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

We consider the problem of ergodicity for the $P(\mathrm{\Phi }{)}_2$ measure of quantum field theory under the flow of the singular stochastic (damped) wave equation $u_{tt}+u_t+(1-\mathrm{\Delta })u+:p(u)\text{mspace}2mu:=\sqrt{2}\xi$, posed on the two-dimensional torus ${\mathbb{T}}^2$. We show that the $P(\mathrm{\Phi }{)}_2$ 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 $H^{1-\epsilon }\times H^{-\epsilon }$. We then exploit the newly developed theory in order to conclude ergodicity and (conditional) uniqueness for the $P(\mathrm{\Phi }{)}_2$ measure.