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We prove the invariance of the Gibbs measure under the dynamics of thethree-dimensional cubic wave equation, which is also known as the hyperbolic$\Phi^4_3$-model. This result is the hyperbolic counterpart to seminal works onthe parabolic $\Phi^4_3$-model by Hairer '14 and Hairer-Matetski '18. The heartof the matter lies in establishing local in time existence and uniqueness ofsolutions on the statistical ensemble, which is achieved by using apara-controlled Ansatz for the solution, the analytical framework of the randomtensor theory, and the combinatorial molecule estimates. The singularity of theGibbs measure with respect to the Gaussian free field brings out a new caloricrepresentation of the Gibbs measure and a synergy between the parabolic andhyperbolic theories embodied in the analysis of heat-wave stochastic objects.Furthermore from a purely hyperbolic standpoint our argument relies on key newingredients that include a hidden cancellation between sextic stochasticobjects and a new bilinear random tensor estimate.

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation