## THE FIELDS INSTITUTE

&

DEPARTMENT OF STATISTICS,
UNIVERSITY OF TORONTO

# Stochastics Seminars

Summer 1998

RANDOM PERTURBATIONS OF SYSTEMS WITH CONSERVATION LAWS

A.D. Wentzell

Department of Mathematics

Tulane
University, New Orleans

This is a joint work with Mark Freidlin,
University of Maryland, College Park.

### Abstract

Small white-noise type perturbations of a dynamical system in
an r-dimensional space having l independent conservation laws can be described
by introducing l coordinates that change slowly, and r - l "fast" coordinates.
This is a situation where an "averaging principle" ought to take place. There
are reasons to expect that, after a suitable time change, the motion of the
"slow" coordinates, i.e., the process on the l-dimensional space Y obtained by
identifying the points along the (r - l)-dimensional surfaces of "fast" motion,
converges to a diffusion on this space. Results in this field can be
reformulated in the language of partial differential equations.
If all fast-motion surfaces are just tori, it leads to a diffusion in an
l-dimensional region, whose characteristics are found by averaging; if some of
them have singular points, the arising space Y is a graph in the case of l = 1
(see Number 523 of Memoirs of AMS), and some structure consisting of pieces of
smooth l-dimensional pieces in the case of l > 1 (in which case it is not
absolutely clear yet how to describe diffusions on Y).

**Organizers:**

D.Dawson (Fields Institute)

N.Reid
(University of Toronto)

V.Vinogradov (University of Toronto & UNBC)