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We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known., Comment: 42 pages

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the $d$-dimensional torus. This class includes the wave equation for $d=1$ and the beam equation for $d\le 3$. We show that the Gibbs measure of the equation without forcing and damping is the unique invariant measure for the flow of this system. Since the flow does not satisfy the Strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible., Comment: The title has being changed from "Unique ergodicity for stochastic hyperbolic equations with additive space-time white noise" to "Unique ergodicity for a class of stochastic hyperbolic equations with additive space-time white noise", many typo corrections, some minor corrections in the proofs