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

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