Transportation cost inequalities for singular SPDEs

TL;DR

This paper establishes transportation cost inequalities for laws of BPHZ random models in singular SPDEs, leading to results on integrability of invariant measures and large deviation principles in the subcritical regime.

Contribution

It proves transportation cost inequalities for BPHZ models under certain conditions, enabling new insights into invariant measures and large deviations for singular SPDEs.

Findings

Invariant measures of singular SPDEs exhibit automatic integrability.

BPHZ models satisfy large deviation principles.

Transportation inequalities hold under no 'variance blowup' condition.

Abstract

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.