Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varphi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. Continuity of $\varphi$ is understood between the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i) and $\mathbb{R}$ with the absolute-value metric; the translations $\tau_{a}$ of $\mathbb{R}^{d}$ for $a\in\mathbb{R}^{d}$ and the push-forwards of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) are those fixed there, and $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i). Being of class $C^{2}$ on $\mathbb{R}^{d}$ and the Hessian matrix $D^{2}\phi(a)$ of such a function are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself, and $0_{\mathbb{R}^{d}}$ is the origin of $\mathbb{R}^{d}$.

1. (Intrinsic test function)(i) The function $\varphi$ is an intrinsic test function on $Q$ if it has the following five properties.

(a)(ii) $\varphi$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

(b)(iii) For every $\mu\in Q$, $\varphi$ is differentiable along couplings at $\mu$ and its gradient along couplings satisfies $\nabla\varphi(\mu)\in T_{\mu}$.

(c)(iv) Let $\mu\in Q$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence with $\pi_{n}\in\Pi(\mu_{n},\mu)$ for every $n\in\mathbb{N}$ such that the sequence $(I(\pi_{n}))_{n\in\mathbb{N}}$ converges to $0$. Then the sequence of real numbers

$$\Bigl(\int_{\mathbb{R}^{d+d}}\bigl\lVert\nabla\varphi(\mu_{n})(x)-\nabla\varphi(\mu)(y)\bigr\rVert^{2}\,\pi_{n}(dz)\Bigr)_{n\in\mathbb{N}}$$

converges to $0$; its $n$-th term is the discrepancy of $\nabla\varphi(\mu_{n})$ and $\nabla\varphi(\mu)$ along $\pi_{n}$, the gradients being those of property (b), that clause being read with $\mu_{n}$ in place of its $\nu$, so that it is a nonnegative real number.

(d)(v) For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the function

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

is of class $C^{2}$ on $\mathbb{R}^{d}$.

(e)(vi) For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and every positive $\varepsilon\in\mathbb{R}$ there is a positive $\theta\in\mathbb{R}$ such that

$$d_{\mathcal{S}(d)}\bigl(D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}}),D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})\bigr)<\varepsilon\qquad\text{for every }\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})\text{ with }W_{2}(\nu,\mu)<\theta ;$$

the Hessian matrices here are defined by property (d) and lie in $\mathcal{S}(d)$ by claim 2 of Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian, and $d_{\mathcal{S}(d)}(A,B)=\lVert A-B\rVert$ is the distance on $\mathcal{S}(d)$.

2. (The translation Hessian)(vii) Let $\varphi$ be an intrinsic test function on $Q$ and let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By property (d) the Hessian matrix $D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$ is defined, and it is symmetric by claim 2 of Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian, hence an element of $\mathcal{S}(d)$. The translation Hessian of $\varphi$ at $\mu$ is

$$H_{\varphi}(\mu)=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})\in\mathcal{S}(d).$$