Wick Product in The Stochastic Burgers Equation: A Curse or a Cure?

TL;DR

This paper investigates the dual role of Wick products in the stochastic Burgers equation, highlighting how they can be both a challenge and a solution depending on the context, through chaos expansion analysis.

Contribution

It provides a detailed analysis of the Wick product's role in the stochastic Burgers equation, revealing conditions under which it acts as a curse or a cure.

Findings

Wick nonlinearity leads to generalized solutions regardless of noise regularity

Certain multiplicative perturbations require Wick interpretation, acting as a cure

Chaos expansion coefficients elucidate the solution structure at different scales

Abstract

It has been known for a while that a nonlinear equation driven by singular 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 coefficients of the chaos expansion of the solution at different stochastic scales.