Transportation Proof Rule in SPDE Analysis

• Transportation Proof Rule is a conceptual framework that transfers transportation-cost inequalities from a base noise law to a derived model law in singular SPDE contexts.
• It employs pathwise translation estimates, coupling arguments, and Hölder-type controls to bridge entropy bounds and regularity properties between noise and model laws.
• The framework underpins results like Gaussian tail estimates and invariant measure proofs, exemplified by applications to models such as Φ⁴₄₋δ.

Transportation Proof Rule denotes, in its most specific current usage, the abstract mechanism by which a transportation-cost inequality for a reference noise law is transferred to the law of a derived random object through a pathwise translation estimate. In "Transportation cost inequalities for singular SPDEs" (Bailleul et al., 24 Nov 2025), this mechanism is formalized as a generalized transportation-cost transfer for BPHZ random models in the full subcritical regime, and then propagated further to laws of singular SPDE solutions and invariant measures. In a broader literature, closely related transport proofs replace a target law or functional inequality by a transported reference object and then exploit change of variables, couplings, entropy duality, or variational structure to conclude; the common feature is that transport is not merely the subject matter, but the proof engine.

1. Formal meaning in the singular-SPDE setting

The paper (Bailleul et al., 24 Nov 2025) starts from a noise law $P$ on a Banach space $\Omega$ of distributions, with a dense Cameron–Martin type Hilbert space $H\subset \Omega$, and cost

$$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$

Its Assumption A requires that the noise is centered, invariant under spacetime shifts, and satisfies the transportation-cost inequality

$$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$

Under Assumption A, the appendix also yields the spectral gap inequality

$$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$

for cylindrical $F$ (Bailleul et al., 24 Nov 2025).

The paper’s key abstraction is Proposition $\ref{PropGeneralisedTCI}$, described there as a refined contraction principle. If $M:\Omega\to\mathsf M$ is a random variable into a metric space and there exist an exponent $r\ge1$ and a random weight $\Omega$0 such that for $\Omega$1-a.e. $\Omega$2,

$\Omega$3

then the law $\Omega$4 satisfies, for every $\Omega$5, an $\Omega$6-TCI, where $\Omega$7 is the convex inverse of a concave function of the form

$\Omega$8

(Bailleul et al., 24 Nov 2025).

In this narrow sense, the Transportation Proof Rule is not a generic slogan. It is an abstract transfer theorem: a TCI for the base noise plus a pointwise translation estimate for a measurable map $\Omega$9 imply a TCI for the pushforward law of $H\subset \Omega$0.

2. Entropy lifting, coupling, and the transfer mechanism

The proof idea in (Bailleul et al., 24 Nov 2025) is an entropy-lifting plus coupling argument. First, any target law $H\subset \Omega$1 on $H\subset \Omega$2 is lifted to a law $H\subset \Omega$3 on $H\subset \Omega$4 by the entropy-image identity

$H\subset \Omega$5

Second, the pointwise estimate for $H\subset \Omega$6 converts transportation in $H\subset \Omega$7 with cost $H\subset \Omega$8 into transportation in $H\subset \Omega$9 with cost $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$0. Third, Hölder’s inequality and the noise TCI are applied. Finally, the argument is pushed back down from $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$1 to $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$2 (Bailleul et al., 24 Nov 2025).

This architecture clarifies what is “transportation” about the rule. The proof does not compute the geometry of $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$3 directly. Instead, it transports entropy control from the reference law $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$4 through the measurable map $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$5, provided the map is sufficiently regular with respect to admissible translations by $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$6. The role of the random weight $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$7 is to permit nonuniform pathwise control while retaining integrability strong enough for the entropy estimates.

A common misconception is that a transportation proof rule must be a classical contraction principle with a deterministic Lipschitz constant. The proposition in (Bailleul et al., 24 Nov 2025) is explicitly more flexible: the control is Hölder type, the exponent $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$8 may be larger than $$c_H(\omega_1,\omega_2)= \begin{cases} \|h\|_H^2, & \omega_1-\omega_2=h\in H,\ +\infty, & \text{otherwise}. \end{cases}$$9, and the prefactor is random but belongs to $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$0. This suggests that the relevant regularity notion is not global Lipschitz continuity on the ambient space, but translation control along the Cameron–Martin directions.

3. BPHZ models and the pathwise translation estimate

The main application in (Bailleul et al., 24 Nov 2025) is the extended BPHZ model $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$1 for general subcritical singular SPDEs in regularity structures, built using a refined regularity-integrability structure and a BPHZ renormalization scheme. The central estimate is Theorem $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$2: $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$3 for some integer $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$4 and $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$5. In the homogeneous model norm $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$6, the theorem is stated as

$$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$7

and for all $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$8, $$a_\circ\, \mathcal T_{c_H}(\mathbb Q,P)\le H(\mathbb Q\mid P) \qquad (\forall \mathbb Q\in \mathsf P(\Omega)).$$9,

$$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$0

The random prefactor is controlled by

$$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$1

The hypotheses behind the BPHZ construction are equally specific. The law $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$2 is centered and shift-invariant on $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$3, satisfies Assumption A with $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$4, and obeys the no-variance-blowup condition

$$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$5

For small $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$6, large enough weights $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$7, all $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$8, and $$\mathbb E[(F-\mathbb E F)^2]\le a_1\,\mathbb E[\|dF\|_{H^*}^2], \qquad a_1=\frac1{2a_\circ},$$9, the paper obtains uniform $F$0-bounds on renormalized models and convergence to a limit model in $F$1 (Bailleul et al., 24 Nov 2025).

The proof of the translation estimate is algebraic and combinatorial. For the deterministic renormalized model, shifting the noise by $F$2 yields the binomial formulas

$F$3

and

$F$4

These identities expand the effect of adding $F$5 to the noise as a sum over subsets of noise leaves. The estimate then follows by counting how many noise leaves are replaced, using homogeneity, and exploiting $F$6, which yields the Hölder-type dependence $F$7 (Bailleul et al., 24 Nov 2025).

4. From model laws to solution laws and invariant measures

Once the pathwise translation estimate is available, the generalized transfer proposition is applied to

$F$8

The resulting corollary states that for every $F$9, the law $\ref{PropGeneralisedTCI}$0 of the BPHZ model satisfies an $\ref{PropGeneralisedTCI}$1-transportation-cost inequality. Taking $\ref{PropGeneralisedTCI}$2 and combining with Gozlan’s entropy–integrability theorem yields the Gaussian tail estimate

$\ref{PropGeneralisedTCI}$3

(Bailleul et al., 24 Nov 2025).

The solution-level transfer requires an a priori estimate controlled by the model. If a singular SPDE solution $\ref{PropGeneralisedTCI}$4 is globally well-posed and

$\ref{PropGeneralisedTCI}$5

with $\ref{PropGeneralisedTCI}$6 continuous, concave, strictly increasing, and $\ref{PropGeneralisedTCI}$7, then the law of $\ref{PropGeneralisedTCI}$8 satisfies an $\ref{PropGeneralisedTCI}$9-TCI with $M:\Omega\to\mathsf M$0. If the SPDE generates a Feller process and a coming down from infinity estimate

$M:\Omega\to\mathsf M$1

holds, then the process has an invariant measure $M:\Omega\to\mathsf M$2, and $M:\Omega\to\mathsf M$3 satisfies the same kind of TCI. The proof uses time-shift invariance of the noise, uniform exponential integrability of $M:\Omega\to\mathsf M$4, tightness of Cesàro averages, and Gozlan’s equivalence between TCI and exponential integrability (Bailleul et al., 24 Nov 2025).

A flagship application is the $M:\Omega\to\mathsf M$5 model on $M:\Omega\to\mathsf M$6,

$M:\Omega\to\mathsf M$7

This dynamics has a unique invariant probability measure, called the $M:\Omega\to\mathsf M$8 measure. The paper proves that there exists $M:\Omega\to\mathsf M$9 such that this measure satisfies an $r\ge1$0-TCI, where

$r\ge1$1

The derivation uses the BPHZ translation estimate, the a priori polynomial bound from Chandra–Moinat–Weber, and the invariant-measure theorem (Bailleul et al., 24 Nov 2025).

5. Broader transport-proof patterns in adjacent literatures

Outside the singular-SPDE setting, transport proofs have a wider methodological meaning. In "Optimal Transportation to the Entropy-Power Inequality" (Rioul, 2017), the entropy-power inequality is proved by choosing Gaussian reference variables $r\ge1$2, transporting them to $r\ge1$3 through increasing maps $r\ge1$4, invoking the entropy change-of-variables formula

$r\ge1$5

and combining a rotation trick with concavity of $r\ge1$6. In that setting, the proof rule is: transport to a Gaussian equality case, rewrite entropy differences through Jacobians, and conclude by Jensen.

In "Transportation Proof of an inequality by Anantharam, Jog and Nair" (Courtade, 2019), the transport device is Brenier’s map from block-Gaussian sources. The key lemma lower-bounds $r\ge1$7 by a determinant expression involving $r\ge1$8, and the final inequality is obtained by optimizing over positive definite block matrices. Here the proof rule is again transfer from a Gaussian reference law, but through Jacobians of convex-gradient maps rather than through couplings on path space.

In "Transport Proofs Of Some Discrete Variants Of The Prékopa-Leindler Inequality" (Gozlan et al., 2019), smooth Jacobian arguments are unavailable. The replacement is monotone coupling plus entropy duality. On $r\ge1$9, compactly supported laws $\Omega$00 are coupled monotonically, pushed through midpoint maps

$\Omega$01

and shown to satisfy

$\Omega$02

Combined with the entropy variational formula, this yields the Klartag–Lehec inequality and, by reduction and tensorization, the Four Functions Theorem. The transport proof rule here is explicitly coupling-and-entropy rather than Jacobian-based.

Two further examples show the same structural theme under different technical realizations. In the variational proof of partial regularity for optimal transport maps (Goldman et al., 2017), the argument goes through the Benamou–Brenier formulation, harmonic approximation, and Campanato iteration: if the transport cost is small, the Eulerian velocity field is close to the gradient of a harmonic function, and this is translated back to the map. In the stochastic heat equation setting (Shang et al., 2019), a $\Omega$03 inequality on path space is derived by Girsanov’s theorem, a coupling of perturbed and unperturbed solutions, a new uniform-metric moment estimate for the stochastic convolution, and Gronwall’s lemma. These examples suggest that “transportation proof rule” can designate either a formal transfer theorem, as in (Bailleul et al., 24 Nov 2025), or a broader proof schema in which transport converts a nonlinear statement into a controllable reference comparison.

6. Scope, structural features, and recurrent misunderstandings

Several features recur across these sources. First, the reference object is usually simpler than the target: a Gaussian law in entropy inequalities, the base noise law in singular SPDEs, a monotone coupling in discrete midpoint inequalities, or a low-excess harmonic approximation in variational regularity theory. Second, the decisive estimate is rarely a direct bound on the target quantity. It is a transport identity or transport inequality: an entropy-image identity, a Jacobian formula, a monotone-coupling entropy convexity bound, or a pathwise translation estimate. Third, the conclusion is typically obtained after pushing the problem through this intermediate structure.

A common misunderstanding is to identify transport proofs exclusively with optimal maps of Monge type. The cited literature does not support that restriction. Some proofs do use Brenier or monotone rearrangement maps (Rioul, 2017, Courtade, 2019, Hindawi et al., 2011), but others rely on couplings (Gozlan et al., 2019), dynamic Eulerian competitors (Goldman et al., 2017), or Girsanov couplings on stochastic path space (Shang et al., 2019). Conversely, not every proof involving optimal transport geometry is a transportation proof rule in the narrow sense of (Bailleul et al., 24 Nov 2025); there, the phrase refers to an abstract extended contraction principle transferring TCI from noise to model law.

The singular-SPDE formulation is distinctive because it isolates a reusable chain of implications:

1. centered, shift-invariant noise satisfying a TCI on $\Omega$04 with respect to $\Omega$05;
2. spectral gap and exponential-integrability consequences for the noise;
3. BPHZ construction under no variance blowup;
4. pathwise translation control for the renormalized model;
5. generalized TCI transfer to the model law;
6. TCI and Gaussian tails for the model norm;
7. TCI for solution laws and invariant measures, including $\Omega$06 on $\Omega$07 (Bailleul et al., 24 Nov 2025).

In that sense, Transportation Proof Rule names both a proposition and a style of reasoning. Its precise technical content depends on context, but the unifying principle is stable: transport furnishes the comparison mechanism that converts entropy, regularity, or cost information at a reference level into a structural statement for a more complicated law or map.