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We consider the problem of ergodicity for the $P(\Phi)_2$ measure of quantum field theory under the flow of the singular stochastic (damped) wave equation $u_{tt} + u_t + (1-\Delta) u + {:}\,p(u)\mspace{2mu}{:} = \sqrt 2 \xi$, posed on the two-dimensional torus $\mathbb T^2$. We show that the $P(\Phi)_2$ measure is ergodic, and moreover that it is the unique invariant measure for (the Markov process associated to) this equation which belongs to a fairly large class of probability measures over distributions. The main technical novelty of this paper is the introduction of the new concepts of asymptotic strong Feller and asymptotic coupling {restricted to the action of a group}. We first develop a general theory that allows us to deduce a suitable support theorem under these hypotheses, and then show that the stochastic wave equation satisfies these properties when restricted the action of translations by shifts belonging to the Sobolev space $H^{1-\varepsilon} \times H^{-\varepsilon}$. We then exploit the newly developed theory in order to conclude ergodicity and (conditional) uniqueness for the $P(\Phi)_2$ measure., Comment: 44 pages

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