The research in this project focuses on several dynamical and numerical aspects of stochastic differential equations. Stochastic ordinary differential equations (sodes) on finite-dimensional manifolds generate stochastic flows on the manifold. One objective of the research is to construct invariant manifolds for such flows near stationary solutions, under suitable regularity and growth conditions on the driving vector fields. In particular, the investigator will prove the existence of stable, unstable and center manifolds near each stationary point on the manifold. In order for sodes to constitute viable physical models, the investigator conjectures that a Kupka-Smale type theorem must hold. An important class of infinite-dimensional semiflows on Hilbert space is generated by dissipative semilinear stochastic partial differential equations (spdes) on smooth compact manifolds. For these semiflows, the investigator intends to prove the existence of finite-dimensional invariant manifolds near stationary points. Important examples of spdes covered by this analysis are Burger's equation, affine linear stochastic evolution equations and stochastic reaction-diffusion equations. The results of the research will reveal new features of the stochastic dynamics of these well-studied models. One encounters models of stochastic systems with memory (sfdes) in many engineering and physical applications. Deterministic smooth constraints on the solutions of such models lead naturally to sfdes on (compact) Riemannian manifolds. The investigator will study the path-space-valued Markov process generated by trajectories of sfdes on the manifold. The investigator will establish an Ito formula for the infinite-dimensional segment process and will attempt to obtain its infinitesimal generator in terms of geometric invariants of the ambient Riemannian manifold. From the pathwise dynamical point of view, the investigator will construct perfect semiflows on the path space which are induced by solutions of sfdes on a compact Riemannian manifold. The existence of invariant manifolds will also be examined.
The project focuses on qualitative and long-term behavior of a large class of probabilistic models known as stochastic differential equations. These equations are widely used by scientists and engineers. Of special interest is a class of models that are used in physics, engineering and biology in order to analyze dynamical systems whose evolution is influenced by random fluctuations and past history. These models are very important in a variety of diverse areas such as signal processing, stock market fluctuations, economic and labor models, aircraft dynamics, materials with memory, population dynamics and fluid flow. The investigator will use the most current probabilistic techniques in order to develop a deeper understanding of these models. The outcome of the research in this project will yield precise information on the long-term behavior of solutions of the underlying stochastic equations on states that are near statistical equilibria. Numerical algorithms will be developed whereby real market data will be used to test option-pricing models where the stock price is governed by its past history. In a different direction, the research will solidify important connections with other areas of modern mathematical research, in particular the theory of dynamical systems and geometry. The results of the project will be compiled in a research monograph targeting graduate students majoring in mathematics, engineering, and finance.
