Stochastic Dynamics: Finite and Infinite Dimensional

Southern Illinois University At Carbondale, Carbondale IL

Investigators

Abstract

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.