Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit

TL;DR

This paper investigates the distributional behavior of stationary solutions to the complex Ginzburg-Landau equation under random forcing, especially in the inviscid limit, revealing density properties of key solution measures.

Contribution

It demonstrates that in the inviscid limit, the distributions of the L^2 norm and energy of stationary solutions have densities, using Itô's formula and local time analysis.

Findings

Distributions of L^2 norm and energy have densities in the inviscid limit.

Stationary measures accumulate as viscosity approaches zero.

Analysis relies on stochastic calculus and properties of semimartingales.

Abstract

We consider the behaviour of the distribution for stationary solutions of the complex Ginzburg-Landau equation perturbed by a random force. It was proved earlier that if the random force is proportional to the square root of the viscosity, then the family of stationary measures possesses an accumulation point as the viscosity goes to zero. We show that if $\mu$ is such point, then the distributions of the L^2 norm and of the energy possess a density with respect to the Lebesgue measure. The proofs are based on It\^o's formula and some properties of local time for semimartingales.