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)