds for Multiscale Stochastic Dynamical Systems

研究者和他的合作者正在开发的相干结构的存在和定性性质的相空间的有限和无限维随机动力系统，编码多个时空尺度的基本理论。随机动力系统出现在物理学、生物学、气候学、经济学等许多现象的建模中，当考虑到不确定性或随机影响时。该项目的应用将集中在尖峰状态的动力学上，被视为缺陷，对于材料科学的Allen-Cahn或Cahn-Hilliard方程，细胞内膜电位和离子浓度的振荡，以及分子运动驱动蛋白的运动及其与微管的相互作用，所有这些都受到随机波动的影响。从理论观点来看，无限维随机动力系统可以由例如随机偏微分方程和随机偏微分方程生成。随机动力系统的研究涉及到随机分析和动力系统的几何理论。研究者和他的合作者正在建立多尺度随机动力系统的基本几何框架。特别是他们正在开发（一）理论的正常双曲不变流形随机动力系统，包括持久性和存在的随机稳定和不稳定的流形和foliations;（二）随机交换引理的快慢系统;（三）理论的近似正常双曲流形在嘈杂的环境。这些都与具体分析的非线性偏微分和积分方程的演变类型，特别注意的持久性和动态的相干结构。虽然许多物理，生物和金融过程似乎受到随机或随机力量的影响，但这些过程也有连贯的结构，这些结构提供了某种程度的可预测性。 该项目正在为确定这些隐藏结构和分析几种应用中出现的具体情况奠定基础。 其中一个是身体每个细胞内的基本运输过程。 在这里，附着在杆状纤维上的分子马达将必需的化学物质和废物运送到细胞内的活性部位，使细胞能够生长、再生、移动和交流。 了解这一过程带来了对衰弱和慢性疾病的理解和治疗。 同样，理解在复杂材料中产生相干结构的机制可以导致设计具有特别有用的磁性，半导体，超导或生物力学特性的先进材料。

The investigator and his collaborators are developing the fundamental theory for the existence and qualitative properties of coherent structures in the phase space of finite and infinite-dimensional stochastic dynamical systems that encode multiple spatio-temporal scales. Random dynamical systems arise in the modeling of many phenomena in physics, biology, climatology, economics, etc. when uncertainties or random influences are taken into account. The applications for this project will center on the dynamics of spike states, viewed as defects, for the Allen-Cahn or Cahn-Hilliard equation of materials science, oscillations in membrane potential and ionic concentrations within cells, and the motion of the molecular motor kinesin and its interactions with microtubules, all of which are subject to stochastic fluctuations. From the theoretical standpoint, Infinite-dimensional random dynamical systems may be generated, for example, by stochastic partial differential equations and random partial differential equations. The study of random dynamical systems involves both stochastic analysis and geometrical theory of dynamical systems. The investigator and his collaborators are establishing much of the basic geometric framework for multiscale, stochastic dynamical systems. In particular they are developing (i) The theory of normally hyperbolic invariant manifolds for stochastic dynamical systems including the persistence and the existence of random stable and unstable manifolds and foliations; (ii) The stochastic Exchange Lemma for fast-slow systems; (iii) The theory of approximate normally hyperbolic manifolds in a noisy environment. These are linked with concrete analysis of nonlinear partial differential and integral equations of evolutionary type, with particular attention paid to the persistence and dynamics of coherent structures. While many physical, biological, and financial processes appear to be subject to random or stochastic forces, there are also coherent structures underlying these processes which give some measure of predictability. This project is laying the groundwork for the determination of these hidden structures and for analyzing specific situations arising in several applications. Among these is a fundamental transport process within each cell of the body. Here, molecular motors attached to rod-like fibers carry essential chemicals and waste products to and from active sites within the cell, allowing for growth, rejuvenation, mobility, and communication. Understanding this process brings understanding of and perhaps therapies for debilitating and chronic diseases. Likewise, understanding the mechanisms that produce coherent structures in complex materials can lead to the design of advanced materials with particularly useful magnetic, semiconducting, superconducting, or biomechanical properties.