Stochastic Differential Equations: A Dynamical Systems Approach

Abstract

The relatively new subject of stochastic differential equations has increasing impor- tance in both theory and applications. The subject draws upon two main sources, prob- ability/stochastic processes and differential equations/dynamical systems. There exists a signifcant \culture gap"" between the corresponding research communities. The objec- tive of the dissertation project is to present a concise yet mostly self-contained theory of stochastic differential equations from the differential equations/dynamical systems point of view, primarily incorporating semigroup theory and functional analysis techniques to study the solutions. Prerequisites from probability/stochastic processes are developed as needed. For continuous-time stochastic processes whose random variables are (Lebesgue) absolutely continuous, the Fokker-Planck equation is employed to study the evolution of the densities, with applications to predator-prey models with noisy coeffcients.