Collaborative Research: Invariant Manifolds for Multiscale Stochastic Dynamical Systems

研究者和他的合作者正在为编码多个时空尺度的有限和无限维随机动力系统相空间中相干结构的存在和定性性质发展基本理论。当考虑不确定性或随机影响时，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。该项目的应用将集中在材料科学的Allen-Cahn或Cahn-Hilliard方程中被视为缺陷的峰态动力学，膜电位和细胞内离子浓度的振荡，分子马达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。从理论上讲，无限维随机动力系统可以由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包括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为多尺度随机动力系统建立许多基本的几何框架。特别是他们正在发展(i)随机动力系统的通常双曲不变流形理论，包括随机稳定和不稳定流形和叶形的持续和存在；快慢系统的随机交换引理；(3)噪声环境下的近似双曲流形理论。这些都与非线性偏微分方程和演化型积分方程的具体分析有关，特别注意连贯结构的持久性和动力学。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，但这些过程背后也存在连贯的结构，这些结构提供了某种程度的可预测性。该项目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了基础。其中包括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携带必需的化学物质和废物往返于细胞内的活性位点，从而实现生长、再生、移动和交流。了解了这个过程，就能理解并治疗衰弱性疾病和慢性疾病。同样，理解复杂材料中产生相干结构的机制可以设计出具有特别有用的磁性、半导体、超导或生物力学性能的先进材料。

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.