Proof of 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

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 (among them the positivity of $\tfrac{\sigma^{2}}{2}$, $\tfrac{\sigma^{2}}{4}$ and their inverses), and the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, are used without further mention. Integrals of nonnegative Borel functions are taken in $[0,\infty]$; linearity and monotonicity of integrals are claim 1 (nonnegative functions) and claim 2 (integrable functions, used only after integrability has been established) of Linearity and Monotonicity of the Lebesgue Integral. Change of variables under a push-forward, including the transfer of integrability, is that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. By The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, $\mathcal{E}$ is real-valued on $\mathcal{D}$, $V$ is $\mu$-integrable for $\mu\in\mathcal{D}$, and $\Sigma(\mu)\in T_{\mu}$ for $\mu\in\mathcal{D}_{\Sigma}$.

Constants. By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §growth fix $v_{0}$ and $C_{g}\ge0$ with $v_{0}\le V$ and $V(x+a)-v_{0}+1\le\exp(C_{g}\lVert a\rVert)(V(x)-v_{0}+1)$; put $u=V-v_{0}+1\ge1$ and $A_{0}=1+|v_{0}-1|$, so that $|V|\le A_{0}u$ and $1+|V|\le(1+A_{0})u$. Taking $x=0_{\mathbb{R}^{d}}$ and $a=y$, and using $\lVert DV\rVert\le C_{s}(1+|V|)$ with $C_{s}\ge0$ from Confining Potentials on Euclidean Space §slope (nonnegative since $0\le\lVert DV(0_{\mathbb{R}^{d}})\rVert$), we get for every $y\in\mathbb{R}^{d}$

using $\exp(s)\exp(s)=\exp(2s)$ (Basic Properties of the Exponential Function). By Confining Potentials on Euclidean Space §curvature with $\varepsilon=1$ fix $C_{1}$ with $\Delta V\le|V|+C_{1}$. Let $c_{1}$ be the constant of 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 and $\gamma=\tfrac{\sigma^{2}}{2}\log c_{1}$. By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §minorant with $M=\tfrac{\sigma^{2}}{4}+1>0$ fix $C_{M}$ with $M\lVert x\rVert^{2}-C_{M}\le V(x)$ for all $x$.

Step E (energy bounds). Let $\mu\in\mathcal{D}$. Integrating the minorant, $M\,M_{2}(\mu)-C_{M}\le\int V\,d\mu$; and $\log c_{1}-\tfrac12M_{2}(\mu)\le\mathrm{Ent}(\mu)$ by 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 §lower. Hence $\mathcal{E}(\mu)\ge\gamma-\tfrac{\sigma^{2}}{4}M_{2}(\mu)+M\,M_{2}(\mu)-C_{M}=\gamma+M_{2}(\mu)-C_{M}$, that is

Next $\int V\,d\mu=\mathcal{E}(\mu)-\tfrac{\sigma^{2}}{2}\mathrm{Ent}(\mu)\le\mathcal{E}(\mu)-\gamma+\tfrac{\sigma^{2}}{4}M_{2}(\mu)$, so by (E1)

Finally $\int V\,d\mu\ge v_{0}$ gives

Step S (smoothing). Let $\rho\in\mathcal{P}(\mathbb{R}^{d})$ satisfy $\rho(\bar{B}(0_{\mathbb{R}^{d}},R))=1$ for some real $R\ge0$, with $\bar{B}$ as in Euclidean Space and Lebesgue Measure: Standing Notation §space. Then $\lVert y\rVert^{2}\le R^{2}$ for $\rho$-almost every $y$, so $M_{2}(\rho)\le R^{2}$ (The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison) and $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For real $s$ with $0<s\le\tfrac12$ let $\rho_{s}$ be the measure $\mu_{s}$ of Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability built from $\rho$. We show $\rho_{s}\in\mathcal{D}_{\Sigma}$ and $W_{2}(\rho_{s},\rho)^{2}\le d\,s$. By Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §density, Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §entropy and Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §score, $\rho_{s}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $W_{2}(\rho_{s},\rho)^{2}\le d\,s$. The functions $V$ (continuous, hence Borel) and $\lVert\nabla V\rVert^{2}$ (Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, $\nabla V$ being Borel by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity) satisfy the exponential bounds (X), so both are $\rho_{s}$-integrable by Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §exponential. Hence $\rho_{s}\in\mathcal{D}$ and then $\rho_{s}\in\mathcal{D}_{\Sigma}$.

Claim 5 (first variation on $\mathcal{D}$). Let $\mu\in\mathcal{D}$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. By The First Variation of the Entropy Along a Gradient Perturbation of the Identity §variation there is $t_{1}>0$ with $(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ for $t\in(-t_{1},t_{1})$ and $t\mapsto\mathrm{Ent}((\mathrm{id}+t\nabla\psi)_{\#}\mu)$ differentiable at $0$ with derivative $-\int\Delta\psi\,d\mu$. By The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §variation, $\nabla V\cdot\nabla\psi$ is $\mu$-integrable, and for every $t$ the function $V\circ(\mathrm{id}+t\nabla\psi)$ is $\mu$-integrable, so $V$ is $(\mathrm{id}+t\nabla\psi)_{\#}\mu$-integrable with $\int V\,d(\mathrm{id}+t\nabla\psi)_{\#}\mu=F_{V}(t):=\int V(x+t\nabla\psi(x))\,\mu(dx)$, and $F_{V}$ is differentiable at $0$ with derivative $\int\nabla V\cdot\nabla\psi\,d\mu$. Put $t_{0}=t_{1}$. For $t\in(-t_{0},t_{0})$, $(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{D}$ and $\mathcal{E}((\mathrm{id}+t\nabla\psi)_{\#}\mu)=\tfrac{\sigma^{2}}{2}\mathrm{Ent}((\mathrm{id}+t\nabla\psi)_{\#}\mu)+F_{V}(t)$. Restricting $F_{V}$ to $(-t_{0},t_{0})$ (claim 2 of Restriction Stability of Continuity and of the Derivative) and using claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, this function is differentiable at $0$ with derivative $\int\nabla V\cdot\nabla\psi\,d\mu-\tfrac{\sigma^{2}}{2}\int\Delta\psi\,d\mu$.

Claim 1 (penalty pair). We verify the five conditions of Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty. Let $\gamma_{d}\in\mathcal{P}(\mathbb{R}^{d})$ be the measure of 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 and $\rho=(P_{1})_{\#}\gamma_{d}$, with $P_{1}$ the radial retraction of Radial Retraction onto a Closed Ball: Compactly Supported Approximation in the Wasserstein Distance (dimension $d$, $R=1$). By Radial Retraction onto a Closed Ball: Compactly Supported Approximation in the Wasserstein Distance §retraction, $\rho(\bar{B}(0_{\mathbb{R}^{d}},1))=1$, and Step S gives $\rho_{1/2}\in\mathcal{D}_{\Sigma}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound. Put $C=|\gamma|+|C_{M}|\ge0$. For $\mu\in\mathcal{D}$, by (E1), $\mathcal{E}(\mu)\ge\gamma-C_{M}+M_{2}(\mu)\ge-C\ge-C(1+M_{2}(\mu))$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation, and the translation Hessian. Let $\mu\in\mathcal{D}$ and $a\in\mathbb{R}^{d}$, with $\tau_{a}(x)=x+a$ (The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants). By 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 §translation, $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $\mathrm{Ent}((\tau_{a})_{\#}\mu)=\mathrm{Ent}(\mu)$. By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §integrable, $x\mapsto V(x+a)$ is $\mu$-integrable, so $V$ is $(\tau_{a})_{\#}\mu$-integrable with integral $\Phi_{\mu}(a)$, the function of The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §translation. Hence $(\tau_{a})_{\#}\mu\in\mathcal{D}$ and $e_{\mu}(a)=\tfrac{\sigma^{2}}{2}\mathrm{Ent}(\mu)+\Phi_{\mu}(a)$. The constant function is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) and $\Phi_{\mu}$ is of class $C^{2}$ (The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §translation), so $e_{\mu}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), with $\partial_{j}\partial_{i}e_{\mu}=\partial_{j}\partial_{i}\Phi_{\mu}$ by claim 1 there, the partial derivatives of a constant being $0$. By Hessian Matrix of a C^2 Function, each entry of $H_{\mathcal{E}}(\mu)=D^{2}e_{\mu}(0_{\mathbb{R}^{d}})$ is a second partial derivative $\partial_{j}\partial_{i}e_{\mu}(0_{\mathbb{R}^{d}})=\partial_{j}\partial_{i}\Phi_{\mu}(0_{\mathbb{R}^{d}})=\int\partial_{j}\partial_{i}V\,d\mu$, and the corresponding entry of $D^{2}V(x)$ is $\partial_{j}\partial_{i}V(x)$; each such function, and $\Delta V$, is $\mu$-integrable by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §integrable. By the definition of the trace, $\mathrm{tr}\,H_{\mathcal{E}}(\mu)=\sum_{i=1}^{d}\int\partial_{i}\partial_{i}V\,d\mu=\int\sum_{i=1}^{d}\partial_{i}\partial_{i}V\,d\mu=\int\Delta V\,d\mu$, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral applied $d-1$ times and The Laplacian of a Twice Continuously Differentiable Function §laplacian. This proves the second sentence of Claim 1.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §variation. Let $\mu\in\mathcal{D}_{\Sigma}$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Claim 5 gives $t_{0}$ and the derivative $\int\nabla V\cdot\nabla\psi\,d\mu-\tfrac{\sigma^{2}}{2}\int\Delta\psi\,d\mu$. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, $\langle\nabla V,\nabla\psi\rangle_{\mu}=\int\nabla V\cdot\nabla\psi\,d\mu$, and by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score, $\langle\xi_{\mu},\nabla\psi\rangle_{\mu}=-\int\Delta\psi\,d\mu$; by bilinearity of the inner product the derivative equals $\langle\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\mu},\nabla\psi\rangle_{\mu}=\langle\Sigma(\mu),\nabla\psi\rangle_{\mu}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §dense. Let $\mu\in\mathcal{D}$ and $\varepsilon>0$. By Radial Retraction onto a Closed Ball: Compactly Supported Approximation in the Wasserstein Distance §approximation choose $R>0$ with $W_{2}(\rho,\mu)<\varepsilon/2$, where $\rho=(P_{R})_{\#}\mu$, and $\rho(\bar{B}(0_{\mathbb{R}^{d}},R))=1$ by Radial Retraction onto a Closed Ball: Compactly Supported Approximation in the Wasserstein Distance §retraction. Let $s$ be the lesser of $\tfrac12$ and $\varepsilon^{2}(8d)^{-1}$, with $d$ read in $\mathbb{R}$; by Step S, $\rho_{s}\in\mathcal{D}_{\Sigma}$ and $W_{2}(\rho_{s},\rho)^{2}\le ds\le\varepsilon^{2}/8<(\varepsilon/2)^{2}$, so $W_{2}(\rho_{s},\rho)<\varepsilon/2$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). By The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, $W_{2}(\rho_{s},\mu)<\varepsilon$.

So $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair, and Claim 1 is proved.

Claim 2 (coercivity and the map property). Let $c\in\mathbb{R}$ and let $(\mu_{n})_{n}$ be a sequence in $S_{c}=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$. By (E2), $\int V\,d\mu_{n}\le\beta_{0}c+b_{0}$ for every $n$. The function $V$ is Borel, bounded below by $v_{0}$, and superquadratic in the sense of Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §compactness by Confining Potentials on Euclidean Space §superquadratic; that clause (with $m=d$) gives $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and a strictly increasing $(n_{k})_{k}$ with $W_{2}(\mu_{n_{k}},\mu)\to0$. By (E3), $\mathrm{Ent}(\mu_{n_{k}})\le\tfrac{2}{\sigma^{2}}(c-v_{0})$, so 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 §closed gives $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$; and The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §closed gives that $V$ is $\mu$-integrable, so $\mu\in\mathcal{D}$. Let $\varepsilon>0$. By 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 §lsc and claim 1 of Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space (the sequence $(\mu_{n_{k}})_{k}$ lies in $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and converges there to $\mu$) there is $N_{1}$ with $\mathrm{Ent}(\mu)-\varepsilon<\mathrm{Ent}(\mu_{n_{k}})$ for $k\ge N_{1}$, and by The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §closed there is $N_{2}$ with $\int V\,d\mu-\varepsilon<\int V\,d\mu_{n_{k}}$ for $k\ge N_{2}$ (the indices $n_{k}\ge k$ being large with $k$). For $k$ at least the larger of $N_{1},N_{2}$, $\mathcal{E}(\mu)-(\tfrac{\sigma^{2}}{2}+1)\varepsilon<\mathcal{E}(\mu_{n_{k}})\le c$. As $\varepsilon>0$ was arbitrary, $\mathcal{E}(\mu)\le c$ (Comparison of Real Numbers with Arbitrary Positive Slack), so $\mu\in S_{c}$. Thus $S_{c}$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, and the pair is Wasserstein-coercive (Wasserstein-Coercive Penalty Pairs §coercive). Every $\mu\in\mathcal{D}$ has finite entropy and is therefore absolutely continuous by 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 $\mathcal{D}$ has the map property by Two Sufficient Conditions for the Map Property: Absolute Continuity, and Atomlessness on the Line §absolutely-continuous.

Claim 3 (semicontinuity and growth). By Claims 1 and 2 the pair is a Wasserstein-coercive penalty pair, so $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by Basic Properties of a Wasserstein-Coercive Penalty Pair §lsc. Let $\mu\in\mathcal{D}$. By (E1), $M_{2}(\mu)\le|\mathcal{E}(\mu)|+|\gamma|+|C_{M}|\le c_{1}'(1+|\mathcal{E}(\mu)|)$ with $c_{1}'=1+|\gamma|+|C_{M}|$. By Claim 1, Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §hessian ($0\le\Delta V$), the choice of $C_{1}$, $|V|\le(V-v_{0})+|v_{0}|$ and (E2),

so $|\mathrm{tr}\,H_{\mathcal{E}}(\mu)|\le c_{2}'(1+|\mathcal{E}(\mu)|)$ with $c_{2}'=\beta_{0}+|b_{0}|+2|v_{0}|+|C_{1}|$. Take $C=c_{1}'+c_{2}'$.

Claim 4 (continuity of the trace of the translation Hessian). Let $R>0$ and $S_{R}=\{\mu\in\mathcal{D}:|\mathcal{E}(\mu)|\le R\}$. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset it suffices to show: if $\mu_{k},\mu\in S_{R}$ and $W_{2}(\mu_{k},\mu)\to0$, then $\mathrm{tr}\,H_{\mathcal{E}}(\mu_{k})\to\mathrm{tr}\,H_{\mathcal{E}}(\mu)$. By (E2), $\int V\,d\mu_{k}\le\beta_{0}R+b_{0}$ for every $k$. The function $\Delta V$ is continuous (Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity) and, being nonnegative, satisfies $|\Delta V|=\Delta V\le\varepsilon|V|+C_{\varepsilon}$ for every $\varepsilon>0$ with $C_{\varepsilon}$ from Confining Potentials on Euclidean Space §curvature. So The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy §convergence with $h=\Delta V$ and $c=\beta_{0}R+b_{0}$ gives $\int\Delta V\,d\mu_{k}\to\int\Delta V\,d\mu$, which by Claim 1 is the assertion. $\blacksquare$