Intrinsic Test Functions on the Torus Wasserstein Space

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mathcal{K}\subseteq\mathcal{P}(\mathbb{T}^{d})$ and $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$. Continuity of $\varphi$ is understood between the metric space $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ of Optimal Transport on the Flat Torus: Standing Notation §measures and $\mathbb{R}$ with the absolute-value metric. Differentiability along couplings and the gradient along couplings $\nabla\varphi(\mu)$ are those of that definition, and $T_{\mu}$ is the tangent space.

(Intrinsic test function)¶ The function $\varphi$ is an intrinsic test function on $\mathcal{K}$ if it has the following three properties.

(a)¶ $\varphi$ is continuous on $\mathcal{P}(\mathbb{T}^{d})$.

(b)¶ For every $\mu\in\mathcal{K}$, $\varphi$ is differentiable along couplings at $\mu$ and $\nabla\varphi(\mu)\in T_{\mu}$.

(c)¶ Let $\mu\in\mathcal{K}$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{K}$, and let $\gamma_{n}\in\Pi(\mu_{n},\mu)$ for $n\in\mathbb{N}$ be such that $(I_{\mathbb{T}}(\gamma_{n}))_{n\in\mathbb{N}}$ converges to $0$. Then the sequence of real numbers

$$\Bigl(\int_{\mathbb{R}^{d+d}}\bigl\lVert\nabla\varphi(\mu_{n})(\mathrm{pr}_{1}(w))-\nabla\varphi(\mu)(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\,\gamma_{n}(dw)\Bigr)_{n\in\mathbb{N}}$$

converges to $0$; each term is a real number, being at most $2\lVert\nabla\varphi(\mu_{n})\rVert_{\mu_{n}}^{2}+2\lVert\nabla\varphi(\mu)\rVert_{\mu}^{2}$ by the inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and the change-of-variables formula, and it does not depend on the representatives chosen, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison.