Notation Table - OT4ML Web

Notation	Meaning	First reference
$\RR^d$	Euclidean ambient space.	Section sec-measures
$\X,\Y$	Source and target spaces.	Eq. eq-monge-continuous
$\Mm(\X)$	Finite signed Radon measures on $\X$ .	Section sec-measures
$\Mm_+(\X),\Mm_+^1(\X)$	Positive finite measures and probability measures.	Section sec-measures
$\Pp(\X),\Pp_p(\X)$	Probability measures, with finite $p$ -moment for $\Pp_p$ .	Section sec-kantorovich-continuous
$\simplex_n$	Probability simplex of histograms of length $n$ .	Definition def-probability-simplex
$\de_x$	Dirac mass at $x$ .	Definition def-discrete-measure
$\al,\be,\ga$	Source, target and auxiliary probability measures.	Eq. eq-monge-continuous
$\density{\al}$	Density of $\al$ with respect to a reference measure.	Definition def-relative-density
$\d\al,\d x$	Integration against $\al$ and against Lebesgue measure.	Section sec-measures
$\EE$	Expectation of a random variable.	Section sec-measures
$\supp(\pi)$	Topological support of a measure.	Definition def:support
$\Supp(\b)$	Index support of a histogram.	Eq. eq-discr-diverg
$\Cc(\X)$	Continuous real-valued functions on $\X$ .	Section sec-measures
$\norm{\cdot}$	Euclidean norm or the norm indicated by a subscript.	Chapter sec-matching
$\dotp{\cdot}{\cdot}$	Euclidean/Frobenius pairing or measure-function pairing.	Section sec-measures

Notation	Meaning	First reference
$(x_i)_i,(y_j)_j$	Source and target point clouds.	Eq. eq-optimal-assignment
$\C=(\C_{i,j})$	Cost matrix between source and target points.	Eq. eq-optimal-assignment
$\sigma\in\Perm(n)$	Permutation encoding a one-to-one matching.	Eq. eq-optimal-assignment
$\P_\sigma,\mathcal P_n^{\mathrm{perm}}$	Permutation matrix and the set of all such matrices.	Definition def-permutation-matrices
$\mathcal B_n$	Birkhoff polytope of bistochastic matrices.	Definition def-birkhoff-polytope
$\a,\b$	Discrete probability histograms.	Eq. eq-discr-couplings
$\P$	Discrete transport/coupling matrix.	Eq. eq-discr-couplings
$\CouplingsD(\a,\b)$	Polytope of discrete couplings with marginals $\a,\b$ .	Eq. eq-discr-couplings
$\ones_n,\transp{\P}$	All-ones vector and transpose of $\P$ .	Eq. eq-discr-couplings
$\MKD_\C(\a,\b)$	Discrete Kantorovich optimal value with cost $\C$ .	Eq. eq-kanto-discr
$\distD$	Ground distance matrix for discrete Wasserstein distances.	Definition def-discrete-wasserstein-distance
$\WassD_p(\a,\b)$	Discrete $p$ -Wasserstein distance.	Definition def-discrete-wasserstein-distance

Notation	Meaning	First reference
$T,\T$	Transport map.	Eq. eq-monge-continuous
$T_\sharp\al$	Push-forward of $\al$ by $T$ .	Definition defn-pushfwd
$T^\sharp g$	Pullback of a test function, $T^\sharp g=g\circ T$ .	Remark rem-pullback-pushforward
$\Id$	Identity map.	Definition defn-pushfwd
$\tilde\Wass_p$	Directed Monge transport distance.	Eq. eq-monge-distance
$\nabla\phi$	Brenier map for quadratic cost.	Theorem thm-brenier
$\cumul{\al}$	Cumulative distribution function of a 1-D measure.	Eq. eq-cumul-defn
$\cumul{\al}^{-1}$	Quantile function of a 1-D measure.	Eq. eq-OT-map-1d
$\Gaussian(\mean,\cov)$	Gaussian law with mean $\mean$ and covariance $\cov$ .	Eq. eq-gauss-pf
$\mean_\al,\cov_\al$	Mean and covariance of a Gaussian measure $\al$ .	Eq. eq-dist-gauss
$\Bb(\cov_\al,\cov_\be)$	Bures covariance distance.	Definition def-bures-metric
$\Phi_2(\al)$	Raw second-moment matrix of a probability measure.	Second-moment Bures quotient
$\tr(\cov)$	Trace of a matrix.	Eq. eq-dist-gauss

Notation	Meaning	First reference
$\pi$	Coupling or transport plan.	Definition def-continuous-couplings
$\Couplings(\al,\be)$	Set of couplings between $\al$ and $\be$ .	Eq. eq-coupling-generic
$\MK_\c(\al,\be)$	Kantorovich optimal value with ground cost $\c$ .	Eq. eq-mk-generic
$\dist$	Ground distance on the underlying metric space.	Eq. eq-defn-wass-dist
$\Wass_p(\al,\be)$	$p$ -Wasserstein distance.	Definition def-wasserstein-distance
$\Wass_\infty(\al,\be)$	Worst-displacement Wasserstein distance.	Eq. eq-wass-infty
$\mathfrak A,\mathfrak B$	Probability laws over probability measures.	Eq. eq-wow-parametric-law
$\widehat{\mathfrak A}_p$	Empirical law of $p$ random probability measures.	Proposition prop-wow-barycenter-lln
$\bar\alpha_{\mathfrak A}$	Collapsed mixture associated with a law over measures.	Definition def-collapsed-barycentric-mixture
$\widetilde\alpha_{\mathfrak A}$	Wasserstein-barycenter flattening of a law over measures.	Section sec-barycenters
$\mathbb W_p$	$p$ -Wasserstein distance on the Wasserstein space.	Eq. eq-wow-distance
$\Gamma$	$c$ -cyclically monotone subset of $\X\times\Y$ .	Definition def:ccm
$\rho$	Glued or composed coupling.	Lemma lem-gluing-general
$\rightharpoonup$	Weak $^*$ convergence of measures.	Definition dfn-weak-conv
$\TV,\norm{\cdot}_{\TV}$	Total variation divergence/norm.	Section sec-measures

Notation	Meaning	First reference
$\fD,\gD$	Discrete dual potentials.	Eq. eq-dual
$\f,\g$	Continuous dual potentials.	Eq. eq-dual-generic
$\Ee_0(f,g),\Ee(g)$	Full dual and semi-dual objectives, for functions or vectors.	Eqs. eq-full-dual-functional-web and eq-semi-dual-web
$\PotentialsD(\C)$	Feasible set of discrete dual potentials for cost $\C$ .	Eq. eq-feasible-potential
$\Potentials(\c)$	Feasible set of continuous dual potentials.	Eq. eq-dfn-pot-dual
$f^c,g^{\bar c}$	$c$ - and $\bar c$ -transforms of dual potentials.	Definition def-c-transform
$\Laguerre_j(\gD)$	Laguerre/power cell in semi-discrete OT.	Eq. eq-laguerre-cells
$\VV_j(Y)$	Voronoi cell of codepoint $y_j$ .	Proposition prop-free-masses-voronoi
$\Qq_m(\al)$	Optimal $m$ -point quantization error.	Eq. eq-optimal-quantization
$\Ff(Y),\Ff_{\rm eq}(Y)$	Free-mass and equal-weight quantization energies.	Section sec-optimal-quantization
$\Lip(f)$	Lipschitz constant of $f$ .	Eq. eq-lip-constant
$\Wass_1$	Kantorovich--Rubinstein distance/norm.	Eq. eq-w1-metric
$\flow,	\flow	$
$\Wass_{1,G}$	Graph Wasserstein-1/transshipment distance.	Proposition prop-graph-w1-beckmann
$d_G,\nabla_G,\operatorname{div}_G$	Graph geodesic distance, gradient and divergence.	Proposition prop-graph-w1-beckmann
$\norm{\cdot}_B$	Extended dual seminorm induced by a discriminator class $B$ .	Eq. eq-dual-norm-cont
$\RKHS,\Krkhs$	Reproducing kernel Hilbert space and its kernel.	Definition def-kernel-mmd-norm
$\MMD_k$	Maximum mean discrepancy/kernel seminorm for $k$ .	Definition def-kernel-mmd-norm
$\Divergm_\phi,\DivergmD_\phi$	Continuous and discrete $\phi$ -divergences.	Eq. eq-phi-div
$\phi'_\infty$	Recession slope of an entropy function.	Definition def_entropy
$\phi^\star$	Legendre transform of $\phi$ .	Eq. eq-legendre
$\KL,\KLD$	Continuous and discrete Kullback--Leibler divergences.	Definitions def-discrete-relative-entropy, def-measure-relative-entropy
$\Hellinger$	Hellinger distance.	Section sec-phi-div
$\JS$	Jensen--Shannon distance.	Section sec-phi-div

Notation	Meaning	First reference
$\epsilon$	Entropic regularization strength.	Eq. eq-regularized-discr
$\HD(\P)$	Shannon--Boltzmann entropy of a matrix.	Definition def-discrete-shannon-boltzmann-entropy
$\MKD_\C^\epsilon(\a,\b)$	Discrete entropic OT value.	Eq. eq-regularized-discr
$\MK_\c^\epsilon(\al,\be)$	Continuous entropic OT value.	Eq. eq-entropic-generic
$\mathcal I(X,Y)$	Mutual information of a coupled pair.	Definition def-mutual-information
$\mathcal I_{\mathrm{geo}}(\al,\be)$	Integrated Fisher information along the quadratic Wasserstein geodesic.	Proposition prop-small-epsilon-expansion
$\K$	Gibbs kernel $e^{-\C/\epsilon}$ .	Eq. eq-scaling-form
$\uD,\vD$	Left and right Sinkhorn scalings.	Eq. eq-scaling-form
$\diag(\uD)\K\diag(\vD)$	Scaling form of the entropic coupling.	Eq. eq-sink-matrix
$\odot$	Entrywise product of vectors.	Eq. eq-dualsinkhorn-constraints2
$\it{\uD},\itt{\uD}$	Current and next Sinkhorn iterates.	Eq. eq-sinkhorn
$f^{c,\epsilon},g^{\bar c,\epsilon}$	Continuous soft $c$ -transforms.	Definition def-continuous-soft-c-transform
$\mathrm{SB}_\epsilon(\al,\be)$	Dynamic Schrodinger bridge value.	Eq. eq-schrodinger-path-space
$\operatorname{prox}_{\mathsf h}^{\KLD}$	KL-proximal map of a marginal penalty.	Eq. eq-kl-prox-marginal
$\norm{\cdot}_V$	Variation seminorm on potentials modulo constants.	Definition def-variation-seminorm
$\Hilbert$	Hilbert projective metric on positive vectors.	Definition def-hilbert-metric
$\Delta_k$	Entropic dual suboptimality after $k$ Sinkhorn cycles.	Proposition prop-sinkhorn-dual-rate
$\eta(K),\lambda(K)$	Projective cross-ratio and Birkhoff contraction factor.	Theorem thm-birkhoff
$\Proj^\KLD$	KL/Bregman projection.	Eq. eq-kl-proj
$\bar\MK_\c^\epsilon(\al,\be)$	Debiased Sinkhorn divergence.	Eq. eq-sinkhorn-divergence
$v_k,S_k,\rho_{k,u}$	Scaled log-Sinkhorn transforms and multiplicative increment.	Proposition prop-scaled-log-sinkhorn-limit
$z=(u,v),Q(z)$	Log-scaling variables and clearing map in the M-function view.	Definition def-mfunctions
$\underline z,\overline z,z^\star$	Lower/upper order barriers and the monotone-clearing fixed point.	Theorem thm-mfunction-jacobi-convergence
$\eta_{ij},\sigma_i,\tau_j$	Loss factors and outside-option coefficients in lossy Sinkhorn clearing.	Example ex-lossy-sinkhorn-clearing
$u_t,F,G,\bar r_t$	Continuous $\varepsilon$ -Sinkhorn potential, log-densities and gauge term.	Definition def-continuous-epsilon-sinkhorn
$B_t,q_t$	Gaussian linear part and mean shift in continuous Sinkhorn closure.	Section sec-continuous-epsilon-sinkhorn

Notation	Meaning	First reference
$\psi_1,\psi_2$	Entropy functions penalizing marginal mismatch.	Eq. eq-unbalanced-primal
$\UW_c,\UW_{c,\tau}$	Relaxed unbalanced OT value with marginal penalties.	Eq. eq-unbalanced-primal
$L_c$	Reverse-formulation local unbalanced cost.	Eq. eq-unbalanced-reverse-local-cost
$H_c$	Homogeneous perspective of the local cost $L_c$ .	Eq. eq-unbalanced-homogeneous-local-cost
$\HW$	Homogeneous unbalanced formulation.	Eq. eq-homogeneous
$\mathfrak{C}[\X]$	Cone over the metric space $\X$ .	Section sec-unbalanced
$\CW$	Cone formulation of unbalanced OT.	Theorem thm-cone-unbalanced-ot
$\Delta_\kappa,\CW_\kappa$	Scaled cone metric and static cone value for Wasserstein--Fisher--Rao transport.	Eqs. eq-wfr-scaled-cone-metric, eq-wfr-scaled-cone-value
$J_\kappa,\mathbb J_\kappa$	Pointwise and measure-valued dynamic unbalanced perspective actions.	Eqs. eq-wfr-momentum-perspective, eq-wfr-measure-action
$\WFR_\kappa$	Wasserstein--Fisher--Rao dynamic distance with growth scale $\kappa$ .	Eq. eq-dynamic-unbalanced-ot
$\be_s,\la_s$	Input measures and weights in barycenter problems.	Eq. eq-barycenter-generic
$\al^\star$	Optimal measure, often a barycenter.	Eq. eq-barycenter-generic
$\mathcal B_c(\mathfrak A)$	Set of barycenters of a law over measures.	Section sec-barycenters
$B,c_{\mathrm{bar}}$	Barycentric map and induced multi-marginal barycenter cost.	Proposition prop-multimarginal-barycenter
$\SW_p$	Sliced Wasserstein distance.	Definition def-sliced-wasserstein
$\Sphere^{d-1}$	Unit sphere of projection directions.	Definition def-sliced-wasserstein
$P_\theta$	Projection on direction $\theta$ .	Definition def-sliced-wasserstein
$\mathfrak R\al$	Measure-valued Radon transform of $\alpha$ .	Remark rem-sliced-radon-viewpoint
$R\rho$	Density Radon transform of $\rho$ .	Remark rem-sliced-radon-viewpoint
$R^\dagger h$	Least-squares Radon pseudoinverse density reconstructed from a sinogram $h$ .	Proposition prop-radon-pseudoinverse
$\gamma_\theta$	One-dimensional projected/Radon-domain barycenter law.	Section sec-barycenters
$\SW_{p,q,k}$	$L^q$ aggregate of $p$ -Wasserstein distances over $k$ -dimensional projections.	Definition def-sliced-variants
$\SW_{p,q},\MaxSW_{p,k}$	Line-sliced and max-sliced abbreviations, with $\MaxSW_{p,k}=\SW_{p,\infty,k}$ .	Definition def-sliced-variants
$\MinSW_2$	Min-SW lifted-plan discrepancy, upper-bounding $\Wass_2$ .	Proposition prop-min-sw-comparison
$\Wass_\gamma$	Spectral Wasserstein distance associated with a matrix gauge $\gamma$ .	Eq. eq-spectral-wasserstein
$\mathcal B_\gamma$	Polar set defining the robust projected form of $\Wass_\gamma$ .	Eq. eq-spectral-polar-set
$\Wass_{2,A}$	Quadratic Wasserstein pseudodistance after projection by $A^{1/2}$ .	Eq. eq-quadratic-projected-cost
$\SRW_{2,k}$	Paty--Cuturi subspace robust Wasserstein distance.	Section sec-spectral-subspace-wasserstein
$\LOT_\rho$	Linear OT distance around reference $\rho$ .	Eq. eq-lot-embedding
$Q,R,g$	Low-rank OT factors and latent mass vector.	Definition def-low-rank-couplings
$\P(\Q,\R,g)$	Coupling induced by a low-rank factored representation.	Eq. eq-low-rank-coupling-factor
$\eta=\sum_k g_k\delta_{z_k}$	Abstract intermediate measure in low-rank OT.	Definition def-low-rank-couplings
$\MK_c^\kappa,\kappa$	Capacity-constrained OT value and capacity density.	Eq. eq-capacity-constrained-ot
$U_{ij}$	Discrete upper-capacity matrix for a capped coupling.	Eq. eq-discrete-capacity-constrained-ot
$\mathcal V_c,\mathcal V_{c,\epsilon}$	Unregularized and KL-normalized entropic OT values used in sensitivity formulas.	Propositions prop-ot-first-variations-unregularized, prop-ot-first-variations-entropic
$\mathcal L_{\mathrm{iOT}}$	Inverse-OT primal--dual gap loss.	Section sec-metric-learning-inverse-ot
$\bar T_\pi$	Barycentric projection of a coupling $\pi$ .	Eq. eq-barycentric-projection
$\bar\beta_\pi$	Pushforward of $\alpha$ by the barycentric projection.	Eq. eq-barycentric-projection
$\WOT_C$	Weak OT value with conditional-law cost $C$ .	Eq. eq-weak-ot
$g^C$	Weak $C$ -transform in weak OT duality.	Proposition prop-weak-ot-duality
$\Couplings_{\mathrm{mart}}(\alpha,\beta)$	Martingale couplings between $\alpha$ and $\beta$ .	Definition def-martingale-coupling
$\preceq_{\mathrm{st}},\preceq_{\mathrm{cx}}$	Stochastic order and convex order.	Section sec-martingale-ot
$C_{\mathrm{bar}}$	Quadratic barycentric weak-transport cost.	Proposition prop-barycentric-weak-ot
$u_t,V_t$	Positive vector-valued density and spatial flux.	Eqs. eq-vector-valued-bb, eq-vector-valued-continuity
$\mathcal W_{\Phi}$	Dynamic vector-valued BB-type cost.	Eq. eq-vector-valued-bb
$\distD,\distD'$	Intra-domain distance matrices in discrete GW.	Eq. eq-gw-def
$\De$	Discrepancy between intra-domain distances.	Eq. eq-gw-def
$\Ee_{\distD,\distD'}(\P)$	Discrete GW distortion energy.	Eq. eq-gw-def
$\GWD$	Discrete Gromov--Wasserstein cost.	Eq. eq-gw-def
$\XX,\YY$	Metric-measure spaces.	Definition def-metric-measure-space
$\GW$	Continuous Gromov--Wasserstein distance.	Eq. eq-gw-generic
$\mathfrak D_\XX$	Law of local distance profiles of a metric-measure space.	Proposition prop-memoli-gw-profile-lower-bound
$\C(\P)$	Half-gradient of the squared discrete GW distortion.	Eq. eq-gw-sinkh
$d_{\mathrm H},d_{\mathrm{GH}}$	Hausdorff and Gromov--Hausdorff distances.	Section sec-gromov-wasserstein
$\operatorname{FGW}_{\lambda,p}$	Fused Gromov--Wasserstein distance.	Section sec-gromov-wasserstein
$\mathbb S^m,\mathbb S_+^m$	Real symmetric matrices and their positive semidefinite cone.	Definition def-positive-matrix-valued-measure
$A_t,P_t$	Positive matrix-valued density and spatial matrix flux.	Eqs. eq-matrix-valued-bb, eq-matrix-valued-continuity
$\mathcal W_{\mathrm{mat}}$	Conservative matrix-valued BB-type cost.	Eq. eq-matrix-valued-bb
$\mathbb H_n,\mathbb H_n^+,\mathbb H_n^{+,1}$	Hermitian matrices, positive semidefinite Hermitian matrices and density matrices.	Definition def-hermitian-density-matrices
$\operatorname{Tr}_A,\operatorname{Tr}_B$	Partial traces of a bipartite matrix.	Eq. eq-qot-partial-traces
$\mathrm{QOT}_C(A,B)$	Finite-dimensional quantum OT value with cost observable $C$ .	Eq. eq-qot-primal
$\mathrm{QOT}_C^\epsilon(A,B)$	Entropically regularized quantum OT value.	Eq. eq-qot-entropic-primal
$D_H(T	K)$	Quantum relative entropy used for Bregman projections.
$T_e(F,G),T_s(F,G)$	Exact Gibbs coupling and symmetric Gurvits-scaling surrogate.	Eqs. eq-qot-gibbs-coupling, eq-qot-symmetric-scaling
$\Omega_{n,m},A_\omega$	Monotone warping paths and their incidence matrices.	Definition def-dynamic-time-warping
$\mathrm{DTW}_c(x,y)$	Dynamic time-warping value between two feature sequences.	Eq. eq-dtw-variational
$\Gamma_\uparrow,\mathrm{CDTW}_c(x,y)$	Monotone clock pairs and continuous time-warping value.	Eq. eq-continuous-dtw
$\mathrm{sDTW}_{c,\epsilon},\PP_\epsilon,E_\epsilon$	Soft-DTW free energy, Gibbs path law and expected alignment matrix.	Eqs. eq-soft-dtw-variational, eq-soft-dtw-expected-alignment

Notation	Meaning	First reference
$\alpha_t$	Time-dependent curve of probability measures.	Eq. eq:eulerian-advection
$\rho_t$	Density of $\alpha_t$ with respect to the relevant reference measure.	Eq. eq:benamou-brenier-convex
$v_t$	Eulerian velocity field transporting $\alpha_t$ .	Eq. eq:eulerian-advection
$\mathcal H_\alpha$	$L^2(\alpha)$ vector fields with zero weighted divergence.	Chapter sec-dynamic-optimal-transport
$\omega_t=\alpha_t v_t$	Vector-valued momentum/flux measure in convex dynamic formulations.	Eq. eq:benamou-brenier-convex
$m_t=\rho_t v_t$	Density of $\omega_t$ when $\alpha_t=\rho_t\,\d x$ .	Eq. eq:benamou-brenier-convex
$J(a,m),\mathbb J(\alpha,\omega)$	Quadratic perspective and its intrinsic measure action.	Eqs. eq-quadratic-perspective, eq-measure-perspective-action
$T_t$	Lagrangian particle flow map.	Eq. eq:lagrangian-advection
$P_t$	Interpolant map in flow matching.	Eq. eq:interp-coupling
$\Ss=C([0,1];\RR^d),e_t$	Path space and evaluation map in the superposition formulation.	Section rem-bb-path-space
$\Wass_2^2$ via action	Benamou--Brenier dynamic formulation.	Eq. eq:benamou-brenier
$	\dot x_t	$
$\mathbb A(\alpha,w)$	Tangent action defining a dynamic length distance.	Eq. eq-generalized-action-length-distance
$\mathsf D_{\mathbb A}$	Length-space distance generated by the tangent action $\mathbb A$ .	Eq. eq-generalized-action-length-distance
$\operatorname{PMO}_{\mathbb A,\alpha}$	Penalized minimization oracle associated with $w\mapsto\mathbb A(\alpha,w)$ .	Eq. eq-local-action-steepest-descent
$Q_\alpha$	Positive operator representing a quadratic local tangent action.	Eq. eq-general-quadratic-tangent-action
$\mathsf D_Q$	Geodesic distance induced by a quadratic tangent operator $Q_\alpha$ .	Eq. eq-general-quadratic-tangent-action
$A(a,w)$	Local velocity action density, with scalar density value $a$ and velocity $w\in\RR^d$ .	Eq. eq-local-velocity-action
$J_A(a,m)$	Momentum perspective of $A$ , with pointwise momentum $m\in\RR^d$ .	Eq. eq-general-momentum-perspective
$\mathbb J_{A,\lambda}(\alpha,\omega)$	Measure-level momentum action associated with $A$ relative to a reference measure $\lambda$ ; written $\mathbb J_A$ only in intrinsic cases.	Eq. eq-general-measure-momentum-action
$\mathsf D_{A,\lambda}$	Dynamic distance induced by the homogeneous momentum action $J_A$ , with the reference measure suppressed only in intrinsic cases.	Proposition prop-homogeneous-dynamic-action-distance
$A_p,J_p,\mathbb A_p$	Velocity action, momentum perspective and tangent action for $\Wass_p$ .	Section sec-generalized-dynamic-wasserstein-distances
$\theta(a),J_\theta,\mathbb A_{\theta,\lambda},\mathsf W_{\theta,\lambda}$	Concave mobility, its momentum action, reference-dependent tangent action and associated mobility distance.	Section sec-generalized-dynamic-wasserstein-distances
$(s_t,g_t)$	Source and relative growth variables in dynamic unbalanced OT.	Eq. eq-dynamic-unbalanced-ot
$\mathbb A_\gamma(\alpha,v)$	Spectral tangent action induced by the gauge $\gamma$ .	Eq. eq-spectral-tangent-action
$\mathsf W_{\gamma,\mathrm{dyn}}$	Dynamic path-length representation of $\Wass_\gamma$ .	Eq. eq-dynamic-spectral-wasserstein
$\Sigma_n$	Finite-state probability simplex in Markov-chain geometries.	Section sec-discrete-wasserstein-markov
$K_{ij},\pi$	Reversible Markov transition rates and invariant law.	Section sec-discrete-wasserstein-markov
$\theta(a,b)$	Logarithmic mean used as Markov/nonlocal mobility.	(75)
$\mathcal K_\rho$	Onsager operator for a discrete reversible Markov chain.	Eq. eq-discrete-markov-onsager
$\mathbb A_K(a,\psi),\mathcal W_K$	Markov-chain tangent action and associated discrete Wasserstein distance.	Eqs. eq-discrete-markov-action, eq-discrete-markov-distance
$\mathfrak m,K(x,\d y),\mathsf J$	Reference measure, reversible jump kernel and symmetric jump measure.	(76)
$\bar\nabla\varphi$	Nonlocal gradient/increment $\varphi(y)-\varphi(x)$ .	Eq. eq-nonlocal-continuity-weak
$\mathbb A_K(\alpha,v),\mathcal W_K$	Nonlocal jump tangent action and associated Wasserstein distance.	Eq. eq-nonlocal-wasserstein-distance
$\mathcal W_k,\RKHS_k^d,\kappa_k$	Kernelized Benamou--Brenier distance, vector-valued velocity RKHS, and uniform evaluation bound.	Eq. eq-kernelized-bb-distance, Proposition prop-kernelized-bb-distance
$\Wgrad f(\alpha)$	Wasserstein gradient of a functional.	Proposition prop-formal-wass-gradient
$\mathcal I(\alpha	\beta)$	Relative Fisher information of $\alpha$ with respect to $\beta$ .
$\Pp_{p,\lambda}(S\times\Omega)$	Conditional probability laws with fixed condition marginal $\lambda$ and finite $p$ th moment.	Section sec-conditional-wasserstein-distances
$\Couplings_\lambda(\alpha,\beta)$	Conditional couplings that keep the condition variable fixed.	Eq. eq-conditional-ot-general
$\MK_c^\lambda,\Wass_{p,\lambda}$	Conditional OT value and, for $c_s=\dist^p$ , its $p$ th-root metric.	Eqs. eq-conditional-ot-general, eq-conditional-wasserstein-distance
$\operatorname{PMO}_{\mathbb A_\gamma,\alpha}(g)$	Spectral specialization of the penalized minimization oracle.	Proposition prop-normalized-spectral-polar
$S_\alpha(g),A_\alpha^\star$	Gradient covariance and active polar preconditioner in spectral flow.	Proposition prop-normalized-spectral-polar
$\Pp_{2,\mathbf m}(\Omega;\RR_+^p)$	Positive multi-species measures with fixed component masses.	Eq. eq-multispecies-space
$\Wass_{2,\oplus}$	Mass-weighted product Wasserstein distance for independent species transport.	Eq. eq-multispecies-product-metric
$\delta f(\alpha)$	First variation of $f$ at $\alpha$ .	Proposition prop-formal-wass-gradient
$	\partial f	(\alpha)$
$\partial_t\alpha+\diverg(\alpha v)=0$	Continuity equation.	Eq. eq:eulerian-advection
$\alpha_{t+\tau}$	One JKO/minimizing-movement step.	Eq. eq:jko-discr
$\Ss=C([0,1];\RR^d)$	Path space in the superposition formulation.	Chapter sec-wasserstein-flows
$X,S,\al_X,F(X)$	Particle configuration, particle velocity, empirical law and lifted energy.	Eq. eq-empirical-momentum-lift
$s,\eta_t,\eta_t^n,\pi_x$	Velocity variable, phase-space laws and spatial projection in inertial flows.	Proposition prop-second-order-liouville

Ambient spaces, measures and elementary objects¶

Discrete matching and discrete Kantorovich OT¶

Monge maps, one-dimensional OT and Gaussians¶

Continuous Kantorovich OT and Wasserstein distances¶

Duality, transforms and weak norms¶

Entropic regularization and Sinkhorn algorithms¶

Extensions of OT¶

Dynamic OT and Wasserstein gradient flows¶