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.
Skip Nav Destination
Research Article| August 01 1994
Long time behavior of nonlinear stochastic oscillators: The one‐dimensional Hamiltonian case
S. Albeverio, A. Klar; Long time behavior of nonlinear stochastic oscillators: The one‐dimensional Hamiltonian case. J. Math. Phys. 1 August 1994; 35 (8): 4005–4027. https://doi.org/10.1063/1.530839
Download citation file: