Lagrangian statistical mechanics applied to non-linear stochastic field equations

We consider non-linear stochastic field equations such as the KPZ equation for deposition and the noise driven Navier-Stokes equation for hydrodynamics. We focus on the Fourier transform of the time dependent two-point field correlation, Φ_k(t). We employ a Lagrangian method aimed at obtaining the distribution function of the possible histories of the system in a way that fits naturally with our previous work on the static distribution. Our main result is a non-linear integro-differential equation for Φ_k(t), which is derived from a Peierls-Boltzmann type transport equation for its Fourier transform in time Φ_k,ω. That transport equation is a natural extension of the steady state transport equation, we previously derived for Φ_k(0). We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of Φ_k(t) is described by Φ_k(t) ∼ exp(-a|k|t^γ), where a is a constant and γ is system dependent.