Stochastic Differential Equations in Duals of Nuclear Spaces with Some Applications.

Abstract

Abstract : These lectures aim at giving an elementary introduction to certain types of stochastic differential equations in infinite dimensional spaces. One lecture introduces countably Hilbertian Nuclear (CHN) spaces and give some examples to illustrate why these infinite dimensional spaces are convenient for the study of some practical problems, e.g. those occuring in stochastic evolutions. This lecture assumes a complete probability spade with a right continuous filtration. It also assumes a given Countably Hilbertian nuclear space. Ornstein-Uhlenbeck stochastic differential equations on duals of nuclear spaces introduces a special class of linear stochastic differential equations with values in duals of nuclear spaces, namely Ornstein-Uhlenbeck type processes with a nuclear valued martingale as a driving term. Weak Convergence of Solutions: now consider the weak convergence of the solutions of to the corresponding stochastic differential equations driven by a Gaussian noise. This last lecture gives an outline of recent works on stochastic evolution equations and nonlinear stochastic differential equations on the dual of a Countably Hilbert nuclear space.