Local existence and uniqueness for a two-dimensional surface growth equation with space--time white noise

TL;DR

This paper investigates the local existence and uniqueness of solutions for a 2D surface growth stochastic PDE driven by space-time white noise, overcoming regularity challenges through spectral Galerkin methods and noise regularization.

Contribution

It provides a rigorous framework for defining solutions and proves their uniqueness despite the failure of standard fixed-point approaches due to irregular noise.

Findings

Solutions are unique when using spectral Galerkin or other noise regularizations.

The stochastic PDE can be given a rigorous meaning despite regularity issues.

Different regularization methods lead to the same solution.

Abstract

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution.