Almost Sure Scattering For The One Dimensional Nonlinear Schrodinger Equation
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Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation
Author: Nicolas Burq
language: en
Publisher: American Mathematical Society
Release Date: 2024-05-15
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Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation
Publisher's description: We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p>1. On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On Rd, the large time dispersion shows that the only invariant measure is the d measure on the trivial solution u=0, and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon-Nikodym derivatives of these measures with respect to each other and we characterise their Lp regularity. We deduce from this precise description the global well-posedness of the equation p>1for and scattering for p>3 (actually even for 1>p>3, we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their Lp+1 norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.
Stochastic Partial Differential Equations and Related Fields
This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.