Ergodicity for the hyperbolic $P(\Phi)_2$-model

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.