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Stochastic Differential Equations: A Dynamical Systems Approach
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145 b
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The relatively new subject of stochastic differential§equations has§increasing importance in both theory and§applications. The subject§draws upon two main sources, probability/stochastic§processes and§differential equations/dynamical systems. There§exists a significant§``culture gap" between the corresponding research§communities. The§objective 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 coefficients.
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