Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

TL;DR

This paper proves the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, advancing the understanding of hyperbolic stochastic PDEs through novel analytical techniques and representations.

Contribution

It establishes local well-posedness and invariance of the Gibbs measure for the hyperbolic $ ext{Φ}^4_3$-model using para-controlled calculus, random tensor theory, and new stochastic estimates.

Findings

Proves invariance of Gibbs measure under the 3D cubic wave dynamics.

Introduces a caloric representation linking parabolic and hyperbolic theories.

Develops new bilinear random tensor estimates for stochastic analysis.

Abstract

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic 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. The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled Ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates. The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate.