Proof of The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for adding, multiplying and comparing inequalities between real numbers, and the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, are used without further mention. Linearity and monotonicity of integrals are claim 1 (nonnegative functions) and claim 2 (integrable functions) of Linearity and Monotonicity of the Lebesgue Integral; for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the inner product of $L^{2}(\nu;\mathbb{R}^{d})$ is $\langle f,g\rangle_{\nu}=\int f\cdot g\,d\nu$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$ (The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings); if $\pi\in\Pi(\nu,\nu')$ and $f$ is Borel, then $\int f(x)\,\pi(dz)=\int f\,d\nu$ and $\int f(y)\,\pi(dz)=\int f\,d\nu'$ whenever either side is defined (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward). By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair and The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, the pair is a penalty pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, $V$ is integrable against every $\nu\in\mathcal{D}$, and for $\nu\in\mathcal{D}_{\Sigma}$, $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$.

Step 0 (bounded products). Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be continuous and compactly supported. By claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set there is $r>0$ with $f(x)=0$ for $\lVert x\rVert>r$. If $w:\mathbb{R}^{d}\to\mathbb{R}$ is continuous, then $|w|$ is bounded on the compact ball $\bar{B}(0_{\mathbb{R}^{d}},r)$ (A Closed Euclidean Ball is Convex and Compact, Extreme Value Theorem on a Compact Subset of a Metric Space applied to the continuous restriction of $|w|$), and $f$ is bounded (A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable); so $wf$ is continuous (claim 3 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space) and bounded, vanishing off that ball. We apply this with $w=\partial_{l}V$, continuous by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity.

Claim 1 (closed score). Let $R\ge0$, let $(\nu_{n})_{n}$ be a sequence in $\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$, let $\nu\in\mathcal{D}$, and let $(\pi_{n})_{n}$ be a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\nu$. Since $W_{2}(\nu_{n},\nu)^{2}\le I(\pi_{n})$ (The Quadratic Wasserstein Distance on Euclidean Space §distance) and $I(\pi_{n})\to0$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\nu_{n},\nu)\to0$, and by Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §weak, $\nu_{n}\Rightarrow\nu$: $\int f\,d\nu_{n}\to\int f\,d\nu$ for every bounded continuous $f$ (Weak Convergence of Finite Borel Measures on a Metric Space).

(a) Gradients; $\nu\in\mathcal{D}_{\Sigma}$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. By The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient and The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §laplacian, each $\partial_{l}\psi$ and $\Delta\psi$ is continuous, compactly supported and bounded. By Step 0, $\nabla V\cdot\nabla\psi=\sum_{l}\partial_{l}V\,\partial_{l}\psi$ and $\lVert\nabla\psi\rVert^{2}=\sum_{l}(\partial_{l}\psi)^{2}$ are bounded and continuous. For $n\in\mathbb{N}$, by the formula for $\Sigma$, bilinearity and Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score,

Put $\ell(\psi)=\int\nabla V\cdot\nabla\psi\,d\nu-\tfrac{\sigma^{2}}{2}\int\Delta\psi\,d\nu$. By weak convergence and claims 1 and 3 of Arithmetic of Limits of Real Sequences, $\ell_{n}(\psi)\to\ell(\psi)$ and $\lVert\nabla\psi\rVert_{\nu_{n}}^{2}\to\lVert\nabla\psi\rVert_{\nu}^{2}$. By The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu_{n};\mathbb{R}^{d})$, $\ell_{n}(\psi)^{2}\le R^{2}\lVert\nabla\psi\rVert_{\nu_{n}}^{2}$; passing to the limit (claim 2 of Arithmetic of Limits of Real Sequences, claim 1 of Order Properties of Limits of Real Sequences), $\ell(\psi)^{2}\le(R\lVert\nabla\psi\rVert_{\nu})^{2}$, so $|\ell(\psi)|\le R\lVert\nabla\psi\rVert_{\nu}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). As $V$ is $\nu$-integrable and this holds for every $\psi$, Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Fisher Information §splitting with $\kappa=\tfrac{\sigma^{2}}{2}$ and $C=R$ gives $\int\lVert\nabla V\rVert^{2}\,d\nu<\infty$ and $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. Hence $\nu\in\mathcal{D}_{\Sigma}$.

(b) Test fields. Let $\eta$ be a test field and $B_{\eta}\ge0$ with $\lVert\eta(x)\rVert\le B_{\eta}$ for all $x$ (Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment). Its components are smooth, hence of class $C^{1}$, and compactly supported, so for every $\nu'\in\mathcal{D}_{\Sigma}\subseteq\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field §parts gives $\langle\xi_{\nu'},\eta\rangle_{\nu'}=-\int\operatorname{div}\eta\,d\nu'$, and so

Here $\nabla V\cdot\eta=\sum_{l}\partial_{l}V\,\eta_{l}$ is bounded and continuous by Step 0, and $\operatorname{div}\eta=\sum_{l}\partial_{l}\eta_{l}$ is bounded and continuous by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient. Applying this to $\nu'=\nu_{n}$ and to $\nu'=\nu$ (by (a)), weak convergence gives $\langle\Sigma(\nu_{n}),\eta\rangle_{\nu_{n}}\to\langle\Sigma(\nu),\eta\rangle_{\nu}$.

Next let $q_{n}$ be a Borel representative of $\Sigma(\nu_{n})$. By The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §pairing, $\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})=\int q_{n}(x)\cdot\eta(y)\,\pi_{n}(dz)$, and $\langle\Sigma(\nu_{n}),\eta\rangle_{\nu_{n}}=\int q_{n}(x)\cdot\eta(x)\,\pi_{n}(dz)$. The components of $\eta$ are uniformly continuous (A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Uniformly Continuous); given $\theta>0$ choose $\delta>0$ with $\lVert\eta(y)-\eta(x)\rVert^{2}<\theta$ whenever $\lVert x-y\rVert<\delta$ (taking the least of the $d$ radii for the tolerance $(\theta/d)^{1/2}$ in each component, with claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Then for every $z$, $\lVert\eta(y)-\eta(x)\rVert^{2}\le\theta+4B_{\eta}^{2}\delta^{-2}\lVert x-y\rVert^{2}$, as $\lVert\eta(y)-\eta(x)\rVert\le2B_{\eta}$ and $1\le\delta^{-2}\lVert x-y\rVert^{2}$ when $\lVert x-y\rVert\ge\delta$. Integrating, $D_{n}=\int\lVert\eta(y)-\eta(x)\rVert^{2}\,\pi_{n}(dz)\le\theta+4B_{\eta}^{2}\delta^{-2}I(\pi_{n})$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost), which is below $2\theta$ for all large $n$; so $D_{n}\to0$. By Cauchy-Schwarz Inequality for the Euclidean Dot Product and Hoelder's Inequality, for Two and for Finitely Many Factors §holder with exponents $2$ and $2$,

since $\int\lVert q_{n}(x)\rVert^{2}\,\pi_{n}(dz)=\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}^{2}\le R^{2}$. With claim 3 of Order Properties of Limits of Real Sequences, $\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})\to\langle\Sigma(\nu),\eta\rangle_{\nu}$.

(c) All fields. Let $\eta_{0}\in L^{2}(\nu;\mathbb{R}^{d})$ and $\varepsilon>0$; put $S_{\nu}=\lVert\Sigma(\nu)\rVert_{\nu}$ and $\varepsilon'=\varepsilon/(3(R+S_{\nu}+1))$. By Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment §dense there is a test field $\eta$ with $\lVert\eta_{0}-\eta\rVert_{\nu}\le\varepsilon'$. By The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §linear and The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §bound, $|\mathcal{K}(\Sigma(\nu_{n}),\eta_{0},\pi_{n})-\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})|\le R\varepsilon'$ for every $n$, and by The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|\langle\Sigma(\nu),\eta_{0}\rangle_{\nu}-\langle\Sigma(\nu),\eta\rangle_{\nu}|\le S_{\nu}\varepsilon'$. By (b) choose $N$ with $|\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})-\langle\Sigma(\nu),\eta\rangle_{\nu}|<\varepsilon/3$ for $n\ge N$. For $n\ge N$, $|\mathcal{K}(\Sigma(\nu_{n}),\eta_{0},\pi_{n})-\langle\Sigma(\nu),\eta_{0}\rangle_{\nu}|<(R+S_{\nu})\varepsilon'+\varepsilon/3<\varepsilon$. So $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n}$ (Strong and Weak Convergence of Vector Fields Along Couplings of Vanishing Cost §weak), and with (a) the pair has closed score along couplings (Penalty Pairs with Closed Score Along Couplings §closed).

Claim 2 (regular penalised maxima). Let $\chi$ be an intrinsic test function on $\mathcal{D}$, let $\lambda>0$, and let $\mu\in\mathcal{D}$ be a point at which $\chi-\lambda\mathcal{E}$ has a local maximum relative to $\mathcal{D}$, with radius $r_{0}>0$. By property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, $\chi$ is differentiable along couplings at $\mu$ with gradient $\zeta=\nabla\chi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, $G_{t}=\mathrm{id}+t\nabla\psi$, let $t_{0}$ be as in The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §variation for $\mu$ and $\psi$, $p_{\psi}=\lVert\nabla\psi\rVert_{\mu}$, and $t_{2}$ the lesser of $t_{0}$ and $r_{0}/(p_{\psi}+1)$. For $t\in(-t_{2},t_{2})$: $(G_{t})_{\#}\mu\in\mathcal{D}$, and $W_{2}(\mu,(G_{t})_{\#}\mu)\le|t|p_{\psi}<r_{0}$ (The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §distance); so $\phi(t)=\chi((G_{t})_{\#}\mu)-\lambda\mathcal{E}((G_{t})_{\#}\mu)\le\phi(0)$, as $(G_{0})_{\#}\mu=\mu$. Thus $\phi:(-t_{2},t_{2})\to\mathbb{R}$ has a local maximum at $0$.

Differentiability of the first term. Put $g_{0}=\langle\zeta,\nabla\psi\rangle_{\mu}$. Let $\varepsilon>0$ and let $\theta>0$ be given by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable for $\varepsilon/(p_{\psi}+1)$. For $0<|t|<\theta/(p_{\psi}+1)$, the coupling $\pi_{t}=(\mathrm{id},G_{t})_{\#}\mu\in\Pi(\mu,(G_{t})_{\#}\mu)$ has $I(\pi_{t})=t^{2}p_{\psi}^{2}<\theta^{2}$ (The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §coupling), and by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=G_{t}$, whose displacement $G_{t}-\mathrm{id}=t\nabla\psi$ is bounded, $\mathcal{J}(\zeta,\pi_{t})=\langle\zeta,t\nabla\psi\rangle_{\mu}=tg_{0}$. Hence $|\chi((G_{t})_{\#}\mu)-\chi(\mu)-tg_{0}|\le\tfrac{\varepsilon}{p_{\psi}+1}|t|p_{\psi}<\varepsilon|t|$, so $t\mapsto\chi((G_{t})_{\#}\mu)$ is differentiable at $0$ with derivative $g_{0}$ (Derivative at an Interior Point), the restriction to $(-t_{2},t_{2})$ being harmless.

By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §variation, claim 2 of Restriction Stability of Continuity and of the Derivative and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\phi$ is differentiable at $0$ with $\phi'(0)=g_{0}-\lambda\bigl(\int\nabla V\cdot\nabla\psi\,d\mu-\tfrac{\sigma^{2}}{2}\int\Delta\psi\,d\mu\bigr)$, and $\phi'(0)=0$ by Vanishing of the Derivative at an Interior Local Extremum. With The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\mu;\mathbb{R}^{d})$,

This holds for every $\psi$, and $V$ is $\mu$-integrable; Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Fisher Information §splitting with $\kappa=\tfrac{\sigma^{2}}{2}$ and $C=\lambda^{-1}\lVert\zeta\rVert_{\mu}$ gives $\int\lVert\nabla V\rVert^{2}\,d\mu<\infty$ and $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, i.e. $\mu\in\mathcal{D}_{\Sigma}$. So the pair has regular penalised maxima (Penalty Pairs with Regular Penalised Maxima §regular).

Claim 3 (displacement convexity). Let $\mu\in\mathcal{D}_{\Sigma}$, $\nu\in\mathcal{D}$ and let $\pi\in\Pi(\mu,\nu)$ be optimal. As $\mu$ has finite entropy it is absolutely continuous (Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §absolutely-continuous), so by Brenier's Theorem: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map §map there is a Borel $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\pi=(\mathrm{id},T)_{\#}\mu$ and $T_{\#}\mu=\nu$; thus $T$ is an optimal map from $\mu$ to $\nu$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, $\int\lVert T-\mathrm{id}\rVert^{2}\,d\mu=I(\pi)<\infty$, so The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=T$ (for which $\pi_{S}=\pi$) gives $\mathcal{J}(\eta,\pi)=\langle\eta,T-\mathrm{id}\rangle_{\mu}$ for every $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, the class $T-\mathrm{id}$ being the difference of the classes $T$ and $\mathrm{id}$.

Entropy. Both $\mu$ and $\nu$ lie in $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $\mu$ has finite Fisher information, so McCann's Tangent Inequality: the Entropy Lies Above its Tangent Along Optimal Maps §tangent gives $\mathrm{Ent}(\mu)+\mathcal{J}(\xi_{\mu},\pi)=\mathrm{Ent}(\mu)+\langle\xi_{\mu},T-\mathrm{id}\rangle_{\mu}\le\mathrm{Ent}(\nu)$.

Potential. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing with the representative $\nabla V$, $\mathcal{J}(\nabla V,\pi)=\int\nabla V(x)\cdot(y-x)\,\pi(dz)$. The functions $z\mapsto V(x)$ and $z\mapsto V(y)$ are $\pi$-integrable with integrals $\int V\,d\mu$ and $\int V\,d\nu$, and $\nabla V(x)\cdot(y-x)\le V(y)-V(x)$ for every $z$ by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §tangent; so $\mathcal{J}(\nabla V,\pi)\le\int V\,d\nu-\int V\,d\mu$.

By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear, $\mathcal{J}(\Sigma(\mu),\pi)=\mathcal{J}(\nabla V,\pi)+\tfrac{\sigma^{2}}{2}\mathcal{J}(\xi_{\mu},\pi)$. Multiplying the entropy inequality by $\tfrac{\sigma^{2}}{2}>0$ and adding,

So the pair is $0$-displacement convex, i.e. displacement convex ($\lambda$-Displacement Convexity of a Penalty Pair on the Wasserstein Space §convex). $\blacksquare$