Random Perturbations of Hamiltonian Systems

Mark Iosifovich Freĭdlin · Alexander D. Wentzell

Jan 1994 · American Mathematical Society: Memoirs of the American Mathematical Society Buch 523 · American Mathematical Soc.

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Random perturbations of Hamiltonian systems in Euclidean spaces lead to stochastic processes on graphs, and these graphs are defined by the Hamiltonian. In the case of white-noise type perturbations, the limiting process will be a diffusion process on the graph. Its characteristics are expressed through the Hamiltonian and the characteristics of the noise. Freidlin and Wentzell calculate the process on the graph under certain conditions and develop a technique which allows consideration of a number of asymptotic problems. The Dirichlet problem for corresponding elliptic equations with a small parameter are connected with boundary problems on the graph.