Transport-Information Inequalities Overview

• Transport-information inequalities are defined as functional inequalities that link optimal transport costs with Fisher or Donsker–Varadhan information in Markov process settings.
• They establish a hierarchy connecting quadratic, Talagrand, Poincaré, and log-Sobolev inequalities through techniques like Lyapunov drift conditions and variational principles.
• These inequalities offer practical insights into regularity estimates for optimal maps and serve as a bridge between geometric, stochastic, and quantum analyses.

A transport-information inequality is a functional inequality that compares an optimal transport cost with an information functional. In the formulation singled out in the survey literature, one writes

$a(T_c(\nu,p)) \le I(\nu\mid p),$

where $T_c(\nu,p)$ is an optimal transport cost from $\nu$ to a reference measure $p$, $a$ is increasing with $a(0)=0$, and $I(\nu\mid p)$ is Fisher or Donsker–Varadhan information associated with a reversible Markov process (Gozlan et al., 2010). In the quadratic case this becomes the transportation-information inequality $W_2I$, typically written as

$W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$

or equivalently $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$ (Liu, 2015). The subject lies at the intersection of optimal transport, Dirichlet forms, entropy dissipation, concentration of measure, curvature, and PDE regularity; related papers also use the same transport-geometric mechanism to control Sobolev norms of optimal maps and to interpolate between entropy, transport, and Fisher information in both classical and quantum settings (Kolesnikov, 2010, Rouzé et al., 2017).

1. General template and dynamical meaning

The general transport-information template is

$T_c(\nu,p)$0

with $T_c(\nu,p)$1 defined from a Dirichlet form. For a $T_c(\nu,p)$2-reversible Markov semigroup $T_c(\nu,p)$3 with Dirichlet form $T_c(\nu,p)$4, the relevant information functional is

$T_c(\nu,p)$5

In the standard diffusion setting $T_c(\nu,p)$6, this becomes

$T_c(\nu,p)$7

and it differs from the more common $T_c(\nu,p)$8 by a factor $T_c(\nu,p)$9 (Gozlan et al., 2010).

The dynamical meaning of this choice of information is central. If $\nu$0 is a stationary ergodic Markov process with invariant law $\nu$1, then the occupation measure

$\nu$2

has a large deviation principle with rate function $\nu$3. In this sense, transport-information inequalities are the Markov-process analogue of transport-entropy inequalities: relative entropy governs i.i.d. empirical measures, while Donsker–Varadhan information governs empirical occupation measures (Gozlan et al., 2010).

This viewpoint yields explicit deviation theory. The survey states that

$\nu$4

is equivalent to exponential Laplace bounds for time averages of 1-Lipschitz observables and to deviation inequalities of the form

$\nu$5

for centered 1-Lipschitz $\nu$6 (Gozlan et al., 2010). The transport side therefore encodes geometry, while the information side encodes dynamical rarity.

2. The quadratic inequality $\nu$7 and its position in the hierarchy

On a connected complete finite-dimensional Riemannian manifold $\nu$8, with

$\nu$9

the quadratic transportation-information inequality is

$p$0

where, for $p$1 with $p$2,

$p$3

and $p$4 otherwise (Liu, 2015). This is the form usually denoted $p$5.

The hierarchy recorded in the supplied literature is precise. It was already known that $p$6 implies Talagrand’s quadratic transport-entropy inequality $p$7, and under lower Bakry–Émery curvature bounds the HWI inequality relates $p$8 and logarithmic Sobolev inequalities. The survey states the implications

$p$9

and, under

$a$0

it adds that $a$1 implies

$a$2

when $a$3 (Gozlan et al., 2010). In the terminology of (Lacker et al., 2020), $a$4 is stronger than Poincaré and weaker than log-Sobolev in diffusion settings.

A related refinement is the restricted quadratic transportation-information inequality. If $a$5 is $a$6-semi-convex and $a$7, the restricted inequality takes the form

$a$8

The paper proves that this restricted version is equivalent to Talagrand’s $a$9, with explicit constant conversion: from $a(0)=0$0 to $a(0)=0$1, one may take $a(0)=0$2, and conversely $a(0)=0$3 (Liu, 2015). This identifies a sharp “restricted” bridge between entropy and Fisher-information transport control.

At the lower end of the hierarchy, the Poincaré inequality is tied to transportation-variance bounds. One paper proves that Poincaré is equivalent to several quadratic transportation-variance inequalities, including

$a(0)=0$4

and uses the same argument to derive further characterizations and a direct route from Lyapunov conditions to $a(0)=0$5 (Liu, 2019). This places transportation-information theory inside a broader ladder of variance, entropy, and Fisher-information inequalities.

3. Characterizations by Lyapunov drift, variational principles, and dimension-free concentration

A central structural result is the Lyapunov characterization of $a(0)=0$6. On a complete Riemannian manifold, the paper (Liu, 2015) proves that the following are equivalent: first, $a(0)=0$7 satisfies $a(0)=0$8; second, there exists a Lyapunov function $a(0)=0$9 with $I(\nu\mid p)$0 locally bounded such that

$I(\nu\mid p)$1

in the sense of distributions, for some $I(\nu\mid p)$2, $I(\nu\mid p)$3, and some point $I(\nu\mid p)$4. This criterion removes curvature assumptions from the characterization and recasts transport-information control as a coercive quadratic drift condition for the generator.

The same paper proves a bounded-perturbation principle of Holley–Stroock type. If $I(\nu\mid p)$5 is absolutely continuous with respect to $I(\nu\mid p)$6 and

$I(\nu\mid p)$7

for some $I(\nu\mid p)$8, then $I(\nu\mid p)$9 satisfying $W_2I$0 implies that $W_2I$1 also satisfies $W_2I$2 (Liu, 2015). This transference principle is one of the main robustness properties of the inequality.

A complementary approach is variational. The paper (Fontbona et al., 2015) studies functionals of the form

$W_2I$3

or in the quadratic setting $W_2I$4. Transport inequalities become statements that $W_2I$5 is the minimizer of $W_2I$6. Nontrivial minimizers satisfy an Euler–Lagrange relation involving a Kantorovich potential $W_2I$7: $W_2I$8 From this framework the paper recovers the implication

$W_2I$9

and records the constant comparison

$W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$0

(Fontbona et al., 2015).

A further characterization identifies $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$1 with a dimension-free concentration property for product Markov processes. For an ergodic reversible Markov process with invariant law $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$2, the paper (Lacker et al., 2020) proves that $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$3 is equivalent to a family of product-space statements, including the Feynman–Kac semigroup bound

$W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$4

for every $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$5, $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$6, $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$7, and every 1-Lipschitz $W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$8, together with the dimension-free deviation estimate

$W_2(\nu,\mu)^2 \le 4C\,I(\nu\mid \mu),$9

This is the Markov-process counterpart of Gozlan’s dimension-free characterization of $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$0.

4. Sobolev regularity of optimal transport as a transport-information inequality

A distinct but closely related line of work interprets transport-information inequalities as regularity estimates for the optimal map itself. In the global Euclidean setting

$W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$1

with optimal map $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$2, the Monge–Ampère equation reads

$W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$3

or equivalently

$W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$4

Assuming uniform convexity of the target,

$W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$5

Theorem 3.1 of (Kolesnikov, 2010) gives the global, dimension-free Hessian bound

$W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$6

This estimate is explicitly presented as a transport-information inequality: the Fisher information of the source controls an $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$7-Sobolev norm of the transport map.

The same paper develops an $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$8-family of estimates for second derivatives of $W_2(\nu,\mu)\le 2\sqrt{C\,I(\nu\mid\mu)}$9, with $T_c(\nu,p)$00, and derives matrix-level bounds such as

$T_c(\nu,p)$01

In the limiting $T_c(\nu,p)$02 regime, the estimates recover Caffarelli’s contraction theorem. In particular, when $T_c(\nu,p)$03 is Gaussian and $T_c(\nu,p)$04, the optimal map from $T_c(\nu,p)$05 to $T_c(\nu,p)$06 is $T_c(\nu,p)$07-Lipschitz (Kolesnikov, 2010).

A key transport-geometric comparison in the paper is

$T_c(\nu,p)$08

valid for every vector $T_c(\nu,p)$09. This is derived from a generalized Talagrand inequality and then differentiated to obtain the Hessian estimate. In this formulation, the cost of translating the source density is bounded below by squared displacement of the optimal transport map, with the curvature of the target producing the factor $T_c(\nu,p)$10 (Kolesnikov, 2010).

The Gaussian specialization is especially explicit. If $T_c(\nu,p)$11 and $T_c(\nu,p)$12, then the paper proves a dimension-free identity/estimate in which the Gaussian relative Fisher information dominates several nonnegative transport terms, including

$T_c(\nu,p)$13

and

$T_c(\nu,p)$14

This identifies the Gaussian log-Sobolev inequality and the Sobolev regularity of the transport map as consequences of a single transport-information structure (Kolesnikov, 2010).

5. Geometric and stochastic realizations

On compact $T_c(\nu,p)$15-dimensional Riemannian manifolds with

$T_c(\nu,p)$16

mass transport yields a curvature-adapted transport inequality

$T_c(\nu,p)$17

where $T_c(\nu,p)$18 is normalized volume, $T_c(\nu,p)$19, and $T_c(\nu,p)$20 is an explicit nonquadratic cost built from

$T_c(\nu,p)$21

Its linearization gives exactly the sharp Poincaré inequality

$T_c(\nu,p)$22

The paper emphasizes that the “naive” quadratic-cost transport approach would only recover the wrong constant $T_c(\nu,p)$23; the modified cost and dimensional entropy restore the sharp spectral gap (Cordero-Erausquin, 2014). It also notes that a transport proof of the sharp log-Sobolev inequality on positively curved manifolds remains open.

For reflected diffusions in a convex domain $T_c(\nu,p)$24, the law $T_c(\nu,p)$25 on path space $T_c(\nu,p)$26 satisfies a dimension-free Talagrand-type inequality

$T_c(\nu,p)$27

provided the drift obeys the one-sided Lipschitz condition

$T_c(\nu,p)$28

The constant is

$T_c(\nu,p)$29

For reflected Brownian motion with constant diffusion matrix $T_c(\nu,p)$30, this simplifies to $T_c(\nu,p)$31; in the standard case $T_c(\nu,p)$32,

$T_c(\nu,p)$33

The proof hinges on the convexity observation

$T_c(\nu,p)$34

which makes the reflection term stabilizing rather than expansive (Pal et al., 2018).

A path-space analogue also holds for a nonlinear hyperbolic SPDE. For the stochastic wave equation in $T_c(\nu,p)$35,

$T_c(\nu,p)$36

with Gaussian noise white in time and correlated in space, the law $T_c(\nu,p)$37 on $T_c(\nu,p)$38 satisfies

$T_c(\nu,p)$39

under the stated Lipschitz, covariance, and initial-data hypotheses. The proof uses Girsanov representation, a coupling estimate, and Gronwall’s inequality, and it yields exponential integrability and Hoeffding-type concentration for Lipschitz functionals of the solution path (Li et al., 2018).

These examples show that the transport-information paradigm is not confined to static Euclidean measures. It persists, with modified costs and norms, on compact manifolds, reflected path spaces, and SPDE path spaces.

6. Discrete, point-process, quantum, and information-constrained extensions

A common misconception is that quadratic $T_c(\nu,p)$40-theory should transfer unchanged to discrete spaces. The discrete literature states the opposite: if a probability measure has support intersecting two sets at positive distance, then it cannot satisfy $T_c(\nu,p)$41 for any $T_c(\nu,p)$42. For Markov chains on countable spaces, the appropriate replacements are $T_c(\nu,p)$43-transport-information and weak transport-information inequalities. Under $T_c(\nu,p)$44,

$T_c(\nu,p)$45

and one also has the weak quadratic bound

$T_c(\nu,p)$46

Under the exponential curvature condition $T_c(\nu,p)$47,

$T_c(\nu,p)$48

and under coarse Ricci curvature $T_c(\nu,p)$49,

$T_c(\nu,p)$50

These inequalities lead further to $T_c(\nu,p)$51-type transport-entropy bounds and to a discrete Bonnet–Myers theorem (Fathi et al., 2015).

On configuration spaces, transport inequalities lift from single-site laws to laws of whole point processes. If the base law $T_c(\nu,p)$52 satisfies Talagrand’s inequality on $T_c(\nu,p)$53, then the mixed binomial point-process law $T_c(\nu,p)$54 satisfies a process-level inequality in which the cost of changing the point count appears as an additional entropy term: $T_c(\nu,p)$55 For Poisson point processes with arbitrary $T_c(\nu,p)$56-finite intensity, the paper proves a universal Marton-type transport-entropy inequality and derives concentration and modified logarithmic Sobolev consequences; taking $T_c(\nu,p)$57 recovers a universal Marton inequality for Poisson processes (Gozlan et al., 2020). Although the right-hand side here is relative entropy rather than Donsker–Varadhan information, the mechanism is unmistakably transport-information in the broader geometric sense.

In noncommutative probability, a quantum HWI inequality provides the analogue of the classical transport-information bridge. For a primitive detailed-balance quantum Markov semigroup with invariant state $T_c(\nu,p)$58, if the Carlen–Maas quantum Ricci lower bound satisfies $T_c(\nu,p)$59, then

$T_c(\nu,p)$60

Here $T_c(\nu,p)$61 is quantum relative entropy, $T_c(\nu,p)$62 the Carlen–Maas quantum Wasserstein distance, and $T_c(\nu,p)$63 quantum Fisher information. This yields modified logarithmic Sobolev and transport-cost inequalities as corollaries (Rouzé et al., 2017).

A further broadening replaces Fisher information by mutual information constraints on couplings. The information-constrained transport cost

$T_c(\nu,p)$64

interpolates between independent coupling at $T_c(\nu,p)$65 and unconstrained optimal transport as $T_c(\nu,p)$66. For Gaussian target $T_c(\nu,p)$67, the paper proves

$T_c(\nu,p)$68

a strict sharpening of Talagrand’s Gaussian transport inequality when $T_c(\nu,p)$69, and uses it in Marton-type concentration and in a converse for Cover’s relay-channel problem (Bai et al., 2020).

Transport-information inequalities thus form a family rather than a single statement. In the strict Markov-semigroup sense they compare transport cost with Fisher or Donsker–Varadhan information; in adjacent frameworks they compare transport with entropy, Sobolev norms of optimal maps, or even mutual information of couplings. Across these variants, the recurring structure is the same: geometry in Wasserstein-type space is constrained by a dissipation functional, and that constraint propagates to concentration, regularity, contraction, and functional inequalities.