Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on T

2020/02/12 by Arnaud Debussche, Yoshio Tsutsumi, Debussche, Arnaud +1
Engineering · Mathematics · #35Q55 #35R60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Ultra-Wideband Communications Technology

paper · pdf · doi:10.48550/arxiv.2002.04899

openalex publication_date 2020/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance (1-∂_x²)^-α on L²(\mathbf T) is quasi-invariant for the associated flow for α>1/2. This is sharp and improves a previous result obtained in \citeOTT where the values α>3/4 were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in \citeCruz1 and \citeCruz2. The arguments are general and can be used to other Hamiltonian equations.

Citations

• Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity

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