Equivariant Optimal Transport Theory

• Equivariant optimal transport is defined as transport under symmetry constraints enforcing invariance via group actions, achieving cost minimization with local-to-global consistency.
• It formulates cost functions per unit volume, ensuring finite mean cost in infinite ergodic systems through bounded-region approximations and mixing techniques.
• It guarantees uniqueness and a map-induced structure for optimal semicouplings, forming a metric space framework for random measures with applications in allocation and matching.

Equivariant optimal transport is the paper of optimal transport phenomena under symmetry or invariance constraints imposed by group actions, spatial or combinatorial patterns, or other forms of equivariance. In this setting, the goal is not merely to minimize a cost functional between measures or distributions, but to do so in a manner that respects underlying symmetries—typically formalized as invariance (or equivariance) under group actions, tiling automorphisms, or prescribed structure. This perspective gives rise to powerful existence, uniqueness, continuity, and structural results, and it allows the transport theory to connect deeply with ergodic theory, topological dynamics, probability, geometry, and representation theory.

1. Equivariant Random Measures and Symmetry Constraints

The foundational framework involves random measures defined on a base space $M$ (typically a Riemannian manifold, most often $\mathbb{R}^d$) equipped with the isometric action of a locally compact group $G$ (e.g., Euclidean motions, translations, or discrete symmetries). A random measure $\lambda^\bullet$ on $M$ is equivariant (sometimes called stationary) with respect to $G$ if for every $g \in G$ and every Borel set $A \subset M$,

$\lambda^{\theta_g\omega}(gA) = \lambda^\omega(A)$

almost surely, where $\theta_g$ encodes the action on the probability space. Equivariance is essential for formulating and solving global transport problems by extending the behavior of finite-volume structures to infinite systems via group invariance.

This framework encompasses transporting Lebesgue measure to a Poisson or compound Poisson process in homogeneous spaces and constructing equivariant allocation (partition) rules. The requirement of equivariance is also critical for meaningful definitions of costs per volume, as it guarantees that transport statistics do not depend on arbitrary anchoring choices or bounded region selection (Huesmann, 2012).

2. Cost Function Formulation and Mean Cost per Volume

The cost functional in equivariant optimal transport is often chosen to be of the form

$c(x, y) = \theta(d(x, y))$

where $\theta$ is strictly increasing, continuous, $\theta(0) = 0$, and $\lim_{r \to \infty} \theta(r) = \infty$. Important examples include polynomial costs $c(x, y) = |x - y|^p$ for $p \geq 1$ and exponential costs $c(x, y) = \exp(\kappa|x - y|^d)$. The focus is on mean transportation cost per unit volume:

$$\mathcal{C}(q^\bullet) = \mathbb{E}\left[\int_{M \times B_0} c(x, y) q^\bullet(dx, dy)\right]$$

where $B_0$ is a fixed fundamental domain for the group action, $q^\bullet$ is an equivariant (semi-)coupling, and the expectation is over the randomness in the measures.

This mean cost per volume replaces the total mass-dependent cost, which is infinite in the presence of infinite ergodic systems (e.g., Lebesgue–Poisson transport). It also aligns with tail-optimal allocations and concentration inequalities for randomized point processes, as in the results of Markó and Timár.

3. Existence, Uniqueness, and Structure of Equivariant Couplings

A central result is that, under natural absolute continuity and finiteness assumptions on the mean cost, there exists a unique equivariant coupling (or semi-coupling) $q^\bullet$ that minimizes $\mathcal{C}(q^\bullet)$. Specifically, if $\lambda^\omega \ll m$ and $\mathcal{C}(q^\bullet) < \infty$, then the optimal semicoupling possesses the following properties (Huesmann, 2012):

• Uniqueness: Any two optimal equivariant semicouplings must coincide almost everywhere.
• Local Optimality: The global minimizer is locally optimal in every bounded region and coincides with minimizers of the classical finite mass OT problem.
• Representation by Transport Map: Every optimal semicoupling is induced by a measurable transport map $T(\cdot, \omega)$ so that

$q^\bullet = (\mathrm{id}, T)_*\lambda^\bullet$

or, more generally,

$$q^\bullet = (\mathrm{id}, T)_*(\rho^\bullet \lambda^\bullet)$$

where $T$ is equivariant under the group action: $T^{\theta_g \omega}(g x) = g T^\omega(x)$.

This generalizes classical single-mass optimal transport to infinite-volume and random equivariant settings, matching the intrinsic symmetry of the measures.

4. Approximation by Bounded-Region Problems and Local-Global Consistency

Because both $\lambda^\bullet$ and $\mu^\bullet$ often have infinite mass, the authors develop methods for approximation via classical OT on bounded regions. For each bounded region $A = gB_r$ (with $B_r$ a union of fundamental regions), the unique semicoupling between $\lambda^\bullet$ and $1_A \mu^\bullet$ is computed; collections of these local solutions are then symmetrically "mixed" to produce equivariant semicouplings with vanishing boundary error as the domain grows (Huesmann, 2012).

Formally, averaging over regions and extracting tightness limits yields a global equivariant semicoupling, where local optimality, boundary mixing, and equivariance force the limiting map to inherit the group's symmetry. The convergence is typically in the sense of vague convergence of measures or local measure-theoretic convergence of transports on expanding balls.

This method is robust and general, allowing the transfer of local geometric properties to global ergodic constructions.

5. Metric Structure on Spaces of Equivariant Random Measures

For cost functions of the form $c(x, y) = d(x, y)^p$, $p \in [1, \infty)$, the optimal transportation cost per unit volume yields a metric on the space $\mathcal{P}_p$ of equivariant random measures of unit intensity and finite cost. The metric is defined as

$$W_p^p(\lambda^\bullet, \mu^\bullet) = \inf_{q^\bullet \in \Pi_{es}(\lambda^\bullet, \mu^\bullet)} \mathcal{C}(q^\bullet)$$

where $\Pi_{es}$ denotes the set of equivariant semicouplings. The associated metric satisfies:

• $W_p(\lambda^\bullet, \mu^\bullet) = 0$ iff $\lambda^\omega = \mu^\omega$ almost surely,
• The triangle inequality is valid: $$W_p(\lambda^\bullet, \mu^\bullet) \leq W_p(\lambda^\bullet, \xi^\bullet) + W_p(\xi^\bullet, \mu^\bullet)$$,
• Symmetry is evident.

This metric structure yields stability, continuity, and comparison results within the ergodic-invariant framework. It also quantifies the structural distance between random measures under optimal transport.

6. Connections to Allocation, Concentration, and Tail-Optimal Costs

Equivariant OT is tightly linked to allocation rules and mass transport schemes with optimal tail properties. For example, Markó and Timár constructed an allocation of Lebesgue measure to a Poisson point process in $d \geq 3$ dimensions that achieves finite mean transportation cost with the exponential distance cost $c(x, y) = \exp(\kappa|x-y|^d)$. Their construction exploits the independence and concentration properties of the Poisson process (specifically, the Ajtai–Komlós–Tusnády matching and strong exponential concentration of points in large cubes), which in turn make possible the derivation of finite mean cost per unit volume (Huesmann, 2012).

The paper extends such arguments to compound Poisson processes with exponential weights, where similar large deviation and concentration inequalities hold (as demonstrated by explicit exponential deviation bounds for Poisson-exponential sums). Consequently, the existence of equivariant allocation rules—matching a reference measure to a point process with optimal or near-optimal mean cost per unit volume—is established for a broad class of random measures.

7. Cohomological and Pattern-Equivariant Generalizations

Pattern-equivariant mass transport is another extension in which the constraint of symmetry is replaced by invariance under the (usually aperiodic) local structure of a tiling or pattern. Here, mass distributions are pattern-equivariant—i.e., functions whose values at $x$ depend on the local configuration within a fixed or arbitrarily large radius. Transport maps are then constrained to respect this pattern equivariance, and cohomological obstructions (calculated in the Čech cohomology of the tiling hull) completely characterize when bounded, weakly PE, or strongly PE transport is possible (Kelly et al., 2018). This generalizes equivariant OT to broader topological and dynamical invariance settings, with existence and uniqueness linked to topological invariants.

Table: Key Elements of Equivariant Optimal Transport

Concept	Mathematical Formulation	Structural Consequence
Equivariance	$\lambda^{\theta_g \omega}(gA) = \lambda^\omega(A)$	Transport maps $T$ satisfy $T^{\theta_g\omega}(g x) = g T^\omega(x)$
Mean cost per volume	$$\mathcal{C}(q^\bullet) = \mathbb{E}\left[\int_{M \times B_0} c(x, y) q^\bullet(dx, dy)\right]$$	Removes location dependence, finite-valued on infinite measure pairs
Uniqueness	Single optimal equivariant semicoupling if cost is finite	Ensures canonical, map-induced structure
Metric structure	$W_p^p(\lambda^\bullet, \mu^\bullet)$ as above	Space of random measures becomes a metric space

8. Broader Significance and Applications

Equivariant optimal transport provides a unifying paradigm for infinite-volume OT, ergodic allocation, geometric measure theory, random point process matching, and topological transport obstructions. In statistical mechanics and spatial statistics, it governs the construction of translation-invariant allocations, the quantitative comparison of random configurations, and the stability of symmetry-respecting transformations.

The theory clarifies why and how global statements—such as the uniqueness and stability of group-invariant mass matching—follow from local or finite-volume analysis when combined with symmetry or ergodic properties. It also underpins advances in geometric probability, optimal matching of random or deterministic tilings, and mass transport in deterministic or random environments.

This framework continues to be extended—particularly toward finer descriptions of mass transport in spaces with lower regularity, complex combinatorial symmetries, or higher-order invariance, including the use of groupoid actions, higher-dimensional tiling spaces, and cohomological transport theory.