Ergodicity of infinite volume ${\mathrm{\Phi }}_3^4$ at high temperature

We consider the infinite volume ${\mathrm{\Phi }}_3^4$ dynamic and show that it is globally well-posed in a suitable weighted Besov space of distributions. At high temperatures / small coupling, we furthermore show that the difference between any two solutions driven by the same realisation of the noise converges to zero exponentially fast. This allows us to characterise the infinite-volume ${\mathrm{\Phi }}_3^4$ measure at high temperature as the unique invariant measure of the dynamic, and to prove that it satisfies all Osterwalder–Schrader axioms, including invariance under translations, rotations, and reflections, as well as exponential decay of correlations.