Linear Stochastic Differential Equations on the Dual of a Countably Hilbert Nuclear Space with Applications to Neurophysiology

摘要

Properties of the Ornstein-Uhlenbeck on the dual of a nuclear space are derived; stationarity and existence of unique invariant measure is proved, Radon-Nikodym derivative exhibited and the OU process is investigated for flicker noise. Existence and uniqueness of solutions to linear stochastic differential equations on the dual of a nuclear spaces established, and general conditions for the weak convergence on Skorohod space of solutions are given. Moreover, solutions are shown to be CADLAG semimartingales (for appropriate initial conditions). The results are applicable to solving stochastic partial differential equations. Finally, the results are applied to giving a rigorous representation and solutions of models in neurophysiology as well as to deriving explicit results for the weak convergence of these solutions. (Author)