Averaging

Abstract

In this chapter we consider random perturbations of Volterra integral equations, of differential equations, and of difference equations, and we develop an averaging method for each. In each case there is an equation of the form F _ε (x(·),ω) = 0, where F _ε is an operator acting on functions x(t). This equation is to be solved for a function x = x _ε (t, ω), where ω is a sample from a probability space on which the random perturbations are defined and ε is a small positive parameter. We average this equation over the probability space by defining F̄(x(·)) = EF(x(·), ω) and we use ergodic theorems to show how the limit as ε → 0 of the perturbed problem is related to the averaged problem F̄(x̄(·)) = 0. The results in this chapter show how to derive, under natural conditions, convergence properties of x _ε (t, ω) - x̄(t) as ε → 0. In general, this error will approach zero in a probabilistic sense that is made precise in each case.