Stochastic representation of tau functions of Korteweg-de Vries equation

TL;DR

This paper provides a stochastic representation of tau functions for the KdV equation using Laplace transforms of iterated Skorohod integrals, extending previous results to non-soliton solutions involving Gaussian processes.

Contribution

It generalizes the stochastic representation of KdV tau functions beyond soliton solutions to include non-soliton solutions linked to Gaussian processes with Cauchy covariance.

Findings

Expresses tau functions as Laplace transforms of Skorohod integrals.

Extends stochastic representation to non-soliton solutions.

Involves Gaussian processes with Cauchy covariance.

Abstract

In this paper we express tau functions for the Korteweg de Vries (KdV) equation, as Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. Our result extends the paper of Ikeda and Taniguchi who obtained a stochastic representation of tau functions for the $N$-soliton solutions of KdV as the Laplace transform of a quadratic functional of $N$ independent Ornstein-Uhlenbeck processes. 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.