Proof of The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score 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 for adding, multiplying and comparing inequalities between real numbers in Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field, and the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, are used without further mention.

Conventions. As in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars, $\lVert s\rVert=|s|$ and $\lVert s\rVert^{2}=s^{2}$ for $s\in\mathbb{R}=\mathbb{R}^{1}$, so $M_{2}(\nu)=\int x^{2}\,\nu(dx)$ for $\nu\in\mathcal{P}(\mathbb{R})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment), and $\langle\xi,\eta\rangle_{\nu}=\int\xi\eta\,d\nu$ on $L^{2}(\nu;\mathbb{R})$; for $\psi\in C_{c}^{\infty}(\mathbb{R})$, $\psi'=\nabla\psi$, $(\psi')'=\Delta\psi$, $\psi'$ and $\Delta\psi$ are continuous and bounded, and $\langle\xi,\nabla\psi\rangle_{\nu}=\int\xi\psi'\,d\nu$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives). Every real number is an interior point of the interval $\mathbb{R}$ (claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line), so a function $\mathbb{R}\to\mathbb{R}$ differentiable at every point is continuous (Differentiability at an Interior Point Implies Continuity There); a continuous function on $\mathbb{R}$ or on $\mathbb{R}^{2}$ is Borel (claims 3(a) and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets). In particular $V$, $V'$ and $V''$ are continuous and Borel. 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 in both directions, is that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and claim 2 of Image Measures, Measures with Densities, and Change of Variables. For $z\in\mathbb{R}^{2}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$; 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, by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling and this change of variables. By The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, $\mathcal{D}\subseteq\mathcal{D}_{\log}\subseteq\mathcal{P}_{2}(\mathbb{R})$ and every $\mu\in\mathcal{D}$ has $V$ $\mu$-integrable.

Constants. By Confining Potentials on the Real Line §slope fix $C_{s}\in\mathbb{R}$ with $|V'(x)|\le C_{s}(1+|V(x)|)$ for every $x$; then $C_{s}\ge0$, since $0\le|V'(0)|\le C_{s}(1+|V(0)|)$ and $1+|V(0)|>0$. By Confining Potentials on the Real Line §curvature with $\varepsilon=1$ fix $C_{1}\in\mathbb{R}$ with $V''(x)\le|V(x)|+C_{1}$ for every $x$. By Confining Potentials on the Real Line §superquadratic with $M=\beta/2>0$ fix a positive $K$ with $\tfrac{\beta}{2}x^{2}\le V(x)$ whenever $K\le|x|$.

Step P1 (A lower bound for $V$ and a quadratic minorant). The restriction of $V$ to $[-K,K]$ is continuous (claim 1 of Restriction Stability of Continuity and of the Derivative), so Extreme Value Theorem on a Closed Real Interval gives $x_{\min}\in[-K,K]$ with $V(x_{\min})\le V(x)$ for $|x|\le K$. Put $m=\min\{V(x_{\min}),0\}$, so $m\le0$ and $-m=|m|$. For $|x|\ge K$, $V(x)\ge\tfrac{\beta}{2}x^{2}\ge0\ge m$; hence $V(x)\ge m$ for every $x\in\mathbb{R}$. Put $b=\tfrac{\beta}{2}K^{2}-m\ge0$. If $|x|\ge K$ then $\tfrac{\beta}{2}x^{2}\le V(x)\le V(x)+b$; if $|x|<K$ then $\tfrac{\beta}{2}x^{2}\le\tfrac{\beta}{2}K^{2}=b+m\le b+V(x)$. Thus

Define $u=1+V-m:\mathbb{R}\to\mathbb{R}$. Then $u\ge1$, $u$ is continuous and Borel, and $u$ is differentiable at every point with $u'=V'$ (claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives). Since $V-m\ge0$,

In particular, for $\nu\in\mathcal{P}(\mathbb{R})$, $V$ is $\nu$-integrable if and only if $u$ is, and then $\int u\,d\nu=1-m+\int V\,d\nu$.

Step P2 (Convexity). (i) Tangent inequality: $V'(x)(y-x)\le V(y)-V(x)$ for all $x,y\in\mathbb{R}$. This is trivial for $y=x$. Let $x\ne y$ and $w=y-x$. For real $t$ with $0<t\le1$, convexity of $V$ (Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ with $n=1$, the points $y,x$ and the weight $t$) gives $V(x+tw)=V(ty+(1-t)x)\le tV(y)+(1-t)V(x)$, so, with $h=tw\ne0$, dividing by $t>0$,

Let $\varepsilon>0$ and let $\delta>0$ be as in Derivative at an Interior Point for $V$ at $x$ with $\varepsilon/|w|$ in place of $\varepsilon$. Choosing $t=\min\{1,\delta/(2|w|)\}$ we get $0<|h|<\delta$, so the difference quotient is within $\varepsilon/|w|$ of $V'(x)$, whence $wV'(x)\le V(y)-V(x)+\varepsilon$. As $\varepsilon>0$ was arbitrary, (i) follows.

(ii) $V''\ge0$. Adding (i) to (i) with $x$ and $y$ exchanged gives $0\le(V'(y)-V'(x))(y-x)$, so $V'(x)\le V'(y)$ whenever $x<y$. Hence every difference quotient $(V'(x+h)-V'(x))/h$ with $h\ne0$ is nonnegative. If $V''(x)<0$ for some $x$, Derivative at an Interior Point for $V'$ at $x$ with $\varepsilon=-V''(x)$ yields $h\ne0$ with a difference quotient below $V''(x)+\varepsilon=0$, a contradiction. So $V''\ge0$.

(iii) Curvature bound. Put $A=1+|m|+|C_{1}|$. By (ii), the choice of $C_{1}$ and (U), $0\le V''\le|V|+C_{1}\le u+|m|+|C_{1}|\le A\,u$.

Step P3 (Slope and growth). Put $B=C_{s}(2+|m|)+1>0$. By the choice of $C_{s}$ and (U), $|u'|=|V'|\le C_{s}(1+|V|)\le B\,u$. We claim

Fix $x$. The map $t\mapsto x+t$ has every difference quotient equal to $1$, so it is differentiable with derivative $1$ (Derivative at an Interior Point), and by Chain Rule for One-Dimensional Derivatives $t\mapsto u(x+t)$ is differentiable at every $t$ with derivative $V'(x+t)$. By claim 1 of Derivative and Continuity of the Scaled Exponential Function and the product rule (claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives), $w_{-}(t)=u(x+t)\exp(-Bt)$ and $w_{+}(t)=u(x+t)\exp(Bt)$ are differentiable at every $t$ with

because $|V'|\le Bu$ and $\exp>0$ (claim 2 of Basic Properties of the Exponential Function). Both are continuous on $\mathbb{R}$, so by The Sign of the Derivative and Monotonicity §nonincreasing and The Sign of the Derivative and Monotonicity §nondecreasing $w_{-}$ is nonincreasing and $w_{+}$ nondecreasing. For $h\ge0$, $u(x+h)\exp(-Bh)=w_{-}(h)\le w_{-}(0)=u(x)$; for $h\le0$, $u(x+h)\exp(Bh)=w_{+}(h)\le w_{+}(0)=u(x)$. Multiplying by $\exp(Bh)$, respectively $\exp(-Bh)$, and using $\exp(v)\exp(-v)=\exp(0)=1$ (claim 1 of Basic Properties of the Exponential Function) gives (G). Since $\exp$ is increasing (claim 4 there), $u(x+h)\le\exp(Br)\,u(x)$ whenever $|h|\le r$, and by (U)

Step P4 (Weak convergence from Wasserstein convergence). Let $\nu_{k},\nu\in\mathcal{P}_{2}(\mathbb{R})$ ($k\in\mathbb{N}$) with $(W_{2}(\nu_{k},\nu))_{k}$ of limit $0$. Then (i) $\nu_{k}\Rightarrow\nu$ on $(\mathbb{R},d_{\mathbb{R}})$ and (ii) $\nu_{k}\boxtimes\nu_{k}\Rightarrow\nu\boxtimes\nu$ on $(\mathbb{R}^{2},d_{E})$, in the sense of Weak Convergence of Finite Borel Measures on a Metric Space.

For each $k$ let $\pi_{k}\in\Pi(\nu_{k},\nu)$ be optimal, $I(\pi_{k})=W_{2}(\nu_{k},\nu)^{2}$ (Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment). (i) For bounded $f:\mathbb{R}\to\mathbb{R}$, Lipschitz with constant $L$, 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 §lipschitz gives $|\int f\,d\nu_{k}-\int f\,d\nu|\le L\,W_{2}(\nu_{k},\nu)$, so $\int f\,d\nu_{k}\to\int f\,d\nu$ by claim 3 of Order Properties of Limits of Real Sequences, and claim 1 of Portmanteau Theorem on a Metric Space gives $\nu_{k}\Rightarrow\nu$. (ii) Let $F:\mathbb{R}^{2}\to\mathbb{R}$ be bounded, $|F|\le B_{F}$, and Lipschitz with constant $\Lambda$ for $d_{E}$. By claim 4 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $d_{E}((x,y),(x,y'))=|y-y'|$ and $d_{E}((x,y),(x',y))=|x-x'|$, so each section $y\mapsto F(x,y)$ is bounded and Lipschitz with constant $\Lambda$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product and the Tonelli part of Tonelli and Fubini Theorems, applied to the nonnegative Borel function $F+B_{F}$ and then subtracting $B_{F}$, the functions $\Psi_{k}(x)=\int F(x,y)\,\nu_{k}(dy)$ and $\Psi(x)=\int F(x,y)\,\nu(dy)$ are Borel with values in $[-B_{F},B_{F}]$, and $\int F\,d(\nu'\boxtimes\nu'')=\int(\int F(x,y)\,\nu''(dy))\,\nu'(dx)$ for $\nu',\nu''\in\{\nu_{k},\nu\}$. 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 §lipschitz, $|\Psi_{k}(x)-\Psi(x)|\le\Lambda\,W_{2}(\nu_{k},\nu)$ for every $x$, and $|\Psi(x)-\Psi(x')|\le\int|F(x,y)-F(x',y)|\,\nu(dy)\le\Lambda|x-x'|$, so $\Psi$ is bounded and Lipschitz. Hence

the last term again 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 §lipschitz; claim 3 of Order Properties of Limits of Real Sequences and claim 1 of Portmanteau Theorem on a Metric Space give (ii).

Step P5 (Semicontinuity of the potential term). Let $\nu_{k},\nu$ be as in Step P4, suppose that $V$ is $\nu_{k}$-integrable for every $k$ and that $\int V\,d\nu_{k}\le a_{0}$ for every $k$, for some $a_{0}\in\mathbb{R}$. Then $V$ is $\nu$-integrable, $\int V\,d\nu\le a_{0}$, and for every $\varepsilon>0$ there is $N$ with $\int V\,d\nu-\varepsilon<\int V\,d\nu_{k}$ for all $k\ge N$.

For $j\in\mathbb{N}$ let $V_{j}=\min\{V,j\}$; it is continuous (claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space) with $m\le V_{j}\le j$, hence bounded and Borel. By Step P4(i), $\int V_{j}\,d\nu_{k}\to\int V_{j}\,d\nu$, and $\int V_{j}\,d\nu_{k}\le\int V\,d\nu_{k}\le a_{0}$, so $\int V_{j}\,d\nu\le a_{0}$ (claim 1 of Order Properties of Limits of Real Sequences). The functions $V_{j}-m\ge0$ are nondecreasing in $j$ with supremum $V-m$ (claim 1 of The Archimedean Property of the Real Numbers), so Monotone Convergence Theorem gives $\int(V-m)\,d\nu=\sup_{j}\int(V_{j}-m)\,d\nu\le a_{0}-m<\infty$. Hence $V-m$, and so $V$, is $\nu$-integrable, and $\int V\,d\nu=\sup_{j}\int V_{j}\,d\nu\le a_{0}$. Given $\varepsilon>0$, choose first $j$ with $\int V_{j}\,d\nu>\int V\,d\nu-\varepsilon/2$ (supremum property), then $N$ with $|\int V_{j}\,d\nu_{k}-\int V_{j}\,d\nu|<\varepsilon/2$ for $k\ge N$; for such $k$, $\int V\,d\nu_{k}\ge\int V_{j}\,d\nu_{k}>\int V\,d\nu-\varepsilon$.

Step P6 (Energy bounds). Let $\mu\in\mathcal{D}$. Since $V\ge m$, $\int V\,d\mu\ge m$, and $\mathcal{E}_{\log}(\mu)\ge-(1+M_{2}(\mu))$ by Basic Properties of the Logarithmic Energy on the Real Line: Lower Bound, Translation Invariance, Closed Sublevel Sets and Lower Semicontinuity §lower-bound; so

Integrating (Q) against $\mu$ gives $\tfrac{\beta}{2}M_{2}(\mu)\le\int V\,d\mu+b$, so $\tfrac{\beta}{4}M_{2}(\mu)\le\tfrac12\int V\,d\mu+\tfrac{b}{2}$, and therefore, writing $\mathcal{E}(\mu)=\tfrac{\beta}{4}\mathcal{E}_{\log}(\mu)+\int V\,d\mu$ and using $\mathcal{E}_{\log}(\mu)\ge-(1+M_{2}(\mu))$ directly rather than (E0), $\mathcal{E}(\mu)\ge-\tfrac{\beta}{4}-\tfrac12\int V\,d\mu-\tfrac b2+\int V\,d\mu$. Hence

Step P7 (First variation on $\mathcal{D}$). Let $\mu\in\mathcal{D}$ and $\psi\in C_{c}^{\infty}(\mathbb{R})$; let $P\ge0$ with $|\psi'(x)|\le P$ for all $x$ (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient), and $G_{t}=\mathrm{id}+t\psi'$, which is Borel with $(G_{t})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $G_{0}=\mathrm{id}$ (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 §borel). We show: there is $t_{0}$ with $0<t_{0}\le1$ such that $(G_{t})_{\#}\mu\in\mathcal{D}$ for every $t\in(-t_{0},t_{0})$, and $t\mapsto\mathcal{E}((G_{t})_{\#}\mu)$ on $(-t_{0},t_{0})$ is differentiable at $0$ with derivative

Let $U=(-1,1)$ and $f(t,x)=V(x+t\psi'(x))$ for $t\in U$, $x\in\mathbb{R}$. (i) For $t\in U$, $x\mapsto f(t,x)=V\circ G_{t}(x)$ is Borel, and by (G$'$) with $r=P$, $|f(t,x)|\le\exp(BP)u(x)+|m|$, which is $\mu$-integrable because $u$ is (Step P1); so $f(t,\cdot)$ is $\mu$-integrable. (ii) For fixed $x$, the map $s\mapsto x+s\psi'(x)$ has all difference quotients equal to $\psi'(x)$, so it is differentiable with derivative $\psi'(x)$; by Chain Rule for One-Dimensional Derivatives and claim 2 of Restriction Stability of Continuity and of the Derivative, $s\mapsto f(s,x)$ is differentiable at every $s\in U$ with $D_{1}f(s,x)=V'(x+s\psi'(x))\psi'(x)$. (iii) By (G$'$), $|D_{1}f(t,x)|\le PB\exp(BP)u(x)$, a $\mu$-integrable function of $x$. By Differentiation under the Integral Sign, $F_{V}(t)=\int f(t,x)\,\mu(dx)$ is differentiable at every point of $U$, with $F_{V}'(0)=\int V'\psi'\,d\mu$. By the change of variables, $V$ is $(G_{t})_{\#}\mu$-integrable with $\int V\,d(G_{t})_{\#}\mu=F_{V}(t)$ for $t\in U$. Let $t_{1}$ be as in The First Variation of the Logarithmic Energy on the Real Line along Gradient Perturbations of the Identity §variation for $\mu$ and $\psi$, and $t_{0}=\min\{t_{1},1\}$. For $t\in(-t_{0},t_{0})$, $(G_{t})_{\#}\mu\in\mathcal{D}_{\log}$ and $V$ is integrable against it, so $(G_{t})_{\#}\mu\in\mathcal{D}$ and $\mathcal{E}((G_{t})_{\#}\mu)=\tfrac{\beta}{4}\mathcal{E}_{\log}((G_{t})_{\#}\mu)+F_{V}(t)$. Restricting both summands to $(-t_{0},t_{0})$ (claim 2 of Restriction Stability of Continuity and of the Derivative) and using claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives with the derivative $-\int F_{\psi}\,d(\mu\boxtimes\mu)$ of The First Variation of the Logarithmic Energy on the Real Line along Gradient Perturbations of the Identity §variation gives (FV). If moreover $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, then by The First Variation of the Logarithmic Energy on the Real Line along Gradient Perturbations of the Identity §score the derivative (FV) equals $\int V'\psi'\,d\mu-\tfrac{\beta}{4}\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}$.

Step P8 (Gaussian smoothing and truncation). Let $\lambda$ be Lebesgue measure on $\mathcal{B}(\mathbb{R})$ and, for $0<s\le1$, $g_{s}$ the Gaussian kernel of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails in dimension $q=1$, which is the weight $\varphi_{s}$ of claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $m=1$, $\eta=s$, and is even (The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives). For $\rho\in\mathcal{P}(\mathbb{R})$ let $p_{\rho,s}=g_{s}*\rho$ be the Gaussian smoothing of Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity; by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality it is nonnegative, Borel and $\lambda$-integrable with $\int p_{\rho,s}\,d\lambda=1$. Let $\nu_{\rho,s}$ be the measure with density $p_{\rho,s}$ with respect to $\lambda$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables), $\nu_{\rho,s}(E)=\int\mathbf{1}_{E}\,p_{\rho,s}\,d\lambda$; since $\nu_{\rho,s}(\mathbb{R})=1$, $\nu_{\rho,s}\in\mathcal{P}(\mathbb{R})$. For Borel $f:\mathbb{R}\to[0,\infty)$ and $x\in\mathbb{R}$ write $(g_{s}*f)(x)=\int g_{s}(x-y)f(y)\,dy\in[0,\infty]$, consistently with the notation of Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity for bounded $f$.

(a) Integrals against $\nu_{\rho,s}$. For bounded Borel $f$, $f\,p_{\rho,s}$ is $\lambda$-integrable and $\int f\,d\nu_{\rho,s}=\int f\,p_{\rho,s}\,d\lambda=\int g_{s}*f\,d\rho$, by claim 3 of Image Measures, Measures with Densities, and Change of Variables and Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality. For Borel $f\ge0$, put $f_{k}=\min\{f,k\}$ ($k\in\mathbb{N}$), bounded and Borel; $f_{k}$ increases to $f$ (claim 1 of The Archimedean Property of the Real Numbers). Three applications of Monotone Convergence Theorem (in $y$ for each fixed $x$, then against $\lambda$, then against $\rho$, the functions $g_{s}*f_{k}$ being Borel by the duality clause) together with claim 3 of Image Measures, Measures with Densities, and Change of Variables give that $g_{s}*f$ is Borel and

(b) Exponential moments. For real $c\ge0$ and $x\in\mathbb{R}$, $\int g_{s}(x-y)\exp(c|y-x|)\,dy\le2\exp(c^{2}/2)$. Indeed $\exp(c|v|)\le\exp(cv)+\exp(-cv)$, one summand being the left side and the other positive. For $\sigma\in\{1,-1\}$, claim 3 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $a=\sigma cs$ (so $Z_{a}(v)=\sigma cv$ and $\kappa_{a}=c^{2}s$) and the evenness of $g_{s}$ give $g_{s}(x-y)\exp(\sigma c(y-x))=\exp(c^{2}s/2)\,g_{s}(y-(x+\sigma cs))$, whose integral in $y$ is $\exp(c^{2}s/2)\le\exp(c^{2}/2)$ by claim 1 there (with $x+\sigma cs$ in place of its $a$), $s\le1$ and the monotonicity of $\exp$. Add the two bounds.

(c) Moments of $u$. Put $\Gamma=2\exp(2B^{2})$. For $j\in\{1,2\}$ and $x,y\in\mathbb{R}$, (G) with $h=y-x$ and claim 1 of Basic Properties of the Exponential Function give $u(y)^{j}\le\exp(jB|y-x|)u(x)^{j}$, so by (b) with $c=jB$, $(g_{s}*u^{j})(x)\le2\exp(j^{2}B^{2}/2)u(x)^{j}\le\Gamma u(x)^{j}$. By (a),

(d) Membership in $\mathcal{D}_{\Sigma}$. Suppose $\rho(\mathbb{R}\setminus[-r,r])=0$ for some real $r\ge0$. By Extreme Value Theorem on a Closed Real Interval applied to the restriction of $u$ to $[-r,r]$ there is $U_{r}$ with $u\le U_{r}$ on $[-r,r]$; so $u^{2}\le U_{r}^{2}$ $\rho$-almost everywhere and $\int u^{2}\,d\rho\le U_{r}^{2}$ (The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison). Let $\nu=\nu_{\rho,s}$. By (c), $\int u^{2}\,d\nu<\infty$, and $\int u\,d\nu\le\int u^{2}\,d\nu$ since $1\le u$. By (Q) and (U), $x^{2}\le\tfrac{2}{\beta}(u(x)+|m|+b)$, so $M_{2}(\nu)<\infty$ and $\nu\in\mathcal{P}_{2}(\mathbb{R})$; by (U), $V$ is $\nu$-integrable; and $(V')^{2}\le B^{2}u^{2}$ (Step P3), so $\int(V')^{2}\,d\nu<\infty$. By Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §regularity, $p=p_{\rho,s}$ is of class $C^{3}$, hence of class $C^{1}$, on $\mathbb{R}^{1}$ (C^k Maps on a Euclidean Open Set), and $\partial_{1}p$ is bounded; by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $p$ is differentiable at every point with $p'=\partial_{1}p$ continuous and bounded. As $\nu(E)=\int_{E}p\,d\lambda$ for every Borel $E$, A Probability Measure on the Real Line with a Continuously Differentiable Density of Bounded Derivative Has Finite Free Fisher Information and Finite Logarithmic Energy §smooth gives $\nu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})\cap\mathcal{D}_{\log}$. By The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, $\nu\in\mathcal{D}$ and then $\nu\in\mathcal{D}_{\Sigma}$.

(e) Weak convergence as $s\to0$. Let $\rho\in\mathcal{P}(\mathbb{R})$, $s_{k}=1/(k+1)$ and $\nu_{k}=\nu_{\rho,s_{k}}$. Then $\nu_{k}\Rightarrow\rho$. Let $f$ be bounded, $|f|\le M_{f}$ with $M_{f}>0$, and Lipschitz with constant $L\ge0$; it is Borel. Given $\varepsilon>0$ put $r=\varepsilon/(L+1)$, so $|f(x)-f(x')|\le Lr\le\varepsilon$ when $|x-x'|\le r$. By Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §approximation (with $q=1$), $|(g_{s}*f)(x)-f(x)|\le\varepsilon+2M_{f}s/r^{2}$ for every $x$, so by (a) $|\int f\,d\nu_{k}-\int f\,d\rho|=|\int(g_{s_{k}}*f-f)\,d\rho|\le\varepsilon+2M_{f}s_{k}/r^{2}$. By claim 2 of The Archimedean Property of the Real Numbers there is $N$ with $2M_{f}<(N+1)r^{2}\varepsilon$, and then the difference is below $2\varepsilon$ for $k\ge N$. Thus $\int f\,d\nu_{k}\to\int f\,d\rho$, and claim 1 of Portmanteau Theorem on a Metric Space gives $\nu_{k}\Rightarrow\rho$.

(f) Wasserstein convergence. Let $\rho$ and $r$ be as in (d) and $\nu_{k}$ as in (e). Then $\rho\in\mathcal{P}_{2}(\mathbb{R})$ (its second moment is at most $r^{2}$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison), $\nu_{k}\in\mathcal{P}_{2}(\mathbb{R})$ by (d), and $(W_{2}(\nu_{k},\rho))_{k}$ has limit $0$. Indeed, let $\varepsilon>0$, put $c_{\rho}=\Gamma\int u\,d\rho+|m|$ and $M=(c_{\rho}+1)/\varepsilon>0$, and let $K_{M}>0$ be given by Confining Potentials on the Real Line §superquadratic. For $|y|\ge K_{M}$, $My^{2}\le V(y)\le u(y)+|m|$ by (U); so, by (c), for every $k$

With (e), Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence (dimension $1$) gives the claim.

(g) Truncation. For $n\in\mathbb{N}$ let $T_{n}(x)=\min\{\max\{x,-n\},n\}$. A case check shows $|T_{n}(x)-T_{n}(x')|\le|x-x'|$, $|T_{n}(x)|\le n$, $T_{n}(x)=x$ for $|x|\le n$ and $|T_{n}(x)-x|\le|x|$ for all $x$; so $T_{n}$ is continuous (A Lipschitz Map is Uniformly Continuous) and Borel. For $\nu\in\mathcal{P}(\mathbb{R})$, $\rho=(T_{n})_{\#}\nu\in\mathcal{P}(\mathbb{R})$ and $\rho(\mathbb{R}\setminus[-n,n])=\nu(\emptyset)=0$. If $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $\rho_{n}=(T_{n})_{\#}\mu$, then $\rho_{n}\in\mathcal{P}_{2}(\mathbb{R})$ and $(W_{2}(\mu,\rho_{n}))_{n}$ has limit $0$: 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, $(\mathrm{id},T_{n})_{\#}\mu\in\Pi(\mu,\rho_{n})$ has cost $I_{n}=\int(x-T_{n}(x))^{2}\,\mu(dx)$; the integrands are dominated by the $\mu$-integrable function $x\mapsto x^{2}$ and vanish at $x$ once $n\ge|x|$ (claim 1 of The Archimedean Property of the Real Numbers), so $I_{n}\to0$ by Dominated Convergence Theorem; and $W_{2}(\mu,\rho_{n})^{2}\le I_{n}$ (The Quadratic Wasserstein Distance on Euclidean Space §distance), so $W_{2}(\mu,\rho_{n})<\varepsilon$ as soon as $I_{n}<\varepsilon^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field).

Claim 1 (Penalty pair). By The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R})$, $\mathcal{E}$ is real-valued on $\mathcal{D}$, and $\Sigma(\mu)\in L^{2}(\mu;\mathbb{R})=T_{\mu}$ for $\mu\in\mathcal{D}_{\Sigma}$ (On the Real Line the Tangent Space is the Whole Space of Square-Integrable Vector Fields §everything). 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. The function $g_{1}$ is nonnegative and Borel with $\int g_{1}\,d\lambda=1$ (claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder), so the measure $\gamma$ with density $g_{1}$ with respect to $\lambda$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables) belongs to $\mathcal{P}(\mathbb{R})$. By Step P8(g), $\rho=(T_{0})_{\#}\gamma$ satisfies $\rho(\mathbb{R}\setminus[0,0])=0$, and by Step P8(d) with $r=0$, $\nu_{\rho,1}\in\mathcal{D}_{\Sigma}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound. Put $C=\tfrac{\beta}{4}+|m|\ge0$. Since $m\ge-|m|\ge-|m|(1+M_{2}(\mu))$, (E0) gives $-C(1+M_{2}(\mu))\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation, and the formula for $H_{\mathcal{E}}$. Let $\mu\in\mathcal{D}$ and $a\in\mathbb{R}$, and let $\tau_{a}(x)=x+a$ (The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants). By Basic Properties of the Logarithmic Energy on the Real Line: Lower Bound, Translation Invariance, Closed Sublevel Sets and Lower Semicontinuity §translation, $(\tau_{a})_{\#}\mu\in\mathcal{D}_{\log}$ with $\mathcal{E}_{\log}((\tau_{a})_{\#}\mu)=\mathcal{E}_{\log}(\mu)$. By (G$'$) with $r=|a|$, $x\mapsto V(x+a)$ is $\mu$-integrable, so by the change of variables $V$ is $(\tau_{a})_{\#}\mu$-integrable with integral $\Phi_{0}(a)=\int V(x+a)\,\mu(dx)$. Hence $(\tau_{a})_{\#}\mu\in\mathcal{D}$ and $e_{\mu}(a)=\tfrac{\beta}{4}\mathcal{E}_{\log}(\mu)+\Phi_{0}(a)$. Put $\Phi_{1}(a)=\int V'(x+a)\,\mu(dx)$ and $\Phi_{2}(a)=\int V''(x+a)\,\mu(dx)$, well defined by (G$'$). Fix $a_{0}\in\mathbb{R}$, let $U_{0}$ be the open interval $(a_{0}-1,a_{0}+1)$ and $r_{0}=|a_{0}|+1$, so $|t|\le r_{0}$ for $t\in U_{0}$. Apply Differentiation under the Integral Sign on $U_{0}$ to $f(t,x)=V(x+t)$: condition (i) holds by (G$'$) with $r=r_{0}$; (ii) holds with $D_{1}f(t,x)=V'(x+t)$, by Chain Rule for One-Dimensional Derivatives for $t\mapsto x+t$ (derivative $1$) and claim 2 of Restriction Stability of Continuity and of the Derivative; (iii) holds with $g=B\exp(Br_{0})u$ by (G$'$). So the restriction of $\Phi_{0}$ to $U_{0}$ is differentiable at $a_{0}$ with derivative $\Phi_{1}(a_{0})$. The same argument with $V'$, $V''$ and the bound $A\exp(Br_{0})u$ of (G$'$) in place of $V$, $V'$ and $B\exp(Br_{0})u$ shows that the restriction of $\Phi_{1}$ to $U_{0}$ is differentiable at $a_{0}$ with derivative $\Phi_{2}(a_{0})$. The condition of Derivative at an Interior Point at $a_{0}$ involves only points $a_{0}+h$ with $0<|h|<\delta$, and $\delta$ may be decreased to at most $1$; so $\Phi_{0}$ and $\Phi_{1}$, as functions on $\mathbb{R}$, are differentiable at $a_{0}$ with derivatives $\Phi_{1}(a_{0})$ and $\Phi_{2}(a_{0})$. Moreover $\Phi_{2}$ is continuous at $a_{0}$: if $(a_{j})_{j}$ converges to $a_{0}$, there is $J$ with $|a_{j}-a_{0}|<1$ for $j\ge J$; for such $j$, $V''(x+a_{j})\to V''(x+a_{0})$ for every $x$ (continuity of $V''$ and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential) with the $\mu$-integrable bound $A\exp(Br_{0})u(x)$, so Dominated Convergence Theorem applied to $(a_{J+i})_{i}$ gives $\Phi_{2}(a_{j})\to\Phi_{2}(a_{0})$, and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion applies. By claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $e_{\mu}$ is differentiable at every point with $e_{\mu}'=\Phi_{1}$, and by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $\partial_{1}e_{\mu}=\Phi_{1}$ and $\partial_{1}\partial_{1}e_{\mu}=\partial_{1}\Phi_{1}=\Phi_{2}$ at every point. The functions $e_{\mu}$ and $\Phi_{1}$ are continuous (being differentiable) and so is $\Phi_{2}$, continuity on $\mathbb{R}^{1}$ for the Euclidean distance being continuity for $d_{\mathbb{R}}$ (preamble of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative). By clauses 1 and 3 of C^k Maps on a Euclidean Open Set, $e_{\mu}$ and $\Phi_{1}$ are of class $C^{1}$, and by clause 2 with $k=1$, $e_{\mu}$ is of class $C^{2}$ on $\mathbb{R}$. Finally, by Hessian Matrix of a C^2 Function, $H_{\mathcal{E}}(\mu)=D^{2}e_{\mu}(0)$ is the $1\times1$ matrix with entry $\partial_{1}\partial_{1}e_{\mu}(0)=\Phi_{2}(0)=\int V''\,d\mu$, where $V''$ is $\mu$-integrable by (G$'$) with $r=0$. This proves the second sentence of the claim.