Global well-posedness for the two-dimensional stochastic complex Ginzburg-Landau equation

TL;DR

This paper establishes the local and global well-posedness of the two-dimensional stochastic complex Ginzburg-Landau equation with additive white noise, employing renormalization techniques and invariant measure arguments.

Contribution

It introduces a novel renormalization approach for the complex-valued SCGL and proves its well-posedness in two dimensions, extending previous results on real-valued equations.

Findings

Proved local well-posedness of the renormalized SCGL.

Established global well-posedness via energy estimates.

Achieved almost sure global well-posedness using invariant measures.

Abstract

We study the stochastic complex Ginzburg-Landau equation (SCGL) with an additive space-time white noise forcing on the two-dimensional torus. This equation is singular and thus we need to renormalize the nonlinearity in order to give proper meaning to the equation. Unlike the real-valued stochastic quantization equation, SCGL is complex valued and hence we are forced to work with the generalized Laguerre polynomials for the sake of renormalization. In handling nonlinearities of arbitrary degree, we derive a useful algebraic identity on the renormalization in the complex-valued setting and prove that the renormalized SCGL is locally well-posed. We prove global well-posedness using an energy estimate and almost sure global well-posedness under different conditions using an invariant measure argument.