The long time behavior of nonlinear, nondissipative systems, which are perturbed by a white noise force are discussed herein. Considering special nonlinear forces and an appropriate scaling, a stochastic convergence theorem is proven. In particular the convergence of the energy process of the system to a limit diffusion is discussed. This corresponds to convergence of the system to a stationary distribution. Furthermore, the limit process is investigated and an explicit formula for its transition probability density is given. An analytic approach to the convergence theorem in terms of a singular perturbation theorem for semigroups is also presented.

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