First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The intrinsic gradient $\nabla\varphi(\mu)$, an element of the tangent space $T_{\mu}$ and hence of $L^{2}(\mu;\mathbb{R}^{d})$, and the translation Hessian $H_{\varphi}(\mu)$, an element of the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices, are those of that definition, and the translations $\tau_{a}$ of $\mathbb{R}^{d}$ and the push-forwards of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward are those fixed there. For $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $t\in\mathbb{R}$, $\mathrm{id}+t\,\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ denotes the map $x\mapsto x+t\,\nabla\psi(x)$ formed from the gradient map of $\psi$, written $G_{t}$ in The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound; it is Borel and its push-forward $(\mathrm{id}+t\,\nabla\psi)_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §borel. The class of $\nabla\psi$ in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $\nabla\psi$, with the inner product $\langle\cdot,\cdot\rangle_{\mu}$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu; here $\nabla\varphi(\mu)$ is the intrinsic gradient of the test function $\varphi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ while $\nabla\psi$ is the gradient map of the test function $\psi$ on $\mathbb{R}^{d}$, the argument determining which is meant. Open intervals $(p,q)$ are those of that definition and differentiability of a real function on an interval at an interior point is that of Derivative at an Interior Point, every point of an open interval being interior to it by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval; being of class $C^{2}$ on $\mathbb{R}^{d}$ and the Hessian matrix $D^{2}$ of such a function are as fixed there. Then the following hold.

1. (Translations)¶ For every $a\in\mathbb{R}^{d}$ the push-forward $(\tau_{a})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ and

2. (First variation along a gradient displacement)¶ Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and let $t_{0}\in\mathbb{R}$ be positive. The function

is differentiable at $0$ with derivative $\langle\nabla\varphi(\mu),\nabla\psi\rangle_{\mu}$.

3. (Second variation along translations)¶ The function

is of class $C^{2}$ on $\mathbb{R}^{d}$ and its Hessian matrix at $0_{\mathbb{R}^{d}}$ is $H_{\varphi}(\mu)$.