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 (iv) 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 (i). 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 (ii); 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 of a Real Function on an Interval, every point of an open interval being interior to it by Basic Facts about Intervals of the Real Line (iv); 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)(i) For every $a\in\mathbb{R}^{d}$ the push-forward $(\tau_{a})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ and

$$W_{2}\bigl((\tau_{a})_{\#}\mu,\mu\bigr)\le\lVert a\rVert .$$

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

$$(-t_{0},t_{0})\to\mathbb{R},\qquad t\mapsto\varphi\bigl((\mathrm{id}+t\,\nabla\psi)_{\#}\mu\bigr),$$

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

3. (Second variation along translations)(iii) The function

$$\mathbb{R}^{d}\to\mathbb{R},\qquad a\mapsto\varphi\bigl((\tau_{a})_{\#}\mu\bigr),$$

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