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

TL;DR

This paper proves ergodicity and uniqueness of the invariant measure for the infinite-volume $\

Contribution

It establishes the global well-posedness and ergodic properties of the $\

Findings

Solutions converge exponentially fast at high temperature

The invariant measure satisfies Osterwalder--Schrader axioms

Exponential decay of correlations

Abstract

We consider the infinite volume $\Phi^4_3$ 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 $\Phi^4_3$ 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.