Quasi-invariance of Gaussian measures transported by the cubic NLS with third-order dispersion on T

Abstract

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance (1 − ∂ 2x ) −α on L 2 (T) is quasi-invariant for the associated flow for α > 1/2. This is sharp and improves a previous result obtained in [20] 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 [4] and [5]. The arguments are general and can be used to other Hamiltonian equations.

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

“…Indeed, many results have appeared regarding the quasiinvariance of Gaussian measures for various different dispersive PDEs. In particular, there are results for quasi-invariance of the BBM and Benjamin-Ono equations [81,32,33], KdV type equations [72], wave equations [66,36,74], and Schrödinger equations [65,62,64,71,29,19,60,30,32]. The key underlying observation in this study is that the quasi-invariance of Gaussian measures is intimately tied to the dispersive character of the equation; see [62,74] for negative results for some dispersionless ODEs.…”

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

“…The problem witnessed recently a resurgence of interest mainly concerning the evolution of the Brownian motion (or related processes) along the flow of dispersive PDEs [33,24,25,26,27,28,14,30,10,7]. A new analytic approach was introduced for flows of dispersive nonlinear equations in [33].…”

“…Otherwise the problem of determining the precise densities given by the gauge map is still open for s = 1. Let us point out that (with the notable exception of [7]) most of the works, appeared recently on the subject in the context of dispersive PDEs, cannot specify the Radon-Nykodim derivative by means of a suitable approximation procedure.…”

Quasi-invariance of Gaussian measures for the periodic Benjamin-Ono-BBM equation

“…This paper triggered a renewed interest in the subject from the viewpoint of dispersive PDEs, which translates into studying the evolution of random initial data (such as Brownian motion and related processes). For recent developments on the topic, see [2,6,7,8,9,10,11,12,14,19,20,21,22,23,24,26], although this list might be not exhaustive.…”

“…However it is only used to prove absolute continuity of the transported Gaussian measure without providing an explicit approximation of the density of the infinite dimensional change of coordinates induced by the flow, which is an important open question for many Hamiltonian PDEs and related models. Recent progresses in this direction have been made in [18], [6], [10], where the techniques developed allowed to get an exact formula for the density.…”