Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation

This setting fixes the standing notation used by results on square-integrable vector fields against a probability measure on Euclidean space, on the tangent space of the quadratic Wasserstein space and on scores of probability measures. It is layered on The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, hence on Probability Measures on Euclidean Space and Random Vectors: Standing Notation, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus and Real Hilbert Spaces: Standing Notation and Background, and on Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, whose notation is in force throughout, the dimensions of the latter being those named by a result adopting this setting, with $d$ among them; it introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it. Symbols bound twice are read as follows: a scalar written $\mu$ in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation is written $t$ here, and $\mu$ denotes a probability measure; for a function on an open subset of a Euclidean space, $Df$, $D^{2}f$ and of class $C^{k}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, while for a function on an open subset of one of the real inner product spaces named by this setting they are those of Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus §calculus; and $\preceq$ and $\lVert B\rVert$ refer to Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background on the symmetric matrices $\mathcal{S}(q)$ and to Real Hilbert Spaces: Standing Notation and Background §forms on $\mathrm{Sym}(E)$.

1. (Measures)¶ 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, $L^{2}(\Omega;\mathbb{R}^{d})$ is the space of classes of square-integrable random vectors on it, $\mathcal{P}(\mathbb{R}^{d})$ is the set of probability measures on $\mathbb{R}^{d}$, with integrals $\int_{\mathbb{R}^{d}}f\,d\mu$ as fixed there, and $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $W_{2}$ are the Wasserstein space and distance. For $\mu\in\mathcal{P}(\mathbb{R}^{d})$, the triple $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ is a probability space, since $\mu$ is a probability measure on $\mathcal{B}(\mathbb{R}^{d})$; the expectation on it is the integral against $\mu$, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §probability-space.

2. (Square-integrable vector fields)¶ For $\mu\in\mathcal{P}(\mathbb{R}^{d})$, $L^{2}(\mu;\mathbb{R}^{d})$ denotes the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, 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 $(\Omega,\mathcal{F},P)$. Its elements are thus the classes of the Borel maps $\xi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int_{\mathbb{R}^{d}}\lVert\xi\rVert^{2}\,d\mu<\infty$ under the relation written $\sim_{P}$ in that definition and $\sim_{\mu}$ here, $\xi\sim_{\mu}\eta$ meaning $\mu(\{\xi=\eta\})=1$; its inner product and norm are written

its distance is written $d_{\mu}(\xi,\eta)=\lVert\xi-\eta\rVert_{\mu}$, 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 it is one of the real Hilbert spaces named by this setting, hence by every result adopting it, so that the notation, the topological vocabulary, including the closure $\overline{A}$ of a subset $A$, and the orthogonal projections $P_{N}$ onto closed linear subspaces of that setting are in force for it, the norm written $|\cdot|$ there being $\lVert\cdot\rVert_{\mu}$ here and the distance $d_{\mu}$.

3. (Test functions, gradients and Laplacians)¶ $C_{c}^{\infty}(\mathbb{R}^{d})$ is the set of test functions on $\mathbb{R}^{d}$, and for $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ the gradient map $\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and the Laplacian $\Delta\psi:\mathbb{R}^{d}\to\mathbb{R}$ are as defined there; The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure is in force. By claims 1 and 2 of that lemma, $\nabla\psi$ is Borel and satisfies $\int_{\mathbb{R}^{d}}\lVert\nabla\psi\rVert^{2}\,d\mu<\infty$ for every $\mu\in\mathcal{P}(\mathbb{R}^{d})$, so that its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\nabla\psi$, and $\Delta\psi$ is integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{d})$.