Propagation of Randomness in Nonlinear Evolution Equations

We are all familiar with dispersive and wave phenomena since we observe it all the time in nature. It could be as simple as when we look at a rainbow: dispersion causes the spatial separation of white light into different colors. Or when we look at the ripples that form when we throw a pebble in the lake: the expanding ring is called a “wave-packet” and we note that waves travel at different speeds, the longest going fastest and the shortest ones slowest.

But wave phenomena also arise in quantum mechanics, plasmas, fiber optics, ferromagnetism, atmospheric and water waves and many other settings. Because waves in nature interact in a nonlinear fashion as they propagate and have different properties such as amplitude, length, oscillation, speed, and position over time, it is important to understand how they may behave under certain noisy conditions or when taking measurements in certain media where small errors are unavoidable.

Understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero (a Bose-Einstein condensate) are two very different phenomena in nature but are both aspects of solutions to the same nonlinear model. Being able to understand and describe the dynamical behavior of solutions to such models given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagates is fundamental to accurately predict wave phenomena when studying the natural world.

This project is aimed at answering several central questions about long-time dynamics and the propagation of randomness in the context of nonlinear dispersive and wave phenomena. At the same time, the work of the project is designed to foster the training of graduate students and junior researchers in the U.S.

The synergy between deterministic and probabilistic approaches in the study of nonlinear evolution equations in the last few years has furthered our understanding of the dynamics of solutions in fundamental ways and opened the door to new paradigms that have moved research forward in various directions. To address important challenges at the cutting edge of current research, aimed at a quantitative understanding of the dynamical properties of generic wave phenomena, the principal investigator (PI) adopts an innovative approach based on the integration of methods and ideas from analysis, probability, statistical mechanics, dynamical systems, combinatorics and analytic number theory coupled with the impetus of recent new methods that were inspired by the spectacular advances in singular stochastic parabolic equations.

As part of this project, the PI will explore several exciting directions in three areas of research at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems aim at studying the long-time dynamics of dispersive flows from a probabilistic viewpoint, the invariance of Gibbs measures for the nonlinear Hartree equation - arising from the mean field limit for the N-body Schrödinger equation - and for the nonlinear wave and Schrödinger equations on tori; and at the development of a new probabilistic quasilinear hyperbolic theory.

The problems to be studied have the advantage that they are graded at different levels of difficulty, each leading to independent partial progress and deeper understanding. Some of the questions that will be pursued as part of this project lead to excellent research problems for graduate doctoral students and postdoctoral fellows. Furthermore, the PI’s work will lead to the development of new graduate topics courses, thus enriching the development of the new generation of researchers.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.