Proof of Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Fisher Information

Each result cited is universally quantified over the data in its own statement. Write $\ell(\psi)=\int\nabla V\cdot\nabla\psi\,d\mu-\kappa\int\Delta\psi\,d\mu$.

Claim 1. By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §integrable, $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\lVert\nabla V\rVert$ and $\Delta V$ are $\mu$-integrable. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. The maps $\nabla V$ (Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity) and $\nabla\psi$ (claim 1 of The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure) are Borel, so $\nabla V\cdot\nabla\psi$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions. With $K$ the bound on $\lVert\nabla\psi\rVert$ from that claim, $|\nabla V\cdot\nabla\psi|\le K\lVert\nabla V\rVert$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product, so $\nabla V\cdot\nabla\psi$ is integrable: a Borel function whose absolute value is dominated by an integrable function has finite integral of its absolute value, by monotonicity of the integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), hence is integrable. The same remark is used below whenever integrability follows from such a bound.

Constants. Let $v_{0}$ be as in Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §growth, so $v_{0}\le V$, and let $U=V-v_{0}+1$, a $C^{2}$ function with $1\le U$ and the same partial derivatives as $V$. Let $C_{1}$ be the constant of Confining Potentials on Euclidean Space §slope, let $C'$ be the constant of Confining Potentials on Euclidean Space §curvature for $\varepsilon=1$, and let $c_{0}=|v_{0}|+1$. Then $|V|\le U+c_{0}\le(1+c_{0})U$, so, with $A=|C_{1}|(2+c_{0})$ and $B=2+c_{0}+|C'|$,

the middle bound by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §hessian; again $1\le U$ is used.

Step 1 (the truncations). For $p\in\mathbb{N}$, read as a positive real number, let $G_{p}:(-p,\infty)\to\mathbb{R}$ be $G_{p}(u)=p-p^{2}(p+u)^{-1}=pu(p+u)^{-1}$. By Reciprocal Rule for One-Dimensional Derivatives and Chain Rule for One-Dimensional Derivatives, $G_{p}$ is twice differentiable on the open interval $(-p,\infty)$, with

both continuous; so $G_{p}$ is of class $C^{2}$ on $(-p,\infty)\subseteq\mathbb{R}^{1}$, its partial derivative being its derivative (Partial Derivative on a Euclidean Open Set). Since $U$ takes values in $[1,\infty)\subseteq(-p,\infty)$, the function $f_{p}=G_{p}\circ U$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, and by its claim 1 and the product rule

Since $U\ge1>0$, one has $0<w_{p}\le1$, $G_{p}''(U)<0$, $p(p+U)^{-1}\le1$ and $U(p+U)^{-1}\le1$. Hence $|\partial_{i}f_{p}|\le p^{2}(p+U)^{-2}AU\le Ap$ and

So the first and second partial derivatives of $f_{p}$ are bounded by $M_{p}=(A+B+2A^{2})p$.

Step 2 (the hypothesis for $f_{p}$). By Approximation of a Twice Continuously Differentiable Function with Bounded First and Second Derivatives by Test Functions §approximation (with $q=d$, $f=f_{p}$, $M=M_{p}$) there are test functions $\psi_{n}$ and $M'\ge0$ whose first and second partial derivatives are bounded by $M'$ and converge pointwise to those of $f_{p}$. Then pointwise $\nabla V\cdot\nabla\psi_{n}\to\nabla V\cdot\nabla f_{p}$, with $|\nabla V\cdot\nabla\psi_{n}|\le\sqrt{d}M'\lVert\nabla V\rVert$; $\Delta\psi_{n}\to\Delta f_{p}$, with $|\Delta\psi_{n}|\le dM'$; and $\lVert\nabla\psi_{n}\rVert^{2}\to\lVert\nabla f_{p}\rVert^{2}$, with $\lVert\nabla\psi_{n}\rVert^{2}\le dM'^{2}$; the limits are by Arithmetic of Limits of Real Sequences. The dominating functions are integrable (Claim 1, and $\mu$ is a probability measure), so Dominated Convergence Theorem and continuity of the square root pass the hypothesis $|\ell(\psi_{n})|\le C\lVert\nabla\psi_{n}\rVert_{\mu}$ to the limit:

Step 3 (dropping the curvature term). By Step 1, $\nabla V\cdot\nabla f_{p}=w_{p}\lVert\nabla V\rVert^{2}$ and $\Delta f_{p}=w_{p}\Delta V+G_{p}''(U)\lVert\nabla V\rVert^{2}$. All of these functions are integrable: $w_{p}\lVert\nabla V\rVert^{2}$ by Claim 1 applied through Step 2, $0\le w_{p}\Delta V\le\Delta V$, and $G_{p}''(U)\lVert\nabla V\rVert^{2}=\Delta f_{p}-w_{p}\Delta V$. Since $\kappa>0$, $G_{p}''(U)\le0$, $0\le\Delta V$ and $w_{p}^{2}\le w_{p}\le1$,

With (1), $X_{p}^{2}\le CX_{p}+D$. Hence $X_{p}\le C+\sqrt{D}$: otherwise $X_{p}>C+\sqrt{D}\ge\sqrt{D}$ would give $X_{p}^{2}>(C+\sqrt{D})X_{p}\ge CX_{p}+D$.

Step 4 (monotone limit). Fix $x$. If $p\le p'$, then $0<p'^{-1}U(x)\le p^{-1}U(x)$, so $1\le1+p'^{-1}U(x)\le1+p^{-1}U(x)$. Squares are monotone on nonnegative numbers (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and inverses reverse the order of positive numbers ($a\le b$ gives $b^{-1}=a\,a^{-1}b^{-1}\le b\,a^{-1}b^{-1}=a^{-1}$ by claim 5 of Elementary Arithmetic in an Ordered Field). Hence $w_{p}(x)\le w_{p'}(x)$. Moreover $p^{-1}\to0$ (The Archimedean Property of the Real Numbers and Limit of a Sequence of Real Numbers), so $w_{p}(x)=(1+p^{-1}U(x))^{-2}\to1$ by Arithmetic of Limits of Real Sequences. So $w_{p}^{2}\lVert\nabla V\rVert^{2}$ increases pointwise to $\lVert\nabla V\rVert^{2}$, and Monotone Convergence Theorem gives $\int\lVert\nabla V\rVert^{2}\,d\mu=\lim_{p}X_{p}^{2}\le(C+\sqrt{D})^{2}<\infty$ (Order Properties of Limits of Real Sequences).

Step 5 (finite Fisher information). Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. By the hypothesis, and the Cauchy–Schwarz inequality in the real Hilbert space $L^{2}(\mu;\mathbb{R}^{d})$ (The Cauchy-Schwarz Inequality in a Real Inner Product Space),

Dividing by $\kappa>0$ gives the inequality of Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §finite with constant $\kappa^{-1}(C+\lVert\nabla V\rVert_{\mu})$. Since $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Claim 1, $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. $\blacksquare$