Let q~ be a weighted Schwartz's space of rapidly decreasing functions, q~' the dual space and ~(t) a perturbed diffusion operator with polynomial coefficients from ~ into itself. It is proven that A~ generates the Kolmogorov evolution operator from q5 into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on q~' :where W(t) is a q~'-valued Brownian motion and ~*(t) is the adjoint of Aq(t) and show a central limit theorem for interacting multiplicative diffusions.