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

Abstract: 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 $\Phi^4_3$-model. This result is the hyperbolic counterpart to seminal works on the parabolic $\Phi^4_3$-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 recent developments 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 inter- play of both parabolic and hyperbolic theories. Then, we briefly discuss the local dynamics of the cubic nonlinear wave equation, focusing on a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.

statistical mechanicsmathematical physicsanalysis of PDEsprobability

Audience: researchers in the topic

Analysis, Quantum Fields, and Probability

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Recorded talk are available on youtube.