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Southern Illinois University Carbondale
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Stochastic Functional Differential Equations On Manifolds (Conference On Probability And Geometry), Salah-Eldin A. Mohammed
Stochastic Functional Differential Equations On Manifolds (Conference On Probability And Geometry), Salah-Eldin A. Mohammed
Miscellaneous (presentations, translations, interviews, etc)
We prove an existence theorem for solutions of stochastic functional differential equations under smooth constraints in Euclidean space. The initial states are semimartingales on a compact Riemannian manifold. It is shown that, under suitable regularity hypotheses on the coefficients, and given an initial semimartingale, a sfde on a compact manifold admits a unique solution living on the manifold for all time. We also discuss the Chen-Souriau regularity of the solution of the sfde in the initial process. The results are joint work with Remi Leandre.
The Stable Manifold Theorem For Stochastic Systems With Memory (Probability Seminar, Université Henri Poincaré Nancy 1), Salah-Eldin A. Mohammed
The Stable Manifold Theorem For Stochastic Systems With Memory (Probability Seminar, Université Henri Poincaré Nancy 1), Salah-Eldin A. Mohammed
Miscellaneous (presentations, translations, interviews, etc)
We state and prove a Local Stable Manifold Theorem for nonlinear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary solutions of sfde's. 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.