Collaborative Research: Invariant Manifolds for Multiscale Stochastic Dynamical Systems

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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.