Transportation cost inequalities for singular SPDEs

TL;DR

This work establishes transportation cost inequalities (TCIs) for the laws of BPHZ random models arising in subcritical singular SPDEs. By leveraging regularity-structure lifting of noise to a model and an extended contraction principle, the authors propagate TCIs from the noise to the model and then to solution laws, even in the presence of renormalization. A key technical achievement is a Hölder-type bound on the BPHZ lift within a refined regularity-integrability framework, enabling explicit TCIs and Gaussian-tail bounds for model norms. As a consequence, TCIs hold for a broad class of singular SPDE solutions, including the full subcritical $\Phi^4_{4-\delta}$ measures on $\mathbf{T}^4$ and, more generally, invariant measures for related dynamics, providing robust probabilistic control in renormalized SPDE settings.

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. As a consequence, the laws of a number of solutions to some singular stochastic partial differential equations also satisfy some transportation cost inequalities. This is in particular the case of the $Φ^4_{4-δ}$ measures over the 4-dimensional torus, for all $0<δ<4$.