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X-WR-CALDESC:Centro de Modelamiento Matemático
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DTSTART:20260616T074232
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DTSTART;TZID=America/Santiago:20260610T161500
DTEND;TZID=America/Santiago:20260610T180000
DTSTAMP:20260608T175628Z
CREATED:20260608
LAST-MODIFIED:20260608
PRIORITY:5
SEQUENCE:12
TRANSP:OPAQUE
SUMMARY:Chilean Probability Seminar: A stochastic collisional picture of the Wave Kinetic Equation
DESCRIPTION:Abstract:\nThe 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.\n\nIn 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.\nFurthermore, 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.\nBased on works in progress with Armand Bernou (U. Lyon 1), Roberto Cortez (UNAB, Chile) and G. Krstulovic (U. Cote d’Azur).\nThe link to join the seminar is:\nhttps://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09\nID de reunión: 845 2183 4914\nCódigo de acceso: 997973\nSpeaker: Joaquín Fontbona (Universidad de Chile)\n \n\n\n
URL:https://www.cmm.uchile.cl/events/chilean-probability-seminar-a-stochastic-collisional-picture-of-the-wave-kinetic-equation/
ORGANIZER;CN=CMM:MAILTO:
CATEGORIES:Seminarios
LOCATION:Sala Maryam Mirzakhani - 6to piso CMM
ATTACH;FMTTYPE=image/jpeg:https://www.cmm.uchile.cl/wp-content/uploads/2026/04/network-mesh-wire-digital-technology-background.jpg
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