Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, the score $\xi$ and the torus Fisher information $\mathcal{I}$ be as in Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite, let $T_{\nu}$ be the tangent space at $\nu\in\mathcal{P}(\mathbb{T}^{d})$, and let convergence along couplings be as defined there. Let $\nu\in\mathcal{P}(\mathbb{T}^{d})$, let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ that converges to $\nu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$, and let $C$ be a real number with $\mathcal{I}(\nu_{n})\le C$ for every $n\in\mathbb{N}$. Then the following hold.

1. (Lower semicontinuity)¶ $\nu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and $\mathcal{I}(\nu)\le C$.

2. (Closedness of the score)¶ Let $\eta\in T_{\nu}$ and, for $n\in\mathbb{N}$, $\eta_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$, and suppose that $((\nu_{n},\eta_{n}))_{n\in\mathbb{N}}$ converges along couplings to $(\nu,\eta)$. Then the real sequence $(\langle\eta_{n},\xi_{\nu_{n}}\rangle_{\nu_{n}})_{n\in\mathbb{N}}$ converges to $\langle\eta,\xi_{\nu}\rangle_{\nu}$.