Effective Equation and Renormalization for a Nonlinear Wave Problem with a Random Source

Abstract

We shall explain how to derive the effective equation for the expected value of the solution of a given nonlinear random partial differential equation. Our method is a generalization of the classical smoothing for linear wave propagation in random media,[1], to nonlinear problems. We also show that the effective equation can be derived from a path integral formulation used by Parisi and Sourlas in the case of a random nonlinear Laplace equation,[2]. They represented the moment-generating functional of the solution as a functional integral over commuting and anticommuting fields. For many problems of physical interest, the meaning of the functional integral, and thus the meaning of the effective equation, is established by a limiting process which involves renormalization. As an illustration we analyze a nonlinear random wave equation, [3]. Our exposition here is based in our joint work, with S. Venakides, [3], where a more detailed analysis can be found.