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

We consider the behaviour of the distribution for stationary solutions of the complex Ginzburg–Landau equation perturbed by a random force. It was proved in S. Kuksin and A. Shirikyan (2004) [4] that if the random force is proportional to the square root of the viscosity ν>0, then the family of stationary measures possesses an accumulation point as ν→0+. We show that if μ is such a point, then the distributions of the L2-norm and of the energy possess a density with respect to the Lebesgue measure. The proofs are based on Itôʼs formula and some properties of local time for semimartingales.