Stochastic Dynamical Systems in Finite and Infinite-Dimensions

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The PI will study the stochastic dynamics of three different classes of differential systems: (1) stochastic ordinary differential equations (sode's) under smooth constraints, (2) constrained stochastic differential systems with long memory and (3) stochastic partial differential equations (spde's). In the first class of problems, the PI will develop a complete characterization of the almost sure behavior of the underlying stochastic flow in the neighborhood of a stationary (non-ergodic) solution. The effect of small perturbations on the almost sure qualitative structure of the stochastic flow will be studied near hyperbolic stationary solutions. Such small perturbations are natural because of unavoidable statistical errors in estimating the parameters of physical models against experimental inaccuracies in the measurement of real data. Issues of genericity and local stability will be addressed. In the second class of problems, a regular class of constrained stochastic systems with long memory will be identified. Such classes allow for the existence of smooth stochastic semiflows and hence a characterization of their invariant manifolds using suitably-modified ergodic theory techniques. The interplay between the geometry of the constraints and the stochastic dynamics will be examined. Weak and strong approximation schemes will be developed for stochastic systems with full memory and then applied to option-pricing models in mathematical finance with delayed stock-dynamics. The dynamics of the third class of problems will be studied by analyzing classical examples such as two-dimensional stochastic Navier-Stokes and Burgers equations. The proposed research is a long-term program that advocates novel links between probability theory/stochastic analysis and traditional mainstream mathematical disciplines such as dynamical systems, differential geometry and numerical analysis. In particular, the research would lead to new interactions between stochastic geometry and dynamical systems.During the past decade, a considerable number of applied mathematicians, engineers and economists have turned their attention to randomly evolving systems with memory for modeling a variety of physical phenomena whose time evolution depends on their past history. In physics, laser dynamics with delayed feedback is often investigated, as well as the dynamics of noisy bi-stable systems with delay. In biophysics, random (viz. stochastic) systems with memory are used to model delayed visual feedback systems or human postural sway and in the design of cardiac pacemaker cells. In mathematical finance, the volatility of the stock may be dependent on its past history and hence the stock dynamics may be best described by a stochastic system with memory. The research in this project is expected to give a complete characterization of the stability structure near equilibria for a large class of infinite-dimensional models called stochastic partial differential equations. Such models are ubiquitous in the study of heat flow, the movement of fluids and modelling of climate change.The PI will complete the preparation of a research monograph on stochastic systems with memory. The monograph is intended to be the basis for a graduate course in mathematics at Carbondale. Stochastic systems with long-memory and their applications to option-pricing in mathematical finance will engage the PI's master's and doctoral graduate students, some of them are females and minorities.