## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 12 - Wahrscheinlichkeitstheorie, Statistik**

Dienstag, 18. September 2001, 17.30, Hörsaal 7

**The Averaging Principle and Diffusion Processes on Graphs**
**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.

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