Chilean Probability Seminar: A stochastic collisional picture of the Wave Kinetic Equation
Abstract:
The wave kinetic equation (WKE) is a fundamental effective equation in wave turbulence theory that describes the evolution of the energy spectrum in a weakly interacting nonlinear wave system, with important applications in fields such as oceanography, plasma physics and nonlinear optics. Recent breakthrough results by Deng and Hani established the first full-range mathematical derivation of the 4-wave (or cubic) WKE from the (dispersive, reversible) nonlinear Schrödinger equation under suitable scaling limits.
In this talk, we introduce an alternative, purely stochastic collisional description of the WKE. In this setting, binary collisions between wave-vectors are proposed analogously to particle’s in the spatially homogeneous Boltzmann equation (hBE) , but are accepted or rejected depending on the local density at the post-collisional configurations, thus reflecting the additional nonlinearity of the cubic WKE. We formalize this interpretation rigorously by constructing a Poisson-driven nonlinear process in the spirit of Tanaka’s process for the hBE, with flow of time-marginal laws equal to a given solution of the WKE.
Furthermore, this stochastic viewpoint allows us to introduce and analyze a regularized mean-field interacting jump-particle system that extends Kac systems for hBE to the WKE framework. This particle system is expected to satisfy the propagation of chaos property and provide an approximation of the WKE as the number of particles tends to infinity and the regularizing parameter vanishes. We provide partial rigorous mathematical results in these directions in dimensions 2 and 3, and discuss the distinct complexities arising in each case. Finally, we present simulations to illustrate the numerical scheme that naturally arises from our approach.
Based on works in progress with Armand Bernou (U. Lyon 1), Roberto Cortez (UNAB, Chile) and G. Krstulovic (U. Cote d’Azur).
The link to join the seminar is:
https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09
ID de reunión: 845 2183 4914
Código de acceso: 997973
Speaker: Joaquín Fontbona (Universidad de Chile)