Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Abstract

Let Φ be a weighted Schwartz's space of rapidly decreasing functions, Φ′ the dual space and ℒ(t) a perturbed diffusion operator with polynomial coefficients from Φ into itself. It is proven that ℒ(t) generates the Kolmogorov evolution operator from Φ into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on Φ′:[Figure not available: see fulltext.] where W(t) is a Φ′ Brownian motion and ℒ*(t) is the adjoint of ℒ(t) and show a central limit theorem for interacting multiplicative diffusions. © 1987 Springer-Verlag.