Proof of The Potential Gradient Paired with the Score, and a Fisher Information Bound on the Domain of the Langevin Free-Energy Pair

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The rules for adding inequalities, for multiplying them by nonnegative reals and for handling absolute values, from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field, are used without further mention; so is the elementary arithmetic of real numbers, in particular $2\cdot\tfrac{\sigma^{2}}{2}=\sigma^{2}$ and $0<\tfrac{\sigma^{2}}{2}$.

Step 0 (Integrability). Since $\nu\in\mathcal{D}_{\Sigma}$, the definition of the Langevin free-energy pair gives $\nu\in\mathcal{D}\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, $V$ integrable with respect to $\nu$, and $\int_{\mathbb{R}^{d}}\lVert\nabla V\rVert^{2}\,d\nu<\infty$. The map $\nabla V$ is 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, so its class belongs to $L^{2}(\nu;\mathbb{R}^{d})$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu); this is the first assertion of clause 1. As $\nu$ is a probability measure and $V$ is integrable with respect to it, Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §integrable shows that $\lVert\nabla V\rVert$ and $\Delta V$ are integrable with respect to $\nu$, and 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, applied at $\nu\in\mathcal{D}$, gives

Moreover $0\le\Delta V(x)$ for every $x$, by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §hessian. Fix a Borel representative of $\xi_{\nu}$, again written $\xi_{\nu}$; then $\int_{\mathbb{R}^{d}}\lVert\xi_{\nu}\rVert^{2}\,d\nu=\mathcal{I}(\nu)<\infty$ by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §information and the formula for the norm in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. By Hoelder's Inequality, for Two and for Finitely Many Factors §holder, with the conjugate exponents $p=q=2$ on the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$ and the square-integrable functions $\lVert\xi_{\nu}\rVert$ and $\lVert\nabla V\rVert$, the product $\lVert\xi_{\nu}\rVert\,\lVert\nabla V\rVert$ is integrable with respect to $\nu$. For $a,b\in\mathbb{R}^{d}$ we have $|a\cdot b|\le\lVert a\rVert\,\lVert b\rVert$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product, and $|a_{i}|\le\lVert a\rVert$ for $i\in[d]$ by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

Step 1 (Cut-off fields). Let $\chi_{R}$ ($R>0$) and the constant $M_{1}\ge0$ be those of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, read with $q=d$. For $m\in\mathbb{N}$ put $R_{m}=m+1$, a real number with $1\le R_{m}$, and let $\eta^{m}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map with components

Regularity. As $V$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (Confining Potentials on Euclidean Space §confining), each $\partial_{i}V$ is of class $C^{1}$ on $\mathbb{R}^{d}$ by clause 2 of C^k Maps on a Euclidean Open Set; $\chi_{R_{m}}$ is smooth by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set each $\eta^{m}_{i}$ is of class $C^{1}$ on $\mathbb{R}^{d}$, and by claim 1 there

so that, by the definition of the Laplacian,

Compact support. The set $\{x:\eta^{m}_{i}(x)\ne0\}$ is contained in $\{x:\chi_{R_{m}}(x)\ne0\}$, hence in the support $\operatorname{supp}\chi_{R_{m}}$, which contains the latter set and is closed by claims 1 and 2 of The Closure is the Smallest Closed Superset. By claim 3 there, $\operatorname{supp}\eta^{m}_{i}\subseteq\operatorname{supp}\chi_{R_{m}}$. The set $\operatorname{supp}\chi_{R_{m}}$ is compact, $\chi_{R_{m}}$ being compactly supported by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, hence bounded by Heine-Borel Theorem in $\mathbb{R}^n$; so $\operatorname{supp}\eta^{m}_{i}$ is bounded, and it is closed by claim 2 of The Closure is the Smallest Closed Superset, hence compact by Heine-Borel Theorem in $\mathbb{R}^n$. Thus $\eta^{m}_{i}$ is compactly supported.

Integration by parts. By The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field §parts, applied to $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and the field $\eta^{m}$, the class of $\eta^{m}$ lies in $L^{2}(\nu;\mathbb{R}^{d})$, $\operatorname{div}\eta^{m}$ is integrable with respect to $\nu$, and

Bounds. For every $x$: $0\le\chi_{R_{m}}(x)\le1$, so $\lVert\eta^{m}(x)\rVert=\chi_{R_{m}}(x)\lVert\nabla V(x)\rVert\le\lVert\nabla V(x)\rVert$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and hence $|\xi_{\nu}(x)\cdot\eta^{m}(x)|\le\lVert\xi_{\nu}(x)\rVert\,\lVert\nabla V(x)\rVert$; $0\le\chi_{R_{m}}(x)\Delta V(x)\le\Delta V(x)$; and, by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff and the coordinate bound of Step 0,

The functions $\chi_{R_{m}}\Delta V$ and $g_{m}$ are continuous, by Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous (claim 3) for $\chi_{R_{m}}$, $\partial_{i}\chi_{R_{m}}$, $\partial_{i}V$ and by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity for $\Delta V$, together with Continuity of Sums and Products of Real-Valued Functions on a Metric Space; hence they are Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and by the bounds just obtained and Step 0 they are integrable with respect to $\nu$. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral, (1a) becomes

Step 2 (Removing the cut-off; clause 1). Fix $x\in\mathbb{R}^{d}$. By claim 1 of The Archimedean Property of the Real Numbers there is $m_{0}\in\mathbb{N}$ with $\lVert x\rVert<m_{0}$; for $m\ge m_{0}$ we have $\lVert x\rVert\le R_{m}$, so $\chi_{R_{m}}(x)=1$ by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, whence $\eta^{m}(x)=\nabla V(x)$ and $\chi_{R_{m}}(x)\Delta V(x)=\Delta V(x)$. Thus, as $m\to\infty$, $\xi_{\nu}(x)\cdot\eta^{m}(x)\to\xi_{\nu}(x)\cdot\nabla V(x)$ and $\chi_{R_{m}}(x)\Delta V(x)\to\Delta V(x)$, the sequences being eventually constant; and $g_{m}(x)\to0$ by (1b), since $R_{m}^{-1}\to0$ (given $\varepsilon>0$, claim 3 of The Archimedean Property of the Real Numbers gives $n$ with $0<n^{-1}<\varepsilon$, and $0<R_{m}^{-1}<n^{-1}$ for every $m\ge n$). The dominating functions $\lVert\xi_{\nu}\rVert\,\lVert\nabla V\rVert$, $\Delta V$ and $d\,M_{1}\lVert\nabla V\rVert$ are integrable with respect to $\nu$ by Step 0. The dominated convergence theorem, applied three times on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$, gives

Passing to the limit in (1c) with Arithmetic of Limits of Real Sequences and the uniqueness of limits (claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences), and using the symmetry of the inner product (condition (a) of Real Inner Product Space §inner-product; $L^{2}(\nu;\mathbb{R}^{d})$ is a real Hilbert space by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, in particular a real inner product space) together with (0a),

This proves clause 1.

Step 3 (Clause 2). In the real inner product space $L^{2}(\nu;\mathbb{R}^{d})$, whose norm satisfies $\lVert w\rVert_{\nu}^{2}=\langle w,w\rangle_{\nu}$ (Real Inner Product Space §norm), the expansion Elementary Identities in a Real Inner Product Space §expansion, bilinearity Elementary Identities in a Real Inner Product Space §bilinear and homogeneity Elementary Identities in a Real Inner Product Space §homogeneity (with $|\tfrac{\sigma^{2}}{2}|=\tfrac{\sigma^{2}}{2}$) give, for $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$ as in The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair,

using clause 1 and $\mathcal{I}(\nu)=\lVert\xi_{\nu}\rVert_{\nu}^{2}$ (Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §information). Since $0\le\lVert\nabla V\rVert_{\nu}^{2}$ (Real Inner Product Space §norm),

which is clause 2.