Stochastic Representation of Tau Functions With an Application to the Korteweg-De Vries Equation

Abstract

In this paper we express the tau functions considered by Pöppe in [23] for the Korteweg de Vries (KdV) equation, as the Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. A stochastic representation of tau functions for the N -soliton solutions of KdV has been proved by Ikeda and Taniguchi in [14]. They express the N -soliton solutions as the Laplace transform of a quadratic functional of N independent Ornstein-Uhlenbeck processes. Our first step is to provide the Wiener chaos decomposition of the underlying functional and to identify the Fredholm determinant of an integral operator in their representation. Our general result goes beyond the N -soliton case and enables us to consider a non soliton solution of KdV associated to a Gaussian process with Cauchy covariance function.