Gibbs measures and propagation of randomness under the flow of nonlinear dispersive PDE

Abstract: In groundbreaking work, Bourgain ’96 put forward a random data theory to study the existence of strong solutions on the statistical ensemble of Gibbs measures associated to dispersive equations. Despite numerous follow-up works to those of Bourgain’s, fundamental questions remained open. How does a given initial random data get transported by a nonlinear flow ? If it is Gaussian initially, how does this Gaussianity propagate? What is the description of the solution beyond the linear evolution?

In recent work, joint with Yu Deng and Haitian Yue, we developed the theory of random tensors, a powerful new framework which allows us to unravel the propagation of randomness under the NLS flow beyond the linear evolution of random data, and answer these questions in an optimal range relative to what we define as the probabilistic scaling. In particular, we establish the invariance of Gibbs measures for 2D NLS and 3D Hartree NLS equations using the method of random averaging operators, a first order approximation to the full random tensor theory.

In this talk we will describe these results, and explain the ideas behind them. We conclude with some open problems and within this context present new work, joint with Bjoern Bringmann, Yu Deng and Haitian Yue establishing the invariance of the 3D Gibbs measure under the flow of the nonlinear wave equation.

Bio Andrea R. Nahmod is Professor of Mathematics at the University of Massachusetts Amherst. Her research lies at the interface of nonlinear Fourier and harmonic analysis, and the theory of partial differential equations. Some of her recent work aims at gaining a quantitative understanding of the propagation of randomness under the nonlinear evolution of dispersive PDE. Nahmod is a Fellow of the American Mathematical Society; a recipient of the Sargent-Faull Fellowship at Harvard’s Radcliffe Institute for Advanced Study, and of a Simons Foundation Fellowship. Nahmod was twice a member of the Institute for Advanced Study at Princeton and held Simons’ Professorships at MSRI, Berkeley and at CRM Montreal. She delivered an invited address at the 2021 Joint Mathematical Meetings of the AMS.

analysis of PDEs

Audience: researchers in the topic

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Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

The talks are open to the public and, due to security reasons, all attendees have to register by following the link siam.zoom.us/webinar/register/WN_wShvDipBQw6WafNN6QO5wg

The link will contain detailed information about the talks. The registration is quick (asks only for your name and email) and, once registered, you will receive an email with the link to the webinars, which is unique to you, so please do not share that email. The registration is valid for all this webinar series, and you will receive a reminder about future talks.

Further, you will receive a follow-up email up to 7 days after each webinar, with a link to the on-demand SIAM PDE YouTube playlist recording.

Updated information and schedule will be available at sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

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