Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schr\"odinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.Comment: Presentation improve

Invariant measures for the dnls equation

We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ data. The measures are absolutely continuous with respect to suitable weighted Gaussian measures supported on Sobolev spaces of increasing regularity. These results have been obtained in collaboration with Giuseppe Genovese (University of Z\"urich) and Daniele Valeri (University of Glasgow)

Invariant measures for the periodic derivative nonlinear Schrödinger equation

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in L2 and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of the Gaussian measure on L2 with covariance (I+(−Δ)k)−1 for any k⩾2

Invariant measures for the periodic derivative nonlinear Schr\"odinger equation

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) for small data in $L^2$ and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of Gaussian measures on $L^2$.Comment: a new result has been included, namely quasi-invariance of Gaussian measures w.r.t. the gauge group (theorem 1.4

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one dimensional periodic derivative nonlinear Schrödinger (DNLS) equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion R hk, k 2 Z+. In each R h2k the term with the highest regularity involves the Sobolev norm _H k(T) of the solution of the DNLS equation. We show that a functional measure on L2(T), absolutely continuous w.r.t. the Gaussian measure with covariance (I + ()k) 1, is associated to each integral of motion R h2k, k 1

Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on L2(T) with covariance [1+(−Δ)s]−1 under these transformations for any [Formula presented]. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter s