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

Each result cited below is universally quantified over the data in its own statement.

Throughout, $\xi_{n}$ abbreviates $\xi_{\nu_{n}}$. Fixing any $n_{0}\in\mathbb{N}$, one has $0\le\lVert\xi_{n_{0}}\rVert_{\nu_{n_{0}}}^{2}=\mathcal{I}(\nu_{n_{0}})\le C$ by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §information, so $C\ge0$ and $\sqrt{C}$ is defined; and $\lVert\xi_{n}\rVert_{\nu_{n}}=\sqrt{\mathcal{I}(\nu_{n})}\le\sqrt{C}$ for every $n$. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ the space $L^{2}(\mu;\mathbb{R}^{d})$ is a real inner product space with norm $\lVert\cdot\rVert_{\mu}$ by The Space of Square-Integrable Random Vectors §inner-product (see Optimal Transport on the Flat Torus: Standing Notation §fields), so the Cauchy-Schwarz inequality $|\langle\alpha,\beta\rangle_{\mu}|\le\lVert\alpha\rVert_{\mu}\lVert\beta\rVert_{\mu}$ holds in it. Since $(\nu_{n})$ converges to $\nu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$, the real sequence $(W_{\mathbb{T}}(\nu_{n},\nu))_{n\in\mathbb{N}}$ converges to $0$.

Step 1 (a Lipschitz bound in the torus distance). Let $u\in C^{1}_{\mathrm{per}}$. We show that there is a real $L_{u}\ge0$ with $|u(y)-u(x)|\le L_{u}\,d_{\mathbb{T}}(x,y)$ for all $x,y\in\mathbb{R}^{d}$. For each $i\in[d]$ the function $\partial_{i}u$ belongs to $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §derivative, so there is a real $M_{i}\ge0$ with $|\partial_{i}u|\le M_{i}$ on $\mathbb{R}^{d}$ by Elementary Properties of Lattice-Periodic Functions §bounded; put $L_{u}=M_{1}+\dots+M_{d}$. Let $x,y\in\mathbb{R}^{d}$, put $v=\varpi(y-x)$ and $z=x+v$. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $y-z=(y-x)-\varpi(y-x)\in\mathbb{Z}^{d}$, so $u(y)=u(z)$ by the periodicity of $u$; and $\lVert v\rVert=d_{\mathbb{T}}(x,y)$ by The Wrapped Displacement and the Flat Torus Distance §distance. If $v=0$, then $u(y)=u(x)$ and there is nothing to prove. Otherwise let $F:[0,1]\to\mathbb{R}$, $F(\tau)=u(x+\tau v)$. Since $\mathbb{R}^{d}$ is open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and $u$ is of class $C^{1}$ on it, Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set (claims 2 and 3, with $J=[0,1]$) shows that $F$ is continuous on $[0,1]$ and differentiable at every $\tau\in(0,1)$ with $F'(\tau)=\sum_{i=1}^{d}\partial_{i}u(x+\tau v)\,v_{i}$. By the mean value theorem there is $c\in(0,1)$ with $u(z)-u(x)=F(1)-F(0)=F'(c)$, whence

$$|u(y)-u(x)|=|F'(c)|\le\sum_{i=1}^{d}M_{i}\,|v_{i}|\le L_{u}\lVert v\rVert=L_{u}\,d_{\mathbb{T}}(x,y).$$

Step 2 (two limits for a fixed smooth periodic function). Let $f\in C^{\infty}_{\mathrm{per}}$. By Optimal Transport on the Flat Torus: Standing Notation §calculus, $\Delta f$ is continuous and periodic, that is, $\Delta f\in C_{\mathrm{per}}$. The partial derivatives $\partial_{i}f$ belong to $C^{\infty}_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §derivative, hence to $C_{\mathrm{per}}$ (a smooth map is of class $C^{1}$ by Smooth Map on a Euclidean Open Set, hence continuous by C^k Maps on a Euclidean Open Set), so $\lVert\nabla f\rVert^{2}=\sum_{i=1}^{d}(\partial_{i}f)^{2}\in C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra. By The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §integrals, applied to these two functions,

$$\int\Delta f\,d\nu_{n}\to\int\Delta f\,d\nu,\qquad\lVert\nabla f\rVert_{\nu_{n}}^{2}=\int\lVert\nabla f\rVert^{2}\,d\nu_{n}\to\int\lVert\nabla f\rVert^{2}\,d\nu=\lVert\nabla f\rVert_{\nu}^{2}$$

as $n\to\infty$. Moreover, by the defining property of the score Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score and the Cauchy-Schwarz inequality, for every $n$,

$$\Bigl|\int\Delta f\,d\nu_{n}\Bigr|=|\langle\xi_{n},\nabla f\rangle_{\nu_{n}}|\le\lVert\xi_{n}\rVert_{\nu_{n}}\lVert\nabla f\rVert_{\nu_{n}}\le\sqrt{C}\,\lVert\nabla f\rVert_{\nu_{n}}.$$

Step 3 (claim 1). Let $f\in C^{\infty}_{\mathrm{per}}$. By Step 2, $|\int\Delta f\,d\nu_{n}|\le\sqrt{C}\,(\lVert\nabla f\rVert_{\nu_{n}}^{2})^{1/2}$ for every $n$; the left side converges to $|\int\Delta f\,d\nu|$ and the right side to $\sqrt{C}\,\lVert\nabla f\rVert_{\nu}$, by continuity of the absolute value (claim 4 of Order Properties of Limits of Real Sequences) and of the square root on $[0,\infty)$ (Properties of Real Powers of Nonnegative Real Numbers §continuity, with $t^{1/2}=\sqrt{t}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement), together with claim 3 of Arithmetic of Limits of Real Sequences for the factor $\sqrt{C}$. Non-strict inequalities between terms of convergent real sequences pass to the limits (claim 1 of Order Properties of Limits of Real Sequences), so

$$\Bigl|\int\Delta f\,d\nu\Bigr|\le\sqrt{C}\,\lVert\nabla f\rVert_{\nu}\qquad\text{for every }f\in C^{\infty}_{\mathrm{per}}.$$

Since $\sqrt{C}\ge0$, this says that $\nu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite. Let $\ell(f)=-\int\Delta f\,d\nu$ for $f\in C^{\infty}_{\mathrm{per}}$; as recorded in Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score, $\ell$ is linear, and $|\ell(f)|\le\sqrt{C}\,\lVert\nabla f\rVert_{\nu}$ by the display. By The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §representation, applied with the constant $\sqrt{C}$, there is exactly one $\zeta\in T_{\nu}$ with $\langle\zeta,\nabla f\rangle_{\nu}=\ell(f)$ for every $f\in C^{\infty}_{\mathrm{per}}$, and $\lVert\zeta\rVert_{\nu}\le\sqrt{C}$. The score $\xi_{\nu}$ lies in $T_{\nu}$ and has the same property by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score, so $\xi_{\nu}=\zeta$, and $\mathcal{I}(\nu)=\lVert\xi_{\nu}\rVert_{\nu}^{2}\le C$ by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §information. In particular $\lVert\xi_{\nu}\rVert_{\nu}\le\sqrt{C}$.

Step 4 (set-up for claim 2). By convergence along couplings there are $\gamma_{n}\in\Pi(\nu_{n},\nu)$ such that $I_{\mathbb{T}}(\gamma_{n})\to0$ and $D_{\gamma_{n}}(\eta_{n},\eta)\to0$; fix them, together with Borel representatives of $\eta_{n}$ and $\eta$. For $f\in C^{\infty}_{\mathrm{per}}$ the components of $\nabla f$ lie in $C^{\infty}_{\mathrm{per}}\subseteq C^{1}_{\mathrm{per}}$ (Step 2), so by Step 1 there are reals $L_{1},\dots,L_{d}\ge0$ with $|\partial_{i}f(y)-\partial_{i}f(x)|\le L_{i}\,d_{\mathbb{T}}(x,y)$; with $L_{f}=L_{1}+\dots+L_{d}$ this gives

$$\lVert\nabla f(y)-\nabla f(x)\rVert\le\sum_{i=1}^{d}|\partial_{i}f(y)-\partial_{i}f(x)|\le L_{f}\,d_{\mathbb{T}}(x,y)\qquad(x,y\in\mathbb{R}^{d}).$$

Step 5 (the estimate for a fixed smooth periodic function). Let $f\in C^{\infty}_{\mathrm{per}}$ and $n\in\mathbb{N}$. By bilinearity of the inner products,

$$\langle\eta_{n},\xi_{n}\rangle_{\nu_{n}}-\langle\eta,\xi_{\nu}\rangle_{\nu}=\langle\eta_{n}-\nabla f,\xi_{n}\rangle_{\nu_{n}}+\bigl(\langle\nabla f,\xi_{n}\rangle_{\nu_{n}}-\langle\nabla f,\xi_{\nu}\rangle_{\nu}\bigr)+\langle\nabla f-\eta,\xi_{\nu}\rangle_{\nu}.$$

The middle term equals $-\int\Delta f\,d\nu_{n}+\int\Delta f\,d\nu$ by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score, applied at $\nu_{n}$ and at $\nu$ (the latter being legitimate by Step 3). By the Cauchy-Schwarz inequality and Step 3, the last term is at most $\lVert\eta-\nabla f\rVert_{\nu}\sqrt{C}$ in absolute value, and the first term is at most $\sqrt{C}\,\lVert\eta_{n}-\nabla f\rVert_{\nu_{n}}$ in absolute value.

We bound $\lVert\eta_{n}-\nabla f\rVert_{\nu_{n}}$ along $\gamma_{n}$. On $\mathbb{R}^{d+d}$ consider the nonnegative functions

$$F_{1}(w)=\lVert\eta_{n}(\mathrm{pr}_{1}w)-\eta(\mathrm{pr}_{2}w)\rVert,\quad F_{2}(w)=\lVert\eta(\mathrm{pr}_{2}w)-\nabla f(\mathrm{pr}_{2}w)\rVert,\quad F_{3}(w)=\lVert\nabla f(\mathrm{pr}_{2}w)-\nabla f(\mathrm{pr}_{1}w)\rVert,$$

and $F_{0}(w)=\lVert\eta_{n}(\mathrm{pr}_{1}w)-\nabla f(\mathrm{pr}_{1}w)\rVert$. They are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (the projections being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, and $\nabla f$ being Borel by Optimal Transport on the Flat Torus: Standing Notation §calculus). Since $(\mathrm{pr}_{1})_{\#}\gamma_{n}=\nu_{n}$ and $(\mathrm{pr}_{2})_{\#}\gamma_{n}=\nu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling), the change-of-variables formula gives

$$\int F_{0}^{2}\,d\gamma_{n}=\lVert\eta_{n}-\nabla f\rVert_{\nu_{n}}^{2},\qquad\int F_{2}^{2}\,d\gamma_{n}=\lVert\eta-\nabla f\rVert_{\nu}^{2},$$

while $\int F_{1}^{2}\,d\gamma_{n}=D_{\gamma_{n}}(\eta_{n},\eta)$ by Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus, and, by Step 4 and the monotonicity of the integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) together with Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost,

$$\int F_{3}^{2}\,d\gamma_{n}\le L_{f}^{2}\int d_{\mathbb{T}}(\mathrm{pr}_{1}w,\mathrm{pr}_{2}w)^{2}\,\gamma_{n}(dw)=L_{f}^{2}\,I_{\mathbb{T}}(\gamma_{n}).$$

So $F_{1},F_{2},F_{3}$ belong to $\mathcal{L}^{2}$ of the measure space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma_{n})$ (Power-Integrable Functions and the p-Seminorm §space). By the triangle inequality of the Euclidean norm, $F_{0}\le F_{1}+F_{2}+F_{3}$ pointwise, so $F_{0}^{2}\le(F_{1}+F_{2}+F_{3})^{2}$, and by monotonicity of the integral and Minkowski's inequality with $p=2$,

$$\lVert\eta_{n}-\nabla f\rVert_{\nu_{n}}=\Bigl(\int F_{0}^{2}\,d\gamma_{n}\Bigr)^{1/2}\le\lVert F_{1}+F_{2}+F_{3}\rVert_{2}\le D_{\gamma_{n}}(\eta_{n},\eta)^{1/2}+\lVert\eta-\nabla f\rVert_{\nu}+L_{f}\,I_{\mathbb{T}}(\gamma_{n})^{1/2}.$$

Collecting the three terms, for every $f\in C^{\infty}_{\mathrm{per}}$ and every $n\in\mathbb{N}$,

$$\bigl|\langle\eta_{n},\xi_{n}\rangle_{\nu_{n}}-\langle\eta,\xi_{\nu}\rangle_{\nu}\bigr|\le\sqrt{C}\Bigl(D_{\gamma_{n}}(\eta_{n},\eta)^{1/2}+L_{f}\,I_{\mathbb{T}}(\gamma_{n})^{1/2}\Bigr)+2\sqrt{C}\,\lVert\eta-\nabla f\rVert_{\nu}+\Bigl|\int\Delta f\,d\nu_{n}-\int\Delta f\,d\nu\Bigr|.$$

Step 6 (claim 2: order of choices). Let $\varepsilon>0$ be given, and put $\delta=\varepsilon/(4\sqrt{C}+2)>0$. First, since $\eta\in T_{\nu}$, the closure of $G_{\nu}$ in $L^{2}(\nu;\mathbb{R}^{d})$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent), Characterization of the Closure in a Metric Space by Open Balls (claim 3, for the metric of the norm $\lVert\cdot\rVert_{\nu}$) and The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients provide $f\in C^{\infty}_{\mathrm{per}}$ with $\lVert\eta-\nabla f\rVert_{\nu}<\delta$. Second, with this $f$ fixed, $L_{f}$ is fixed by Step 4. Third, since $D_{\gamma_{n}}(\eta_{n},\eta)\to0$, $I_{\mathbb{T}}(\gamma_{n})\to0$ and, by Step 2, $\int\Delta f\,d\nu_{n}\to\int\Delta f\,d\nu$, and since the square root is continuous at $0$ (Properties of Real Powers of Nonnegative Real Numbers §continuity, with $t^{1/2}=\sqrt{t}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement), there is $N\in\mathbb{N}$ such that for every $n\ge N$

$$D_{\gamma_{n}}(\eta_{n},\eta)^{1/2}\le\delta,\qquad L_{f}\,I_{\mathbb{T}}(\gamma_{n})^{1/2}\le\delta,\qquad\Bigl|\int\Delta f\,d\nu_{n}-\int\Delta f\,d\nu\Bigr|\le\delta.$$

By Step 5, for every $n\ge N$,

$$\bigl|\langle\eta_{n},\xi_{n}\rangle_{\nu_{n}}-\langle\eta,\xi_{\nu}\rangle_{\nu}\bigr|\le\sqrt{C}(\delta+\delta)+2\sqrt{C}\,\delta+\delta=(4\sqrt{C}+1)\,\delta<\varepsilon.$$

As $\varepsilon>0$ was arbitrary, the real sequence $(\langle\eta_{n},\xi_{\nu_{n}}\rangle_{\nu_{n}})_{n\in\mathbb{N}}$ converges to $\langle\eta,\xi_{\nu}\rangle_{\nu}$. This proves claim 2.