The Intrinsic Calculus on the Wasserstein Space: Standing Notation

This setting fixes the standing notation used by results on the intrinsic theory of second-order equations and their viscosity solutions on the quadratic Wasserstein space, that is, by results phrased in terms of couplings rather than of random vectors. It is layered on Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, hence on Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, Probability Measures on Euclidean Space and Random Vectors: Standing Notation, Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, Real Hilbert Spaces: Standing Notation and Background and Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation, whose notation is in force throughout; it introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it. In particular the dimension $d$, the spaces $\mathcal{P}_{2}(\mathbb{R}^{m})$ with the distance $W_{2}$ and the second moment $M_{2}$, the couplings $\Pi(\rho,\rho')$ with the quadratic cost $I$, the spaces $L^{2}(\rho;\mathbb{R}^{r})$ of square-integrable vector fields with their inner products and norms, the tangent spaces $T_{\mu}$, and the symmetric matrices $\mathcal{S}(d)$ with the ordering $\preceq$ and the norm $\lVert\cdot\rVert$ are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions, Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields and Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices.

1. (Couplings, the displacement pairing and the discrepancy)¶ For $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ the cost $I(\pi)$ is a nonnegative real number by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite. The coordinate projections of $\mathbb{R}^{d+d}$ are $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$, and for $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures. For $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, $\mathcal{J}(\eta,\pi)$ is the displacement pairing of $\eta$ along $\pi$, and The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling is in force; for $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ and $\pi\in\Pi(\nu,\mu)$, the discrepancy $\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)$ is the nonnegative real number of that clause. The coordinate maps $\mathrm{q}_{1},\mathrm{q}_{2},\mathrm{q}_{3}$ of $\mathbb{R}^{3d}$, the notation $\iota_{3}$, the gluing of two couplings with a common middle marginal, and the results Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal and Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling are in force.

2. (Optimal maps and the map property)¶ $\mathrm{id}$ denotes the identity map of $\mathbb{R}^{d}$, whose class in $L^{2}(\mu;\mathbb{R}^{d})$ is that of Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity. Optimal maps from $\mu$ to $\nu$ and uniquely mapped ordered pairs are those of that definition; the class in $L^{2}(\mu;\mathbb{R}^{d})$ of a Borel map $T$ with $T_{\#}\mu=\nu$ is the one supplied by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable and is again written $T$, and The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class and Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost are in force. That a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ has the map property is the property of that definition.

3. (Differentiability along couplings)¶ That a function $\varphi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ is differentiable along couplings at $\mu$, and its gradient along couplings $\nabla\varphi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$ there, are those of that definition. In this setting, and in every result adopting it, the symbol $\nabla\varphi(\mu)$ applied to a function $\varphi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ always denotes that gradient; the test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$ of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian and the intrinsic gradient written $\nabla\varphi(\mu)$ there, which are defined through the lift and require a rich probability space, are not used. The test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and their gradient maps $\nabla\psi$ keep the meaning fixed there, the argument determining which is meant.

4. (Operators, penalty pairs and envelopes)¶ For $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, the bundle $\mathcal{V}(Q)$ of vector fields over $Q$, a second-order equation operator over $Q$ and, relative to a penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$, the $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ of such an operator are those of those definitions, as are the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of a function $u$ relative to a penalty pair, functions on $\mathcal{D}$. Being Wasserstein-coercive is the property of that definition. In a result adopting this setting the letter $q$ denotes a square-integrable vector field unless that result says otherwise; the dimension written $q$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is then not used.

5. (Background)¶ The results The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions, with the mean $m(\mu)$ and the centred measure $\bar{\mu}$ of those clauses, Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment and The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space are in force by reference, each for the dimensions named by the result adopting this setting. As in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §background, this setting does not assume that $(\Omega,\mathcal{F},P)$ is rich; no notation fixed above refers to that probability space, and a result that needs richness says so.