Weak convergence and approximations for partial differential equations with random process coefficients

Abstract

For a parabolic equation with wide bandwidth coefficients, it is shown that the solution converges weakly to that of a stochastic PDE driven by an infinite dimensional Wiener process as the bandwidth tends to infinity. The treatment is novel and purely probabilistic. The solution to the "wide band" coefficient system is represented as a conditional expectation of a functional of a certain diffusion. By a weak convergence argument, the conditional expectation (and its mean square derivatives) converges weakly to a conditional expectation of a functional of a "limit" diffusion. It is then shown that this "limit" functional satisfies the appropriate stochastic PDE. The infinite dimensional Wiener process is represented explicitly in terms of the original system noise. No coercivity or strict ellipticity conditions are required. The result provides a partial justification for the use of infinite dimensional Wiener processes in distributed systems. Since the method is based on weak convergence arguments for Itô-type equations with wide bandwidth coefficients and "PDE methods" are avoided, it is likely that the technique will find greater use in the analysis of infinite dimensional stochastic systems. The methods have already proved to be very useful in studying approximations to non-linear filtering problems with wide bandwidth observation noise [10].