Three types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated... Show moreThree types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated to study stochastic dynamical systems. Invariant structures for stochastic dynamical systems are reviewed and detailed techniques for their simulation, approximation and construction are presented with several illustrative examples. First, a numerical approach for the simulation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. This numerical approach is tested in two illustrative examples: one is for a stochastic differential equation and the other is for a stochastic partial differential equation. Second, invariant foliations for dynamical systems with small white noisy perturbation are approximated via asymptotic analysis. In other words, random invariant foliations are represented as a perturbation of the corresponding deterministic invariant foliations, with deviation errors estimated. The escape probability is a deterministic concept making methods of partial differential equations theory attainable to stochastic dynamics. Finally, the escape probability p(x) for dynamical systems driven by non-Gaussian L´evy motions, especially symmetric α-stable L´evy motions, is considered and characterized. More precisely, it is represented as the solution of the Balayage-Dirichlet problem of a certain partial differential-integral equation. This issue has been investigated previously for dynamical systems driven by Wiener process. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted. Ph.D. in Applied Mathematics, July 2012 Show less
