  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.
