Matthias Weber, TU Dresden
The long time behavior of randomly perturbed Hamiltonian systems is, under suitable conditions, described by a diffusion process on a graph related to the Hamiltonian of the system.
We present an overview of recent results for non-linear oscillators with one degree of freedom, especially for the non-linear pendulum perturbed by white noise, and results for dynamical systems with many degrees of freedom. The differential operators which govern the diffusion process inside the edges of the graph and the gluing conditions at the vertices of the graph can be calculated explicitly and are the result of an averaging of the slow components of the perturbed system.
We show how these results can be used to study special classes of elliptic, hypoelliptic, and parabolic partial differential equations with small coefficients in the second order terms. Similar methods can be used to study the spectrum of elliptic differential operators with small coefficients in the second order terms.
All results are joint work with Mark Freidlin from the University of Maryland, U.S.A.