Wasserstein Spaces: Geometry & Applications

• Wasserstein spaces are metric spaces comprised of probability measures defined via optimal transport and displacement interpolation.
• They exhibit intricate geometric structures, including complete geodesic properties, curvature characteristics, and finite moment conditions tied to the underlying space.
• Their study reveals a balance between isometric rigidity and flexible symmetries, with significant applications in statistical inference, PDEs, and multiscale data analysis.

A Wasserstein space is a metric space whose points are probability measures on a reference metric space, equipped with a distance derived from the theory of optimal transport. These spaces serve as a central structure in several domains of geometric analysis, probability theory, and the mathematics of data science. Classical Wasserstein spaces, as well as their generalizations and variants over non-Euclidean, singular, or hierarchical spaces, exhibit a diverse range of geometric, topological, and analytic properties that are deeply influenced by the geometry of the underlying space and the order of the Wasserstein metric.

1. Foundations: Definition, Metric, and Geodesic Structure

Let $(X,d)$ be a Polish metric space. For $p \geq 1$, the $p$-Wasserstein space $W_p(X)$ consists of all Borel probability measures $\mu$ on $X$ with finite $p$-th moment: $$\int_X d(x_0, x)^p \, \mu(dx) < +\infty \quad \text{for some (hence any) } x_0 \in X.$$ The $p$-Wasserstein distance between $\mu, \nu \in W_p(X)$ is

$$W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p \, \pi(dx dy) \right)^{1/p},$$

where $\Pi(\mu, \nu)$ is the set of couplings (transport plans) with $\mu, \nu$ as marginals. Existence of an optimal plan $\pi$ minimizing the transport cost is standard under the given assumptions.

$(W_p(X), W_p)$ is a complete, separable (Polish) geodesic metric space. Constant-speed geodesics can be explicitly constructed via displacement interpolation: choose an optimal transference plan $\pi$ between $\mu_0$ and $\mu_1$, select measurable geodesics $[x, y]_t$ in $X$ for each pair $(x, y)$, and push forward $\pi$ along the evaluation at time $t$ to construct $\mu_t$.

In the case $p=2$, the geometry of $W_2(X)$ exhibits especially rich structure, with connections to Riemannian geometry and curvature (see (Bachoc et al., 16 Nov 2024, Gomes et al., 7 Jun 2024)). If $X$ is a Hadamard space (simply connected, complete, and globally nonpositively curved, i.e., CAT(0)), $W_2(X)$ inherits a geodesic structure and exhibits a form of displacement convexity paralleling the CAT(0)-inequality in $X$ (Bertrand et al., 2010).

2. Geometry of Wasserstein Spaces and Curvature

The curvature properties of $W_p(X)$ are intimately linked to those of the base space $X$:

• Nonnegative curvature: When $X$ is Euclidean ($\mathbb{R}^n$), $(W_2(\mathbb{R}^n), W_2)$ possesses nonnegative Alexandrov curvature, exhibits nonunique geodesics, and admits "exotic" isometries beyond those induced by isometries of $X$ (Bertrand et al., 2014).
• Strict negative curvature: If $X$ is a strictly negatively curved Hadamard space (simply connected, nonpositively curved without flats), the Wasserstein space $W_2(X)$ demonstrates isometric rigidity: every isometry of $W_2(X)$ is induced by an isometry of $X$. This is in sharp contrast to the Euclidean case, where the isometry group of $W_2(\mathbb{R}^n)$ strictly contains the canonical push-forward isometries (Bertrand et al., 2014).

The proof of rigidity in negatively curved settings depends crucially on properties of transport geodesics—Dirac masses are mapped to Dirac masses, and measures supported on geodesics are rigidly characterized. Negative curvature eliminates the possibility of propagating the "exotic" one-dimensional isometries present in $W_2(\mathbb{R})$ to more general settings, as flat strips cannot exist (Bertrand et al., 2014).

Generalized curvature statements:

• $(W_p(\mathbb{R}^d),W_p)$ for $d \geq 2$ is not locally Busemann nonpositively curved or NPC (Busemann NPC) for any $p > 1$ (Adve et al., 2020).
• For ultrametric base spaces, $(W_p(X),W_p)$ is affinely isometric to a convex subset of $\ell^1$ with snow-flaking, and admits precise metric dimension-theoretic characterizations (Kloeckner, 2013).

3. Isometries, Flexibility, and Rigidity

The structure of the isometry group of $W_p(X)$ is a major theme:

• For general $X$, push-forwards by isometries $\varphi:X \to X$ define a subgroup of $\operatorname{Iso}(W_p(X))$, but in many settings, $W_p(X)$ may admit nontrivial or "exotic" isometries.
• Rigidity results: For strictly negatively curved $X$, the isometry group of $W_2(X)$ consists exactly of those induced by $X$ (Bertrand et al., 2014). For suitable products $Y = [0,1]\times X$ with specific metrics, $W_1(Y)$ can be made isometrically rigid even when $X$ is not (Balogh et al., 13 Feb 2025).
• Flexible constructions: There exist constructions where $W_p(Y)$ admits mass-splitting isometries, i.e., isometries that do not preserve Dirac masses or the "shape" of measures. For example, $W_1([0,1])$ admits a nontrivial "flip" that sends Dirac masses to genuine mixtures, and such flexibility can be lifted to product spaces (Balogh et al., 13 Feb 2025).
• The dichotomy between rigidity and flexibility can be tuned by the geometry of the base space and the metric exponent parameter, as well as by passing to suitable ambient spaces.

Rigidity Regime	Example	Isometries
Strict negative curvature	$W_2(X)$, $X$ CAT$(\kappa<0)$	Only from $X$
Euclidean	$W_2(\mathbb{R}^n)$	Push-forwards plus exotic
Product with flexible metric	$W_1([0,1]\times X)$, $d_Y$ with $\alpha<1$	Only from $Y$
Product with flexible metric	$W_q([0,1]\times X)$, $d_Y$ with $\alpha=1/q$	Mass-splitting isometries

4. Riemannian-Like and Metric Geometry

The Wasserstein space $(W_2(X), W_2)$ for sufficiently regular $X$ carries a formal infinite-dimensional Riemannian geometry:

• Tangent spaces: At an absolutely continuous measure $P$, the tangent space consists of $L^2$ vector fields with respect to $P$, identified with closure of gradients of smooth functions (Bachoc et al., 16 Nov 2024, Gomes et al., 7 Jun 2024).
• Riemannian metric: The Otto metric defines

$$g_P(v, w) := \int \left\langle v(x), w(x) \right\rangle \, dP(x).$$

• Geodesics: Displacement interpolation gives constant-speed geodesics. For absolutely continuous measures, the initial velocity corresponds to the unique (Brenier) optimal transport map.
• Benamou-Brenier formula: The length of a curve $(\mu_t)$ in $\mathcal{P}_2$ is defined via the minimal kinetic energy among velocity fields $v_t$ solving the continuity equation $\partial_t \mu_t + \nabla \cdot (\mu_t v_t) = 0$ (Gomes et al., 7 Jun 2024).
• Convexity: Many energy functionals are convex along Wasserstein geodesics, enabling the extension of gradient-flow and variational frameworks (Vauthier, 3 Dec 2025).

5. Large-Scale and Infinite-Dimensional Properties

Wasserstein spaces typically display infinite-dimensional geometric complexity:

• Dimension and "largeness": For a compact $d$-manifold $X$, the critical "power-exponential parameter" (a bi-Lipschitz invariant generalizing Hausdorff dimension) of $W_2(X)$ equals $d$ (Kloeckner, 2011, Kloeckner, 2013). This reflects dimension-robustness.
• Effects of ultrametricity: For compact ultrametric $X$, $(W_p(X),W_p)$ is isometrically embeddable into a convex set of $(\ell^1, \|\cdot\|_1^{1/p})$, making its geometry and dimension properties explicitly analyzable (Kloeckner, 2013).
• Embeddings and universality: $W_p(\mathbb{R}^3)$ can embed any finite metric (after taking the $1/p$ power for $p>1$); $W_1(\ell^1)$ is universal for all finite metric spaces (Frogner et al., 2019).

6. Stability, Quotients, and Limits

Recent results show that Wasserstein spaces preserve or reflect many stability and convergence properties of their underlying metric spaces:

• Gromov-Hausdorff stability: For broad classes of spaces (e.g., compact Alexandrov spaces), Gromov–Hausdorff convergence of the base spaces is equivalent to convergence of the $W_2$-spaces. This underpins an infinite-dimensional analogue of Perelman's topological stability and finiteness (Alattar, 10 Jun 2024).
• Quotients and group actions: If a compact group $G$ acts isometrically on $(X,d)$, Wasserstein spaces of $G$-invariant measures are isometric (for generalized/unbalanced Wasserstein metrics) to the corresponding space over the metric quotient $X/G$ (Chung et al., 2019).
• Functoriality and higher Wasserstein hierarchy: Iterated Wasserstein spaces ($P_2^{(n)}(M)$) admit a systematically lifted variational structure, with explicit gradient and geodesic characterizations at each level (Vauthier, 3 Dec 2025).

7. Application Directions and Open Problems

Applications of Wasserstein spaces range from statistics and probability to data science, PDEs, and stochastic processes:

• Statistical tools: Concepts such as Wasserstein spatial depth provide robust statistical depth measures on the space of distributions, crucial for order-based inference, clustering, and outlier detection in distributional data (Bachoc et al., 16 Nov 2024).
• Minimal surfaces: There is an active program in extending geometric measure theory to Wasserstein spaces, including minimal surface problems for families of distributions (Li et al., 2023).
• Stochastic processes: Variants like the adapted Wasserstein distance $AW_p$ are tailored for filtered processes, yielding a metric space of stochastic processes with optimal transport features (Bartl et al., 2021).
• Multiscale analysis: Hierarchical transforms and decompositions, such as those based on McCann interpolants and optimality numbers, allow for multiscale, wavelet-like analysis and anomaly detection in measure-valued data (Mattar et al., 12 Sep 2025).
• Current challenges: Key open problems include precise characterizations of rigidity/flexibility regimes for diverse $(p,X)$ in $W_p(X)$, general isometric rigidity conjectures, analytic and topological invariants of Wasserstein spaces over singular or branching spaces, and the extension of geometric analytic techniques to infinite-dimensional, non-linear metric geometries.

In summary, Wasserstein spaces encode a nonlinear, measure-theoretic extension of the geometry of their base spaces, featuring a wide tapestry of structures—ranging from infinite-dimensional Riemannian geometry to flexible and rigid isometry groups, refined large-scale dimension theory, and applications to both theory and practice at the intersection of analysis, geometry, and data science. Negative curvature rigidifies $W_2(X)$ to encode exactly the metric structure of $X$, while Euclidean and product constructions allow for “wild” symmetries and universal embedding properties. The field remains rich with avenues for further exploration and generalization (Bertrand et al., 2014, Balogh et al., 13 Feb 2025, Kloeckner, 2013, Bachoc et al., 16 Nov 2024, Gomes et al., 7 Jun 2024, Vauthier, 3 Dec 2025, Mattar et al., 12 Sep 2025, Adve et al., 2020).