Approximation of solutions to differential equations with random inputs by diffusion processes

Let yɛ(·) denote a random process whose bandwidth, loosely speaking, goes to ∞ as ɛ → 0. Consider the family of differential equations xɛ=g(xɛ,yɛ)+f(xɛ,yɛ)/α(ɛ), where α(ɛ) → 0 as ɛ → 0. The question of interest is: does the sequence {xɛ(·)} converge in some sense and if so which, if any, ordinary or Ito differential equation does it satisfy? Normally, the limit is taken in the sense of weak convergence. The problem is of great practical importance, for such questions arise in many practical situations arising in many fields. Often the limiting equation is nice and can be treated much more easily than can the xɛ(·). In any case, in practice approximations to properties of the xɛ(·) are usually sought in terms of ɛ and some limit. To illustrate these points, as well as a related stability problem, we give a practical example which arises in the theory of adaptive arrays of antennas.