Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: approximate invariant measures

Abstract

Abstract : Let (x superscript epsilon (dot)) be a sequence of solutions to an ordinary differential equation with random right sides (due to input noise (xi superscript epsilon (dot)) and which converges weakly to a diffusion x(dot) with unique invariant measure mu(dot). Let mu(t, dot) denote the measure of x(t), and suppose that mu(t, dot) approaches mu (dot) weakly. The paper shows, under reasonable conditions, that the measures of x superscript epsilon (t) are close to mu (dot) for large t and small epsilon. In applications, such information is often more useful than the simple fact of the weak convergence. The noise xi superscript epsilon (dot) need not be bounded, the pair (x superscript epsilon (dot), xi(dot)) need not be Markovian (except for the unbounded noise case), and the dynamical terms need not be smooth. The discrete parameter case is treated, and several examples arising in control and communication theory are given.