Proof of The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition

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

Conventions. $L$ is the constant of the datum $(b,f)$. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $L^{2}(\mu;\mathbb{R}^{d})$ and its norm $\lVert\cdot\rVert_{\mu}$ are those of Optimal Transport on the Flat Torus: Standing Notation §fields; for a Borel representative $\xi$ of a class one has $\lVert\xi\rVert_{\mu}^{2}=\int\lVert\xi\rVert^{2}\,d\mu$ by The Space of Square-Integrable Random Vectors §inner-product, and the class of a pointwise combination $\xi+c\zeta$ of representatives is the combination of the classes, by The Space of Square-Integrable Random Vectors §classes. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $p\in T_{\mu}$ with a Borel representative, again written $p$, put

$$h_{\mu,p}(x)=\tfrac12\lVert p(x)\rVert^{2}-b(x,\mu)\cdot p(x)-f(x,\mu)\qquad(x\in\mathbb{R}^{d});$$

by Control Data and the Control Hamiltonian on the Torus Wasserstein Space §hamiltonian this function is Borel and integrable with respect to $\mu$, $H(\mu,p)=\int h_{\mu,p}\,d\mu$, and this value does not depend on the representative. Two estimates are used repeatedly. First, let $\kappa$ be a probability measure on $\mathbb{R}^{m}$ for some $m\ge1$ and let $u,v:\mathbb{R}^{m}\to[0,\infty)$ be Borel with $\int u^{2}\,d\kappa<\infty$ and $\int v^{2}\,d\kappa<\infty$. Then $u$ and $v$ are square-integrable random variables on the probability space $(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m}),\kappa)$, and the Cauchy-Schwarz inequality, claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, gives

$$\int uv\,d\kappa\le\Bigl(\int u^{2}\,d\kappa\Bigr)^{1/2}\Bigl(\int v^{2}\,d\kappa\Bigr)^{1/2};\qquad\text{in particular}\qquad\int u\,d\kappa\le\Bigl(\int u^{2}\,d\kappa\Bigr)^{1/2}\qquad(\ast)$$

(take $v=1$). Second, for $x,y,z\in\mathbb{R}^{d}$ and $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product and the bounds of Control Data and the Control Hamiltonian on the Torus Wasserstein Space §data give

$$|b(x,\mu)\cdot z|\le L\lVert z\rVert,\qquad\bigl|(b(x,\mu)-b(y,\nu))\cdot z\bigr|\le L\bigl(d_{\mathbb{T}}(x,y)+W_{\mathbb{T}}(\mu,\nu)\bigr)\lVert z\rVert .\qquad(\ast\ast)$$

Integrals of nonnegative Borel functions are compared and split using the monotonicity and additivity of claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and integrals of integrable functions using claim 2 there.

Step 1 (Continuity along couplings). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $p\in T_{\mu}$, and let $\mu_{n}\in\mathcal{P}(\mathbb{T}^{d})$ and $p_{n}\in T_{\mu_{n}}$ be such that $((\mu_{n},p_{n}))_{n\in\mathbb{N}}$ converges along couplings to $(\mu,p)$. By Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus §convergence there are $\gamma_{n}\in\Pi(\mu_{n},\mu)$ such that $I_{n}=I_{\mathbb{T}}(\gamma_{n})$ and $D_{n}=D_{\gamma_{n}}(p_{n},p)$ both converge to $0$. Fix Borel representatives of $p_{n}$ and $p$, and define on $\mathbb{R}^{d+d}$ the functions

$$e_{n}(w)=\bigl\lVert p_{n}(\mathrm{pr}_{1}(w))-p(\mathrm{pr}_{2}(w))\bigr\rVert,\qquad a(w)=\bigl\lVert p(\mathrm{pr}_{2}(w))\bigr\rVert,\qquad\tau(w)=d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr).$$

They are Borel and nonnegative: $e_{n}$ and $a$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, and $\tau$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz. By the definition of $D_{\gamma}$ in the preamble of Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus and of the torus cost in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost, $\int e_{n}^{2}\,d\gamma_{n}=D_{n}$ and $\int\tau^{2}\,d\gamma_{n}=I_{n}$. Since $(\mathrm{pr}_{1})_{\#}\gamma_{n}=\mu_{n}$ and $(\mathrm{pr}_{2})_{\#}\gamma_{n}=\mu$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, the change-of-variables formula gives $\int a^{2}\,d\gamma_{n}=\int\lVert p\rVert^{2}\,d\mu=\lVert p\rVert_{\mu}^{2}$. Finally, $W_{n}=W_{\mathbb{T}}(\mu_{n},\mu)$ satisfies $W_{n}^{2}\le I_{n}$, because $W_{n}^{2}$ is the greatest lower bound of the torus costs of the members of $\Pi(\mu_{n},\mu)$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance and $\gamma_{n}$ is one of them; hence $W_{n}\le\sqrt{I_{n}}$.

Write $h_{n}=h_{\mu_{n},p_{n}}$ and $h=h_{\mu,p}$. By the same change-of-variables formula, $h_{n}\circ\mathrm{pr}_{1}$ and $h\circ\mathrm{pr}_{2}$ are integrable with respect to $\gamma_{n}$, with $H(\mu_{n},p_{n})=\int h_{n}\circ\mathrm{pr}_{1}\,d\gamma_{n}$ and $H(\mu,p)=\int h\circ\mathrm{pr}_{2}\,d\gamma_{n}$. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

$$\bigl|H(\mu_{n},p_{n})-H(\mu,p)\bigr|\le\int\bigl|h_{n}(\mathrm{pr}_{1}(w))-h(\mathrm{pr}_{2}(w))\bigr|\,\gamma_{n}(dw).$$

Fix $w$ and write $x=\mathrm{pr}_{1}(w)$, $y=\mathrm{pr}_{2}(w)$, $q=p_{n}(x)$ and $s=p(y)$, so that $e_{n}(w)=\lVert q-s\rVert$ and $a(w)=\lVert s\rVert$. Since $\lVert q\rVert^{2}-\lVert s\rVert^{2}=(q-s)\cdot(q+s)$ and $q+s=(q-s)+2s$, the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product and the triangle inequality give $\bigl|\lVert q\rVert^{2}-\lVert s\rVert^{2}\bigr|\le e_{n}(w)\bigl(e_{n}(w)+2a(w)\bigr)$. Next, $b(x,\mu_{n})\cdot q-b(y,\mu)\cdot s=b(x,\mu_{n})\cdot(q-s)+(b(x,\mu_{n})-b(y,\mu))\cdot s$, so by $(\ast\ast)$ its absolute value is at most $Le_{n}(w)+L(\tau(w)+W_{n})a(w)$. Finally $|f(x,\mu_{n})-f(y,\mu)|\le L(\tau(w)+W_{n})$ by Control Data and the Control Hamiltonian on the Torus Wasserstein Space §data. Adding,

$$\bigl|h_{n}(\mathrm{pr}_{1}(w))-h(\mathrm{pr}_{2}(w))\bigr|\le\tfrac12e_{n}^{2}+e_{n}a+Le_{n}+L\tau a+LW_{n}a+L\tau+LW_{n},$$

all functions evaluated at $w$. Integrating against the probability measure $\gamma_{n}$ and bounding each term by $(\ast)$ with the three integrals computed above, together with $W_{n}\le\sqrt{I_{n}}$,

$$\bigl|H(\mu_{n},p_{n})-H(\mu,p)\bigr|\le\tfrac12D_{n}+\sqrt{D_{n}}\,\bigl(\lVert p\rVert_{\mu}+L\bigr)+2L\sqrt{I_{n}}\,\bigl(\lVert p\rVert_{\mu}+1\bigr).$$

Let $\varepsilon>0$, put $K=\tfrac12+\lVert p\rVert_{\mu}+L+2L(\lVert p\rVert_{\mu}+1)>0$ and $t=\min\{1,\varepsilon/(2K)\}$. As $(D_{n})$ and $(I_{n})$ converge to $0$, there is $N$ with $D_{n}<t^{2}$ and $I_{n}<t^{2}$ for all $n\ge N$; then $\sqrt{D_{n}}<t$, $\sqrt{I_{n}}<t$ and $D_{n}<t^{2}\le t$, so $|H(\mu_{n},p_{n})-H(\mu,p)|\le Kt\le\varepsilon/2<\varepsilon$. Thus $(H(\mu_{n},p_{n}))_{n\in\mathbb{N}}$ converges to $H(\mu,p)$, and $H$ is continuous along couplings. This proves The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §continuous.

Step 2 (Score perturbation). Put $\delta_{0}=\sigma^{2}/4>0$ and, for real $R>0$, $C_{R}=2(R+L)^{2}/\sigma^{2}\ge0$. Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $p,\zeta\in T_{\mu}$ with $\lVert p\rVert_{\mu}\le R$, and $0<\delta\le\delta_{0}$. Fix Borel representatives $p$ and $\zeta$; then $p+\delta\zeta$ is a Borel representative of the class $p+\delta\zeta$. For every $x$,

$$h_{\mu,p+\delta\zeta}(x)-h_{\mu,p}(x)=\delta\,p(x)\cdot\zeta(x)+\tfrac{\delta^{2}}{2}\lVert\zeta(x)\rVert^{2}-\delta\,b(x,\mu)\cdot\zeta(x),$$

and both functions on the left are integrable, so by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product and $(\ast\ast)$,

$$\bigl|H(\mu,p+\delta\zeta)-H(\mu,p)\bigr|\le\delta\int\lVert p\rVert\,\lVert\zeta\rVert\,d\mu+\tfrac{\delta^{2}}{2}\int\lVert\zeta\rVert^{2}\,d\mu+\delta L\int\lVert\zeta\rVert\,d\mu .$$

By $(\ast)$ for $\kappa=\mu$, with $z=\lVert\zeta\rVert_{\mu}$ this is at most $\delta Rz+\tfrac{\delta^{2}}{2}z^{2}+\delta Lz=\delta(R+L)z+\tfrac{\delta^{2}}{2}z^{2}$. Since $\delta\le\sigma^{2}/4$, one has $\tfrac{\delta^{2}}{2}z^{2}\le\tfrac{\sigma^{2}\delta}{8}z^{2}$; and $(R+L)z=\bigl(\tfrac{\sigma z}{2}\bigr)\bigl(\tfrac{2(R+L)}{\sigma}\bigr)\le\tfrac12\bigl(\tfrac{\sigma^{2}z^{2}}{4}+\tfrac{4(R+L)^{2}}{\sigma^{2}}\bigr)=\tfrac{\sigma^{2}}{8}z^{2}+C_{R}$, by the elementary inequality $uv\le\tfrac12(u^{2}+v^{2})$. Hence

$$\bigl|H(\mu,p+\delta\zeta)-H(\mu,p)\bigr|\le\frac{\sigma^{2}\delta}{8}z^{2}+C_{R}\,\delta+\frac{\sigma^{2}\delta}{8}z^{2}=\frac{\sigma^{2}\delta}{4}\lVert\zeta\rVert_{\mu}^{2}+C_{R}\,\delta ,$$

which is the score-perturbation bound with these $\delta_{0}$ and $C_{R}$. This proves The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §score-perturbation.

Step 3 (The structure estimate). Let $\mu,\nu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, let $T$ be an optimal map from $\mu$ to $\nu$, $S$ one from $\nu$ to $\mu$, let $\alpha>0$, and put $W=W_{\mathbb{T}}(\mu,\nu)$. The Borel maps $v_{T}$ and $v_{S}$ of Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement serve as representatives. By The Wrapped Displacement and the Flat Torus Distance §distance and Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement,

$$d_{\mathbb{T}}(x,T(x))=\lVert\varpi(T(x)-x)\rVert=\lVert v_{T}(x)\rVert\qquad(x\in\mathbb{R}^{d}).$$

By Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map, $T_{\#}\mu=\nu$ and $(\mathrm{id},T)_{\#}\mu$ is optimal, that is, its torus cost equals $W^{2}$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal. Since $\mathrm{pr}_{1}\circ(\mathrm{id},T)=\mathrm{id}$ and $\mathrm{pr}_{2}\circ(\mathrm{id},T)=T$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, the change-of-variables formula applied to the torus cost of Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost gives

$$W^{2}=I_{\mathbb{T}}\bigl((\mathrm{id},T)_{\#}\mu\bigr)=\int d_{\mathbb{T}}(x,T(x))^{2}\,\mu(dx)=\int\lVert v_{T}\rVert^{2}\,d\mu,\qquad\text{hence}\qquad\int\lVert v_{T}\rVert\,d\mu\le W$$

by $(\ast)$ for $\kappa=\mu$.

By McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §inverse, applicable as $\mu$ and $\nu$ are absolutely continuous (Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures), there is $D'\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(D')=1$ and $v_{S}(T(x))=-v_{T}(x)$ for $x\in D'$. The function $g=h_{\nu,\alpha v_{S}}$ is Borel and $\nu$-integrable with $H(\nu,\alpha v_{S})=\int g\,d\nu$; since $T_{\#}\mu=\nu$, the change-of-variables formula gives that $g\circ T$ is $\mu$-integrable with $\int g\,d\nu=\int g\circ T\,d\mu$. For $x\in D'$,

$$g(T(x))=\tfrac{\alpha^{2}}{2}\lVert v_{T}(x)\rVert^{2}+\alpha\,b(T(x),\nu)\cdot v_{T}(x)-f(T(x),\nu)=:G(x),$$

and $G$ is Borel (compositions of the Borel maps $T$, $v_{T}$, $b(\cdot,\nu)$, $f(\cdot,\nu)$ (Borel by Control Data and the Control Hamiltonian on the Torus Wasserstein Space §hamiltonian) with the functions of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions), so $G$ is $\mu$-integrable with $\int G\,d\mu=\int g\circ T\,d\mu=H(\nu,\alpha v_{S})$, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. On the other hand $-\alpha v_{T}$ represents $-\alpha v_{T}\in T_{\mu}$, so

$$H(\mu,-\alpha v_{T})=\int\Bigl(\tfrac{\alpha^{2}}{2}\lVert v_{T}(x)\rVert^{2}+\alpha\,b(x,\mu)\cdot v_{T}(x)-f(x,\mu)\Bigr)\,\mu(dx).$$

Subtracting by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the quadratic terms cancel:

$$H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})=\int\Bigl(\alpha\bigl(b(T(x),\nu)-b(x,\mu)\bigr)\cdot v_{T}(x)+f(x,\mu)-f(T(x),\nu)\Bigr)\,\mu(dx).$$

By $(\ast\ast)$, applied with the pairs $(x,\mu)$ and $(T(x),\nu)$, and by Control Data and the Control Hamiltonian on the Torus Wasserstein Space §data, together with $d_{\mathbb{T}}(x,T(x))=\lVert v_{T}(x)\rVert$, the absolute value of the integrand is at most

$$\alpha L\bigl(\lVert v_{T}(x)\rVert+W\bigr)\lVert v_{T}(x)\rVert+L\bigl(\lVert v_{T}(x)\rVert+W\bigr).$$

Integrating with claims 1 and 2 of Linearity and Monotonicity of the Lebesgue Integral, and using $\int\lVert v_{T}\rVert^{2}\,d\mu=W^{2}$ and $\int\lVert v_{T}\rVert\,d\mu\le W$,

$$H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})\le\alpha L\bigl(W^{2}+W\cdot W\bigr)+L(W+W)=2L\bigl(\alpha W^{2}+W\bigr).$$

This is the inequality of The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §structure.

Step 4 (The structure condition). Let $\eta>0$. If $L>0$ put $r=\eta/(2L)$; if $L=0$ put $r=1$. Let $\mu,\nu,T,S,\alpha$ be as in The Structure Condition Along Optimal Maps for a Hamiltonian on the Torus Wasserstein Space §structure with $\alpha W^{2}+W<r$, where $W=W_{\mathbb{T}}(\mu,\nu)$. By Step 3, if $L>0$ then $H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})\le2L(\alpha W^{2}+W)<2Lr=\eta$, and if $L=0$ then $H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})\le0<\eta$. Hence $H$ satisfies the structure condition, completing the proof of The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §structure.