Diffusion Approximation

Abstract

In this chapter we consider randomly perturbed systems of differential and difference equations whose averaged systems are static. So, the solution of the perturbed equation is expected to converge as ε → 0 to a constant determined by the initial position of the system. It follows from the theorem on normal deviations that the deviations of such solutions from the initial position are asymptotically Gaussian random variables whose variance is of order εt. Here we consider the problem farther out in time. We prove, under some reasonable assumptions, that the stochastic process x _ε (t/ε), where x _ε (t) is the solution of the perturbed system at time t, is asymptotically a diffusion process as ε → 0. We derive here a detailed description of these processes. Moreover, if the averaged equation has a first integral (i.e., a function φ exists that is constant on any trajectory of the averaged equation), then for any fixed time, say t ₁, the function φ(x _ε (t ₁)) is asymptotically constant, but the function φ(x _ε (t/ε)) is approximately a diffusion process.