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 §dimensions 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 §pushforward 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 §square-integrable. 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 §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $0_{\mathbb{R}^{d}}$ is the origin of $\mathbb{R}^{d}$.

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

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

(b) 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) For every nonnegative $R\in\mathbb{R}$ there is a nonnegative $L\in\mathbb{R}$ such that

for all $\mu,\nu\in Q$ with $M_{2}(\mu)\le R$ and $M_{2}(\nu)\le R$ and every $\pi\in\Pi(\mu,\nu)$; the left-hand side is the discrepancy of $\nabla\varphi(\mu)$ and $\nabla\varphi(\nu)$ along $\pi$, that clause being read with $\mu$ in place of its $\nu$ and $\nu$ in place of its $\mu$, so that it is a nonnegative real number.

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

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

2. (The translation Hessian)¶ 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 Second Partial Derivatives and Symmetry of the Hessian, hence an element of $\mathcal{S}(d)$. The translation Hessian of $\varphi$ at $\mu$ is