Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation

This setting fixes the standing notation used by results on second-order equations and their viscosity solutions on the quadratic Wasserstein space. It is layered on Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, hence on 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 and Real Hilbert Spaces: Standing Notation and Background, and on 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.

The dimension $d$ and the probability space $(\Omega,\mathcal{F},P)$ are those of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data. The letters $m$, $k$, $q$ and $r$ denote natural numbers with $1\le m$, $1\le k$, $1\le q$ and $1\le r$, and the notation below is introduced for all of them simultaneously, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. Two symbols bound elsewhere are read as follows here and in every result adopting this setting: a dimension written $p$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces or in Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §numbers, and the dimension written $d$ in The Space of Square-Integrable Random Vectors where clause 2 below instantiates it, is written $r$, the letter $p$ being reserved for the second component of a point of $\mathbb{R}^{d+d}$ as in clause 4; and a scalar written $\mu$ in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation is written $t$, as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation.

1. (Measures, Wasserstein spaces and random vectors in every dimension)¶ For each $m$: $\mathcal{P}(\mathbb{R}^{m})$ is the set of probability measures on $\mathbb{R}^{m}$; $M_{2}(\rho)$ is the second moment of $\rho\in\mathcal{P}(\mathbb{R}^{m})$ and $\mathcal{P}_{2}(\mathbb{R}^{m})$ the set of those with finite second moment; $\Pi(\rho,\rho')$ is the set of couplings of $\rho,\rho'\in\mathcal{P}(\mathbb{R}^{m})$ and $I(\pi)$ the quadratic cost of $\pi\in\Pi(\rho,\rho')$; a coupling is optimal as defined there; and $W_{2}$ is the quadratic Wasserstein distance on $\mathcal{P}_{2}(\mathbb{R}^{m})$, a metric by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric, so that $(\mathcal{P}_{2}(\mathbb{R}^{m}),W_{2})$ is a metric space. Each of these is the object of the definition cited, read with $m$ in place of the dimension written $d$ there. Likewise $L^{2}(\Omega;\mathbb{R}^{m})$ is the space of classes of square-integrable random vectors in $\mathbb{R}^{m}$ on $(\Omega,\mathcal{F},P)$, with the inner product and norm $\langle\cdot,\cdot\rangle_{L^{2}}$, $\lVert\cdot\rVert_{L^{2}}$, the law $\mathcal{L}(Z)$ and the notational convention fixed there; it is a real Hilbert space by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §hilbert, $\mathcal{L}(Z)\in\mathcal{P}_{2}(\mathbb{R}^{m})$ with $M_{2}(\mathcal{L}(Z))=\lVert Z\rVert_{L^{2}}^{2}$ for every $Z\in L^{2}(\Omega;\mathbb{R}^{m})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §law, and for $a\in\mathbb{R}^{m}$ the constant class $c_{a}$ and the translation $\tau_{a}$ are those of The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §constants. For $m=d$ all of these are the objects fixed in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §space, The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §wasserstein and The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants. Every $L^{2}(\Omega;\mathbb{R}^{m})$ is one of the real Hilbert spaces named by this setting, hence by every result adopting it, so that the notation, the topological vocabulary and the differential calculus on open subsets of it are in force for it, with the norm written $\lVert\cdot\rVert_{L^{2}}$, the distance $d_{L^{2}}(Z,Z')=\lVert Z-Z'\rVert_{L^{2}}$, and $L^{2}(\Omega;\mathbb{R}^{m})$ open in itself by Real Hilbert Spaces: Standing Notation and Background §topology.

2. (Square-integrable vector fields against a measure)¶ For $\rho\in\mathcal{P}(\mathbb{R}^{q})$, $L^{2}(\rho;\mathbb{R}^{r})$ denotes the space of classes of square-integrable random vectors in $\mathbb{R}^{r}$ on the probability space $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}),\rho)$, that definition and The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations being applied with this probability space in place of $(\Omega,\mathcal{F},P)$ and with $r$ in place of $d$, exactly as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu in the case $q=r=d$. Its elements are the classes of the Borel maps $\eta:\mathbb{R}^{q}\to\mathbb{R}^{r}$ with $\int_{\mathbb{R}^{q}}\lVert\eta\rVert^{2}\,d\rho<\infty$ under the relation $\eta\sim_{\rho}\eta'$ meaning $\rho(\{\eta=\eta'\})=1$, its inner product, norm and distance are written

and the convention that a class and a representative of it are denoted by the same symbol is in force. It is a real Hilbert space by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §hilbert and is one of the real Hilbert spaces named by this setting. In the case $q=r=d$ it is the space $L^{2}(\mu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, for which the tangent space $T_{\mu}$ at $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ is as defined there.

3. (First marginals)¶ For $\gamma\in\mathcal{P}(\mathbb{R}^{d+k})$, the first marginal of $\gamma$ is the push-forward

of $\gamma$ by the coordinate projection $\mathrm{pr}^{d,k}_{1}:\mathbb{R}^{d+k}\to\mathbb{R}^{d}$.

4. (Plans and the variables $x$ and $p$)¶ $\mathrm{pr}_{1}=\mathrm{pr}^{d,d}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{d,d}_{2}$ are the coordinate projections of $\mathbb{R}^{d+d}$ fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. A plan is an element of $\mathcal{P}_{2}(\mathbb{R}^{d+d})$; the first marginal of a plan $\pi$ is $\pi^{(1)}=(\mathrm{pr}_{1})_{\#}\pi$, clause 3 with $k=d$. For a point $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $p=\mathrm{pr}_{2}(z)$, and for a Borel $f:\mathbb{R}^{d+d}\to\mathbb{R}$ or $f:\mathbb{R}^{d+d}\to[0,\infty]$ we write $\int f(x,p)\,\pi(dz)$ for the integral $\int_{\mathbb{R}^{d+d}}f\,d\pi$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and likewise for a Borel map of the two variables built from the projections; in particular $\int\lVert p\rVert^{2}\,\pi(dz)$ denotes the integral of the function $z\mapsto\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, which is nonnegative and Borel, being the composition of the Borel projection $\mathrm{pr}_{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections with the Borel map $y\mapsto\lVert y\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, compositions of Borel maps being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

5. (Symmetric matrices, indices and origins)¶ For each $m$, the initial segment $[m]$ and the origin $0_{\mathbb{R}^{m}}$ of $\mathbb{R}^{m}$ are as fixed there, and $\mathcal{S}(m)$ is the set of symmetric real $m\times m$ matrices, with the positive semidefinite ordering $\preceq$, the norm $\lVert\cdot\rVert$ and distance $d_{\mathcal{S}(m)}$, the identity matrix $I_{m}$ and zero matrix $0_{m}$ and the standard basis vectors $e_{i}$; $\mathrm{tr}\,A$ denotes the trace of a square real matrix $A$. For $Z\in\mathcal{S}(d+k)$, $Z^{11}\in\mathcal{S}(d)$ denotes the upper-left block of $Z$, that lemma being read with $d$ and $k$ in place of its $m$ and $n$. These are the same objects that Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background puts in force, that clause adopting the clauses of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation cited here. As already fixed in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, the symbols $\preceq$ and $\lVert B\rVert$ refer to these matrices and to the bounded symmetric bilinear forms $\mathrm{Sym}(E)$ of a real inner product space $E$ as in Real Hilbert Spaces: Standing Notation and Background §forms, the argument determining which is meant.

6. (Background)¶ The results 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, Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix, Basic Properties of the Trace, Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients, Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity and The Wasserstein Distance and the Mean-Square Distance of Random Vectors are in force by reference, each for the dimensions named by the result adopting this setting. That $(\Omega,\mathcal{F},P)$ is rich is the property of that definition, which quantifies over every dimension; as in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §rich, this setting does not assume it, and a result that needs it says so.