Diffusion processes on an open book and the averaging principle

Abstract

Asymptotic problems for classical dynamical systems, stochastic processes, and PDEs can lead to stochastic processes and differential equations on spaces with singularities. We consider the averaging principle for systems with conservation laws perturbed by small noise, where, after a change of time scale, the limiting slow motion is a diffusion process on a space which is called in topology an open book: the space consisting of a number of n-dimensional manifold pieces (pages) that are glued together, sometimes several at a time, at the “binding”, which is made up of manifolds of lower dimension. A diffusion process on such a space is determined by differential operators governing the process inside the pages, and gluing conditions, which determine its behavior after hitting the binding. We prove weak convergence of measures in the function space that correspond to the slow-motion process in our averaging problem, and calculate the characteristics of the limiting process.