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Hyperbolic & Dispersive PDE Seminar

Invariant Gibbs measures for the three-dimensional cubic nonlinear wave equation

Bjoern Bringmann

Location:  Hill 705
Date & time: Thursday, 02 February 2023 at 3:30PM - 4:30PM

In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic ?43-model. This resultis the hyperbolic counterpart to seminal works on the parabolic ?43-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. We also connect our theorem with classical and recentdevelopments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a non-technical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we discuss our para-controlled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.