Transportation cost inequalities for singular SPDEs

1. Problem Statement

The paper addresses the probabilistic properties of solutions to singular stochastic partial differential equations (SPDEs), specifically those requiring renormalization (e.g., the $\Phi^4_3$ and $\Phi^4_4$ models). While the theory of Regularity Structures (Hairer) and Rough Paths provides robust pathwise solution theories, establishing quantitative probabilistic estimates (such as concentration of measure and large deviations) for the laws of these solutions has been challenging.

The specific goal is to prove Transportation Cost Inequalities (TCI) for the laws of the BPHZ (Bogoliubov–Parasiuk–Hepp–Zimmermann) random models associated with these singular SPDEs. A TCI relates the relative entropy (Kullback-Leibler divergence) between two probability measures to the optimal transport cost between them. Proving such inequalities for the model implies strong concentration properties for the solution fields and enables the derivation of Large Deviation Principles (LDP).

2. Methodology and Functional Setting

The authors work within the framework of Regularity Structures, specifically utilizing a refined Regularity-Integrability Structure to handle the probabilistic lifting of noise to models.

A. Assumptions on the Noise

The noise $\xi$ is defined on a separable Banach space $\Omega$ containing a dense Hilbert subspace $H$. The law $P$ of the noise is assumed to satisfy Assumption A:

1. Translation Invariance: The law is invariant under spacetime shifts.
2. Transportation Cost Inequality: The law satisfies a TCI with respect to the cost function $c_H(\omega_1, \omega_2) = \|\omega_1 - \omega_2\|_H^2$ if $\omega_1 - \omega_2 \in H$, and $\infty$ otherwise.
   $$a_\circ T_{c_H}(Q, P) \leq H(Q|P)$$
   This assumption is stronger than the standard Spectral Gap inequality but is satisfied by Gaussian measures (where $H$ is the Cameron-Martin space) and is robust under perturbations.

B. The BPHZ Model Construction

The authors construct the BPHZ model $\hat{M}_{\infty; \varepsilon}$ using a limiting procedure involving a spectral gap assumption. A key innovation is the introduction of a refined Regularity-Integrability Structure that differs from previous works (e.g., Bailleul & Hoshino [1], Hairer & Steele [26]) in two ways:

1. Higher-Order Malliavin Calculus: Instead of restricting to first-order derivatives, the structure includes trees corresponding to all orders of Malliavin derivatives. This is necessary to obtain polynomial estimates in the perturbation parameter $h \in H$.
2. Extended Type Set: The structure includes two copies of the Hilbert space $H$ (denoted $\mathbf{H}$ and $\bar{\mathbf{H}}$) to distinguish between elements estimated in $B_{\infty, \infty}$ and $B_{p, \infty}$ norms, allowing for flexible control of integrability.

C. The Core Technical Tool: Extended Contraction Principle

The proof relies on an extension of the contraction principle for TCIs (Gasteratos & Jacquier). The strategy involves proving a pathwise Hölder estimate for the map lifting the noise to the model:
$$\|\hat{M}_{\infty; \varepsilon}(\omega + h) - \hat{M}_{\infty; \varepsilon}(\omega)\|_{\mathcal{M}} \leq L(\omega) \max\left(\|h\|_H, \|h\|_H^{1/r}\right)$$
where $L(\omega)$ has Gaussian tails and $r \geq 1$.

• Key Insight: Unlike previous works where first-order calculus sufficed, proving this specific polynomial bound requires higher-order Malliavin calculus. The authors show that the difference between models driven by $\omega$ and $\omega+h$ can be expanded using a Taylor-like series involving derivatives of the model with respect to the noise, where the coefficients are multilinear in $h$.

3. Key Contributions and Results

A. Main Theorem: TCI for BPHZ Models

Theorem 2 & 15: Under Assumption A and the absence of "variance blow-up," the law of the BPHZ model $\hat{M}_{\infty; \varepsilon}$ satisfies a family of transportation cost inequalities. Specifically, for any $1 \leq \alpha < 2$:
$$a_\alpha T_{c_M^\alpha}(Q, P_M) \leq H(Q|P_M)^{\alpha/(2r)} + H(Q|P_M)^{\alpha/2}$$
This implies that the norm of the BPHZ model has Gaussian tails:
$$\mathbb{E}\left[\exp(a \|\hat{M}_{\infty; \varepsilon}\|_{\mathcal{M}}^2)\right] < \infty$$

B. Application 1: Integrability of Invariant Measures

The authors apply the TCI for the model to the solutions of singular SPDEs. If the solution $u$ satisfies a pathwise bound of the form $g(\|u\|) \leq 1 + \|\hat{M}_{\infty; \varepsilon}\|_{\mathcal{M}}$ (where $g$ is concave and increasing), then the invariant probability measure $\mu$ of the dynamics satisfies a TCI.

• Specific Result: For the $\Phi^4_{4-\nu}$ measure on the 4-dimensional torus (for $0 < \nu < 4$), the invariant measure satisfies:
  $$\int \exp\left(a \|\phi\|_{C^{-1-\kappa+\nu}}^{2/p(\nu)}\right) \mu(d\phi) < \infty$$
  This establishes exponential integrability for these measures, a crucial property for Euclidean Quantum Field Theory (Osterwalder-Schrader axioms), derived without assuming log-Sobolev inequalities directly.

C. Application 2: Large Deviation Principle (LDP)

Theorem 19: The paper establishes a Large Deviation Principle for the BPHZ model $\hat{M}_{\infty; \varepsilon}^\kappa$ associated with scaled noise $\kappa \xi$ as $\kappa \to 0$.

• Rate Function: The rate function $J(M)$ is given by the infimum of the rate function of the noise over all $h \in H$ that map to the model $M$ via the "naive" lifting map $L(h)$.
  $$J(M) = \inf \{ I(h) : h \in H, L(h) = M \}$$
  This extends the seminal result of Hairer & Weber (2014) from the Allen-Cahn equation to a general class of singular SPDEs, using a pathwise approach rather than Wiener chaos decompositions.

4. Significance and Impact

1. Robustness to Noise: The results hold for a broad class of noises (not just Gaussian) provided they satisfy a spectral gap or TCI. This includes non-Gaussian perturbations.
2. Pathwise vs. Probabilistic: The proof is fundamentally pathwise. By establishing a deterministic Hölder continuity of the solution map with respect to the noise in the $H$-norm, the authors transfer probabilistic properties (TCI, LDP) from the noise to the solution. This avoids the need for explicit moment calculations or chaos expansions which are often intractable for highly singular equations.
3. Higher-Order Calculus: The introduction of a regularity structure capable of handling arbitrary order Malliavin derivatives is a significant technical advancement. It allows for the derivation of polynomial bounds in the perturbation parameter, which is essential for proving the specific form of the TCI required for the applications.
4. New Integrability Results: The paper provides the first proof of exponential integrability for $\Phi^4$ measures in the full subcritical regime (specifically in 4D) under minimal assumptions on the noise, bridging a gap between stochastic analysis and constructive quantum field theory.

In summary, the paper provides a unified, robust framework for establishing concentration of measure and large deviations for singular SPDEs by leveraging the pathwise structure of Regularity Structures combined with higher-order Malliavin calculus.