A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, intrinsic test functions on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$, their gradients along couplings and their translation Hessians are those of that definition; optimal maps are those of the setting and the map property is that of that definition, and for an optimal map $S$ from $\mu$ to $\nu$ the class $\mathrm{id}-S\in L^{2}(\mu;\mathbb{R}^{d})$ is that of The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable; convergent series of real numbers and convergent series in the real Hilbert space $L^{2}(\mu;\mathbb{R}^{d})$ are those of those items; and $I_{d}$ is the identity matrix of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices.

Let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ have the map property, let $B\in\mathbb{R}$, let $(\mu_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $M_{2}(\mu_{k})\le B$ for every $k\in\mathbb{N}$, and let $(c_{k})_{k\in\mathbb{N}}$ be a sequence of positive real numbers whose series converges, with sum $c=\sum_{k=1}^{\infty}c_{k}$. Then the following hold.

1. (Convergence)¶ For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the series $\sum_{k=1}^{\infty}c_{k}\,W_{2}(\mu,\mu_{k})^{2}$ and $\sum_{k=1}^{\infty}c_{k}\,W_{2}(\mu,\mu_{k})$ converge.

In the remaining clauses $\psi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ is the function, defined by clause 1, given by

$$\psi(\mu)=\sum_{k=1}^{\infty}c_{k}\,W_{2}(\mu,\mu_{k})^{2}.$$

2. (Test function)¶ $\psi$ is an intrinsic test function on $Q$.

3. (Gradient)¶ Let $\mu\in Q$ and, for every $k\in\mathbb{N}$, let $S_{k}$ be an optimal map from $\mu$ to $\mu_{k}$. Then the series $\sum_{k=1}^{\infty}2c_{k}\,(\mathrm{id}-S_{k})$ converges in $L^{2}(\mu;\mathbb{R}^{d})$ to the gradient along couplings $\nabla\psi(\mu)$, which exists by clause 2, and

$$\lVert\nabla\psi(\mu)\rVert_{\mu}\le2\sum_{k=1}^{\infty}c_{k}\,W_{2}(\mu,\mu_{k}).$$

4. (Translation Hessian)¶ $H_{\psi}(\mu)=2c\,I_{d}$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.