Proof of The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy

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, the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, and the vector-space operations and the norm properties of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ (claim 4: $|x_{l}|\le\lVert x\rVert$; claim 5: homogeneity; claim 6: triangle inequality) 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; a Borel function dominated in absolute value by an integrable one is integrable (Integrable Function and the Lebesgue Integral). For $a\in\mathbb{R}^{d}$, the translation $\tau_{a}(x)=x+a$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants is Borel: each component $x\mapsto x_{l}+a_{l}$ is the sum of a coordinate function, Borel by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and a constant, hence Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets applies; so $x\mapsto f(x+a)$ is Borel for Borel $f$, preimages of Borel sets under a composition being iterated preimages.

Constants and the bound (G). 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 $v_{0}\in\mathbb{R}$ and $C_{g}\ge0$ with $v_{0}\le V(x)$ and $V(x+a)-v_{0}+1\le\exp(C_{g}\lVert a\rVert)(V(x)-v_{0}+1)$ for all $x,a$. Put $u=V-v_{0}+1$; then $u\ge1$, $u$ is continuous and Borel, and for $\nu\in\mathcal{P}(\mathbb{R}^{d})$, $V$ is $\nu$-integrable if and only if $u$ is, constants being integrable against a probability measure. By Confining Potentials on Euclidean Space §slope fix $C_{s}$ with $\lVert DV(x)\rVert\le C_{s}(1+|V(x)|)$ for all $x$; then $C_{s}\ge0$, as $0\le\lVert DV(0_{\mathbb{R}^{d}})\rVert$ and $1+|V(0_{\mathbb{R}^{d}})|>0$. By Confining Potentials on Euclidean Space §curvature with $\varepsilon=1$ fix $C_{1}$ with $\Delta V(x)\le|V(x)|+C_{1}$ for all $x$, $\Delta V$ being the Laplacian. Since $V=u+(v_{0}-1)$ and $u\ge1$, $|V|\le u+|v_{0}-1|\le A_{0}u$ with $A_{0}=1+|v_{0}-1|$, and $1+|V|\le(1+A_{0})u$. For $r\ge0$ put $K_{r}=(A_{0}+|C_{1}|+C_{s}(1+A_{0}))\exp(C_{g}r)$. Let $x\in\mathbb{R}^{d}$ and $a\in\mathbb{R}^{d}$ with $\lVert a\rVert\le r$. Since $\exp$ is increasing and $\exp(0)=1$ (Basic Properties of the Exponential Function), $u(x+a)\le\exp(C_{g}r)u(x)$ and $1\le\exp(C_{g}r)$. Using Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §hessian ($0\le\Delta V$ and $|\partial_{j}\partial_{i}V|\le\Delta V$) we get, for all $i,j\in[d]$,

the last because $\Delta V(x+a)\le A_{0}u(x+a)+|C_{1}|u(x+a)$.

Claim 1. Let $\mu$ be as stated; then $u$ is $\mu$-integrable. For $a\in\mathbb{R}^{d}$ the functions $x\mapsto V(x+a)$, $\partial_{i}V(x+a)$, $\partial_{j}\partial_{i}V(x+a)$ are Borel and, by (G) with $r=\lVert a\rVert$, dominated by $K_{r}u$; so they are $\mu$-integrable and

are real numbers. In particular ($a=0_{\mathbb{R}^{d}}$) $\partial_{j}\partial_{i}V$ is $\mu$-integrable and $\Phi_{ij}(0_{\mathbb{R}^{d}})=\int\partial_{j}\partial_{i}V\,d\mu$.

Partial derivatives. Fix $a_{0}\in\mathbb{R}^{d}$ and $i\in[d]$, let $e_{i}$ be the $i$th standard basis vector, $r=\lVert a_{0}\rVert+1$ and $U=(-1,1)$. For $t\in U$, $\lVert a_{0}+te_{i}\rVert\le\lVert a_{0}\rVert+|t|<r$. Apply Differentiation under the Integral Sign on $U$ to $f(t,x)=V(x+a_{0}+te_{i})$. Condition (i) holds by the above. For (ii), fix $x$ and $s\in U$ and put $y=x+a_{0}+se_{i}$; for real $h\ne0$ the point $x+a_{0}+(s+h)e_{i}$ is $y$ with its $i$th coordinate increased by $h$, so the difference quotient $(f(s+h,x)-f(s,x))/h$ is the difference quotient of $V$ at $y$ in the $i$th variable of Partial Derivative on a Euclidean Open Set. As the partial derivative $\partial_{i}V(y)$ exists, the condition of Derivative at an Interior Point holds for $s\mapsto f(s,x)$ at $s$ with the value $\partial_{i}V(y)$, the two conditions being the same after decreasing $\delta$ so that $s\pm\delta\in U$; so $D_{1}f(s,x)=\partial_{i}V(x+a_{0}+se_{i})$. Condition (iii) holds with the integrable function $K_{r}u$, by (G). Hence $t\mapsto\Phi_{\mu}(a_{0}+te_{i})$ is differentiable at $0$ with derivative $\Phi_{i}(a_{0})$, which, by the same comparison of difference quotients, says that the partial derivative of $\Phi_{\mu}$ with respect to the $i$th variable exists at $a_{0}$ with value $\Phi_{i}(a_{0})$. The same argument with $\partial_{i}V$, $\partial_{j}\partial_{i}V$ and $j$ in place of $V$, $\partial_{i}V$ and $i$, using the second and third bounds of (G), shows that $\partial_{j}\Phi_{i}(a_{0})$ exists and equals $\Phi_{ij}(a_{0})$.

Continuity. Let $\Psi\in\{\Phi_{\mu},\Phi_{i},\Phi_{ij}\}$ with integrand $w\in\{V,\partial_{i}V,\partial_{j}\partial_{i}V\}$, which is 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. Let $a_{0}\in\mathbb{R}^{d}$, $r=\lVert a_{0}\rVert+1$, and let $(a_{k})_{k}$ converge to $a_{0}$ in $(\mathbb{R}^{d},d_{E})$; choose $J$ with $\lVert a_{k}-a_{0}\rVert<1$, hence $\lVert a_{k}\rVert<r$, for $k\ge J$. For each $x$, $\lVert(x+a_{k})-(x+a_{0})\rVert=\lVert a_{k}-a_{0}\rVert\to0$, so $w(x+a_{k})\to w(x+a_{0})$ (Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential); the functions $x\mapsto w(x+a_{J+k})$ are dominated by $K_{r}u$ (G), and Dominated Convergence Theorem gives $\Psi(a_{k})\to\Psi(a_{0})$. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, $\Psi$ is continuous at $a_{0}$ for $d_{E}$ and the absolute-value metric. This is continuity at $a_{0}$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces: $\sum_{l}(x_{l}-a_{0,l})^{2}=\lVert x-a_{0}\rVert^{2}$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), $(\Psi(x)-\Psi(a_{0}))^{2}=|\Psi(x)-\Psi(a_{0})|^{2}$, and for nonnegative reals $s,\delta$ one has $s<\delta$ if and only if $s^{2}<\delta^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field).

Class $C^{2}$. By clause 1 of C^k Maps on a Euclidean Open Set (with clause 3), each $\Phi_{i}$ is of class $C^{1}$ on $\mathbb{R}^{d}$, its partial derivatives $\Phi_{ij}$ existing and being continuous, and so is $\Phi_{\mu}$; by clause 2 with $k=1$, $\Phi_{\mu}$ is of class $C^{2}$. In the notation of clause 4, $\partial_{j}\partial_{i}\Phi_{\mu}(0_{\mathbb{R}^{d}})=\partial_{j}\Phi_{i}(0_{\mathbb{R}^{d}})=\Phi_{ij}(0_{\mathbb{R}^{d}})=\int\partial_{j}\partial_{i}V\,d\mu$.

Claim 2. By claim 1 of The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure, $\nabla\psi$ is Borel and there is $K_{\psi}\ge0$ with $\lVert\nabla\psi(x)\rVert\le K_{\psi}$ for all $x$. For $t\in\mathbb{R}$ the map $\mathrm{id}+t\nabla\psi$ is Borel (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), so $x\mapsto V(x+t\nabla\psi(x))$ is Borel, and by (G) with $a=t\nabla\psi(x)$ and $r=|t|K_{\psi}$ it is dominated by $K_{|t|K_{\psi}}u$, hence $\mu$-integrable. The function $\nabla V\cdot\nabla\psi$ is Borel and $\mu$-integrable by Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Fisher Information §integrable.

Apply Differentiation under the Integral Sign on $U=(-1,1)$ to $f(t,x)=V(x+t\nabla\psi(x))$. Condition (i) was just shown. For (ii), fix $x$ and let $\ell:\mathbb{R}^{1}\to\mathbb{R}^{d}$, $\ell(s)=x+s\nabla\psi(x)$, whose components $s\mapsto x_{l}+s\,\partial_{l}\psi(x)$ are of class $C^{1}$ on the open set $\mathbb{R}^{1}$ (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), being sums of constants and multiples of the coordinate function (claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), with $\partial_{1}\ell_{l}=\partial_{l}\psi(x)$ by claim 1 there, the coordinate function having every difference quotient equal to $1$. As $V$ is of class $C^{1}$ (clause 2 of C^k Maps on a Euclidean Open Set), claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ gives, for every $s$,

The difference quotients of $s\mapsto f(s,x)$ in Derivative at an Interior Point are those of $V\circ\ell$ in Partial Derivative on a Euclidean Open Set with $n=1$, so $D_{1}f(s,x)$ exists and equals this value. For (iii), by Cauchy-Schwarz Inequality for the Euclidean Dot Product and (G) with $r=K_{\psi}$ (as $\lVert s\nabla\psi(x)\rVert\le K_{\psi}$ for $s\in U$), $|D_{1}f(s,x)|\le K_{\psi}\lVert DV(x+s\nabla\psi(x))\rVert\le K_{\psi}K_{K_{\psi}}u(x)$, an integrable function of $x$. So $F(t)=\int V(x+t\nabla\psi(x))\,\mu(dx)$ is differentiable at $0$ as a function on $U$, with $F'(0)=\int DV(x)\cdot\nabla\psi(x)\,\mu(dx)=\int\nabla V\cdot\nabla\psi\,d\mu$. The condition of Derivative at an Interior Point at $0$ involves only points $h$ with $0<|h|<\delta$, and $\delta$ may be decreased to at most $1$; so the function on $\mathbb{R}$ is differentiable at $0$ with the same derivative.

Claim 3. For $k\in\mathbb{N}$, read in $\mathbb{R}$, let $w_{k}=\min\{V-v_{0},k\}$; it is continuous (claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space) with $0\le w_{k}\le k$, so bounded and Borel. By Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §weak and Weak Convergence of Finite Borel Measures on a Metric Space, $\int w_{k}\,d\mu_{n}\to\int w_{k}\,d\mu$; and $\int w_{k}\,d\mu_{n}\le\int(V-v_{0})\,d\mu_{n}=\int V\,d\mu_{n}-v_{0}\le c-v_{0}$, so $\int w_{k}\,d\mu\le c-v_{0}$ (claim 1 of Order Properties of Limits of Real Sequences). The functions $w_{k}$ are nondecreasing in $k$ and $w_{k}(x)=V(x)-v_{0}$ once $k\ge V(x)-v_{0}$ (claim 1 of The Archimedean Property of the Real Numbers), so Monotone Convergence Theorem gives $\int(V-v_{0})\,d\mu=\sup_{k}\int w_{k}\,d\mu\le c-v_{0}<\infty$. Hence $V-v_{0}$, and so $V$, is $\mu$-integrable, and $\int V\,d\mu\le c$. Given $\varepsilon>0$, choose first $k$ with $\int w_{k}\,d\mu>\int(V-v_{0})\,d\mu-\varepsilon/2$ (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$), then $N$ with $|\int w_{k}\,d\mu_{n}-\int w_{k}\,d\mu|<\varepsilon/2$ for $n\ge N$. For such $n$, $\int(V-v_{0})\,d\mu_{n}\ge\int w_{k}\,d\mu_{n}>\int(V-v_{0})\,d\mu-\varepsilon$; adding $v_{0}$ gives $\int V\,d\mu-\varepsilon<\int V\,d\mu_{n}$.

Claim 4. By Claim 3, $V$ is $\mu$-integrable and $\int V\,d\mu\le c$. Put $a_{c}=c-v_{0}$; then for $\nu\in\{\mu\}\cup\{\mu_{n}:n\in\mathbb{N}\}$, $0\le\int(V-v_{0})\,d\nu\le a_{c}$, so $a_{c}\ge0$. Since $V-v_{0}\ge0$, $|V|\le(V-v_{0})+|v_{0}|$. The function $h$ is continuous, hence Borel, and with $\varepsilon=1$ in the hypothesis, $|h|\le(V-v_{0})+|v_{0}|+|C_{1}'|$, where $C_{1}'$ is the constant provided for $\varepsilon=1$; so $h$ is integrable with respect to every such $\nu$.

Let $\varepsilon>0$. First put $\eta=\varepsilon/(4(a_{c}+1))>0$ and let $C_{\eta}$ be given by the hypothesis for $\eta$; then put $L=\eta|v_{0}|+|C_{\eta}|\ge0$ and $h_{L}=\max\{-L,\min\{h,L\}\}$, continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, with $|h_{L}|\le L$, hence bounded and Borel. Pointwise, $|h-h_{L}|\le\eta(V-v_{0})$: if $|h(x)|\le L$ then $h_{L}(x)=h(x)$; if $h(x)>L$ then $h(x)-h_{L}(x)=h(x)-L=|h(x)|-L$; if $h(x)<-L$ then $h_{L}(x)-h(x)=-L-h(x)=|h(x)|-L$; and in the last two cases

Hence $|\int h\,d\nu-\int h_{L}\,d\nu|\le\int|h-h_{L}|\,d\nu\le\eta a_{c}<\varepsilon/4$ for every such $\nu$. Finally, by Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §weak and Weak Convergence of Finite Borel Measures on a Metric Space choose $N$ with $|\int h_{L}\,d\mu_{n}-\int h_{L}\,d\mu|<\varepsilon/2$ for $n\ge N$. For $n\ge N$,

so $\int h\,d\mu_{n}\to\int h\,d\mu$ by Limit of a Sequence of Real Numbers. $\blacksquare$