Infinite Dimensional Forward-Backward Stochastic Differential Equations and the KPZ Equation

TL;DR

This paper establishes existence and uniqueness results for infinite dimensional forward-backward stochastic differential equations with non-Lipschitz nonlinearities and uses these to provide a new probabilistic representation of the KPZ equation.

Contribution

It introduces a novel connection between infinite dimensional forward-backward SDEs and the KPZ equation, extending classical finite-dimensional results.

Findings

Proved existence and uniqueness of solutions for the infinite dimensional forward-backward SDEs.

Established a new probabilistic representation for the KPZ equation.

Extended the theory to include non-Lipschitz quadratic nonlinearities.

Abstract

Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial differential equation(SPDE) driven by a space-time white noise. In recent years there have been several works directed towards giving a rigorous meaning to a solution of this equation. Bertini, Cancrini and Giacomin have proposed a notion of a solution through a limiting procedure and a certain renormalization of the nonlinearity. In this work we study connections between the KPZ equation and certain infinite dimensional forward-backward stochastic differential equations. Forward-backward equations with a finite dimensional noise have been studied extensively, mainly motivated by problems in mathematical finance. Equations considered here differ from the classical works in that, in addition to having an infinite dimensional driving noise, the associated SPDE involves a non-Lipschitz (namely a quadratic) function of the gradient. Existence and uniqueness of solutions of such infinite dimensional forward-backward equations is established and the terminal values of the solutions are then used to give a new probabilistic representation for the solution of the KPZ equation.