Wick product in the stochastic Burgers equation: A curse or a cure?

Abstract

It has been known for a while that a nonlinear equation driven by singu- lar noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coe-cients of the chaos expansion of the solution at difierent stochastic scales.