Dynamics of Stochastic Systems with Memory (Mathematics and Statistics Colloquium, Wright State University)

We describe an approach to the dynamics of stochastic systems with finite memory using multiplicative cocycles in Hilbert space. We introduce the notion of hyperbolicity for stationary solutions of the stochastic differential system. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses ideas from infinite-dimensional multiplicative ergodic theory and interpolation arguments

The Stable Manifold Theorem for SDE\u27s (Probability Seminar, University of California, Irvine)

In this talk, we formulate a local stable manifold theorem for stochastic differential equations in Euclidean space, driven by multi-dimensional Brownian motion. We introduce the concept of hyperbolicity for stationary trajectories of a SDE. This is done using the Oseledec muliplicative ergodic theorem on the linearized SDE along the stationary solution. Using methods of (non-linear ergodic theory), we construct a stationary family of stable and unstable manifolds in a stationary neighborhood around the hyperbolic stationary trajectory of the non-linear SDE. The stable/unstable manifolds are dynamically characterized using anticipative stochastic calculus

The Stable Manifold Theorem for Stochastic Systems with Memory (Probability Seminar, Université Henri Poincaré Nancy 1)

We state and prove a Local Stable Manifold Theorem for nonlinear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde\u27s)). We introduce the notion of hyperbolicity for stationary solutions of sfde\u27s. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments

On the Dynamics of Stochastic Differential Equations (Ellis B. Stouffer Colloquium, University of Kansas)

We formulate and outline a proof of the Local Stable Manifold Theorem for stochastic differential equations (SDE\u27s) in Euclidean space (joint work with M. Scheutzow). This is a central result in dynamical systems with noise. Starting with the existence of a stochastic flow for an SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution. For Stratonovich noise, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE

Stochastic Dynamics of Infinite-Dimensional Systems (Stochastic and Non-linear Analysis Seminar, University of Illinois)

We describe an approach to the dynamics of non-linear stochastic differential systems with finite memory using multiplicative cocycles in Hilbert space. We introduce the notion of hyperbolicity for stationary solutions of stochastic systems with memory. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses ideas from infinite-dimensional multiplicative ergodic theory and interpolation arguments

The Stable Manifold Theorem for SDE\u27s (Stochastic Analysis Seminar, MSRI)

I gave the following two talks at the Stochastic Analysis Seminar at the Mathematical Sciences Research Institute, Berkeley, California. The two talks describe recent joint work with Michael Scheutzow. FIRST TALK: THE STABLE MANIFOLD THEOREM FOR SDE\u27S , Part I Wednesday, December 3, 1997, 11:00-12:00 am, MSRI Lecture Hall. In this talk, we formulate a local stable manifold theorem for stochastic differential equations in Euclidean space, driven by multi-dimensional Brownian motion. We introduce the concept of hyperbolicity for stationary trajectories of a SDE. This is done using the Oseledec muliplicative ergodic theorem on the linearized SDE around the stationary solution. Using methods of (non-linear ergodic theory), we construct a stationary family of stable and unstable manifolds in a stationary neighborhood around the hyperbolic stationary trajectory of the non-linear SDE. The stable/unstable manifolds are dynamically characterized using anticipating stochastic calculus. SECOND TALK : THE STABLE MANIFOLD THEOREM FOR SDE\u27S, Part II Friday, December 5, 1997, 11:00-12:00 am, MSRI Lecture Hall. This is a continuation of the talk given on Wednesday, December 3, 1997. We outline the basic ideas underlying the proof of the Stable Manifold Theorem for SDE\u27s. We discuss the linearization of the SDE along a hyperbolic stationary solution. The stable and unstable manifolds are constructed using ideas and techniques from multiplicative ergodic theory that were developed by David Ruelle in the late seventies. In particular, we develop estimates of the stochastic flow in a neighborhood of the hyperbolic stationary solution. Finally, we discuss generalizations to semimartingale noise, related open problems, and conjectures

Retarded functional differential equations: a global point of view

This work deals with some of the fundamental aspects of retarded functional differential equations (RFDE's) on a differentiable manifold. We start off by giving a solution of the Cauchy initial value problem for a RFDE on a manifold X. Conditions for the existence of global solutions are given. Using a Riemannian structure on the manifold X, a RFDE may be pulled back into a vector field on the state space of paths on X. This demonstrates a relationship between vector fields and RFDE's by giving a natural embedding of the RFDE's on X as a submodule of the module o* vector fields on the state space. For a given RFDE it is shown that a global solution may level out asymptotically to an equilibrium path. Each differentiable RFDE on a Riemannian manifold linearizes in a natural way, thus generating a semi-flow on the tangent bundle to the state space. Sufficient conditions are given to smooth out the orbits and to obtain the stable bundle theorem for the semi-flow There are examples of RFDE's on a Riemannian manifold. These include the vector fields, the differential delay equations, the delayed Cartan development and equations of Levin-Nohel type. The retarded heat equation on a compact manifold provides an example of a partial RFDE on a function space. We conclude by making suggestions for further research