Stochastic complex Ginzburg-Landau equation with space-time white noise

TL;DR

This paper establishes local well-posedness for the highly singular stochastic complex Ginzburg-Landau equation driven by space-time white noise in three dimensions, using two advanced mathematical frameworks.

Contribution

It provides the first rigorous analysis of this equation's well-posedness in three dimensions via regularity structures and paracontrolled distributions.

Findings

Proves local well-posedness using regularity structures.

Proves local well-posedness using paracontrolled distributions.

Bridges two modern analytical approaches for singular SPDEs.

Abstract

We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be under- stood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.