Transportation cost inequalities for singular SPDEs

We prove that the laws of the BPHZ random models satisfy some transportation cost inequalities in the full subcritical regime if there is no 'variance blowup' and the law of the noise is translation invariant and satisfies some transportation cost inequality. We emphasize two consequences of this result or its proof: The automatic integrability properties of the invariant probability measures of a number of singular stochastic partial differential equations, including the $\Phi^4_{4-\delta}$ measures over the $4$-dimensional torus, for all $0 < \delta < 4$, and a general large deviation principle satisfied by the BPHZ models.