Canonical phase-space approach to the noisy Burgers equation: Probability distributions

We present a canonical phase-space approach to stochastic systems described by Langevin equations driven by white noise. Mapping the associated Fokker -Planck equation to a Hamilton-Jacobi equation in the nonperturbative weak noise limit we invoke a principle of least action for the determination of the probability distributions. We apply the scheme to the noisy Burgers and Kardar-Parisi-Zhang equations and discuss the time-dependent and stationar y probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short-time region we discuss heuristically the nonlinear soliton contr ibutions and derive an expression for the distribution in accordance with t he directed polymer-replica and asymmetric exclusion model results. We also comment on the distribution in higher dimensions. [S1063-651X(99)09705-6].