Proof of The First Variation of the Logarithmic Energy on the Real Line along Gradient Perturbations of the Identity

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 and scaling inequalities of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field, and the elementary properties of the absolute value in Properties of the Absolute Value in an Ordered Field (claim 4, multiplicativity; claim 5, the triangle inequality; claim 9, the strict two-sided bound), are used without further mention.

Data. Fix $\mu\in\mathcal{D}_{\log}$ and $\psi\in C_{c}^{\infty}(\mathbb{R})$. By One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §quotient fix a real number $L\ge0$ with $|\Delta\psi(t)|\le L$ for every $t\in\mathbb{R}$; by the same clause $F_{\psi}$ is Borel and $\mu\boxtimes\mu$-integrable, and $|F_{\psi}(x,y)|\le L$ for all $x,y\in\mathbb{R}$. Put

a positive real number, and let $U=(-t_{0},t_{0})$; by An Open Interval is an Interval All of Whose Points Are Interior it is an interval every point of which is an interior point. For $s\in\mathbb{R}$ let $G_{s}=\mathrm{id}+s\psi'$, so that $G_{s}(x)=x+s\psi'(x)$; since $\psi'=\nabla\psi$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives, 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 (with $d=1$) shows that $G_{s}$ is Borel, that $\nu_{s}=(G_{s})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R})$, and that $G_{0}=\mathrm{id}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections every $z\in\mathbb{R}^{2}$ equals $\iota(x,y)$ with $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, and we write $\ell(x,y)$, $F_{\psi}(x,y)$ for the values at this point, as in the statements; $\ell$ is the kernel of The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure and $\Delta\subseteq\mathbb{R}^{2}$ is its diagonal.

Step 1 (a factorisation). Let $s\in U$ and $x,y\in\mathbb{R}$, and put $a_{s}(x,y)=1+sF_{\psi}(x,y)$. Since $|s|<t_{0}$ and $0\le L$, we have $|sF_{\psi}(x,y)|=|s|\,|F_{\psi}(x,y)|\le t_{0}L<\tfrac12$, the last inequality because $2L<2L+2$. Hence

in particular $a_{s}(x,y)$ is positive. Now let $x\ne y$. By the definition of $F_{\psi}$, $\psi'(x)-\psi'(y)=(x-y)F_{\psi}(x,y)$, hence

As $x-y\ne0$ and $a_{s}(x,y)\ne0$, the product is nonzero, so $G_{s}(x)\ne G_{s}(y)$: the map $G_{s}$ is injective. Moreover $|G_{s}(x)-G_{s}(y)|=|x-y|\,a_{s}(x,y)$, a product of two positive numbers, so by the definition of $\ell$ and the identity $\log(uv)=\log u+\log v$ for $u,v>0$ of The Natural Logarithm,

while $\ell\bigl(G_{s}(x),G_{s}(x)\bigr)=0$ by the definition of $\ell$ on the diagonal. $\quad(4)$

Step 2 (integrals against the product of two push-forwards). Let $T:\mathbb{R}\to\mathbb{R}$ be Borel, let $\nu=T_{\#}\mu\in\mathcal{P}(\mathbb{R})$ as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, and let $T^{(2)}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the pairing $z\mapsto\iota\bigl(T(\mathrm{pr}_{1}(z)),T(\mathrm{pr}_{2}(z))\bigr)$ of the Borel maps $T\circ\mathrm{pr}_{1}$ and $T\circ\mathrm{pr}_{2}$ (compositions of Borel maps, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); it is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $T^{(2)}(\iota(x,y))=\iota(T(x),T(y))$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. We claim: (a) for every Borel $\Phi:\mathbb{R}^{2}\to[0,\infty]$,

(b) a Borel $\Phi:\mathbb{R}^{2}\to\mathbb{R}$ is $\nu\boxtimes\nu$-integrable if and only if $\Phi\circ T^{(2)}$ is $\mu\boxtimes\mu$-integrable, and then the two integrals in (a) are equal real numbers.

For (a): by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, $\Phi\circ\iota$ is measurable with respect to $\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})$ and $\int_{\mathbb{R}^{2}}\Phi\,d(\nu\boxtimes\nu)=\int_{\mathbb{R}\times\mathbb{R}}\Phi\circ\iota\,d(\nu\otimes\nu)$. By the Tonelli part of Tonelli and Fubini Theorems (probability measures are $\sigma$-finite), for every $x$ the section $y\mapsto\Phi(\iota(x,y))$ is Borel, the function $H(x)=\int_{\mathbb{R}}\Phi(\iota(x,y))\,\nu(dy)$ is Borel from $\mathbb{R}$ to $[0,\infty]$, and $\int\Phi\circ\iota\,d(\nu\otimes\nu)=\int_{\mathbb{R}}H\,d\nu$. The change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied first to each section and then to $H$, gives $H(x)=\int_{\mathbb{R}}\Phi(\iota(x,T(y)))\,\mu(dy)$ for every $x$ and $\int_{\mathbb{R}}H\,d\nu=\int_{\mathbb{R}}H\circ T\,d\mu$; hence

The map $\Phi\circ T^{(2)}:\mathbb{R}^{2}\to[0,\infty]$ is Borel (a composition of Borel maps) and $(\Phi\circ T^{(2)})(\iota(x,y))=\Phi(\iota(T(x),T(y)))$; so the same two results, 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, now for $\mu\boxtimes\mu$ and $\mu\otimes\mu$, show that $\int_{\mathbb{R}^{2}}\Phi\circ T^{(2)}\,d(\mu\boxtimes\mu)$ equals the same iterated integral. This proves (a). For (b): $\Phi\circ T^{(2)}$ is Borel, and its positive and negative parts, in the sense of Integrable Function and the Lebesgue Integral, are $\Phi^{+}\circ T^{(2)}$ and $\Phi^{-}\circ T^{(2)}$. By (a) applied to $\Phi^{+}$ and to $\Phi^{-}$, the integrals of $\Phi^{\pm}$ against $\nu\boxtimes\nu$ equal those of $\Phi^{\pm}\circ T^{(2)}$ against $\mu\boxtimes\mu$; so both pairs are finite together, and then the integrals, being the differences of these, agree, by Integrable Function and the Lebesgue Integral.

Step 3 (the integrand and its derivative in $s$). For $s\in U$ let $G_{s}^{(2)}$ be the map $T^{(2)}$ of Step 2 for $T=G_{s}$, and define $f:U\times\mathbb{R}^{2}\to\mathbb{R}$ by $f(s,z)=\ell\bigl(G_{s}^{(2)}(z)\bigr)$. Fix $z=\iota(x,y)\in\mathbb{R}^{2}$, so that $f(s,z)=\ell(G_{s}(x),G_{s}(y))$.

If $x=y$, then $f(s,z)=0$ for every $s\in U$ by (4), so $s\mapsto f(s,z)$ is constant and, by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, differentiable at every point of $U$ with derivative $D_{1}f(s,z)=0$.

If $x\ne y$, put $c=F_{\psi}(x,y)$ and $\gamma(s)=1+sc$ for $s\in U$. The identity map of $U$ is differentiable at every point with derivative $1$, directly from Derivative at an Interior Point, since all its difference quotients equal $1$; hence, by claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\gamma$ is differentiable at every $s\in U$ with $\gamma'(s)=c$. By (1), $\gamma(s)=a_{s}(x,y)$ lies in the interval $(0,\infty)$, and it is an interior point of it in the sense of Interior Point of an Interval, since $\tfrac12\gamma(s)<\gamma(s)<\gamma(s)+1$ with both outer points in $(0,\infty)$; by The Natural Logarithm, $\log$ is differentiable there with derivative $1/\gamma(s)$. By the chain rule Chain Rule for One-Dimensional Derivatives, $s\mapsto\log\gamma(s)$ is differentiable at every $s\in U$ with derivative $c/\gamma(s)$. By (3), $f(s,z)=\ell(x,y)-\log\gamma(s)$ for $s\in U$, so by claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives

By (1), $2a_{s}(x,y)>1$, and multiplying by the positive number $1/a_{s}(x,y)$ gives $1/a_{s}(x,y)<2$; with $|F_{\psi}|\le L$ this yields $|D_{1}f(s,z)|\le2L$. Together with the diagonal case:

Step 4 ($\nu_{s}\in\mathcal{D}_{\log}$ and its energy). Fix $s\in U$ and $z\in\mathbb{R}^{2}$. As $G_{0}=\mathrm{id}$, $G_{0}^{(2)}(z)=\iota(\mathrm{pr}_{1}(z),\mathrm{pr}_{2}(z))=z$, so $f(0,z)=\ell(z)$. If $s\ne0$, let $a<b$ be the numbers $0$ and $s$ in increasing order; by the mean value theorem Mean Value Theorem on an Open Interval, applied to $s'\mapsto f(s',z)$ on the open interval $(-t_{0},t_{0})$ (differentiable at every point by Step 3), there is $c\in(a,b)$ with $f(s,z)-f(0,z)=D_{1}f(c,z)\,s$, whence by (5) $|f(s,z)-\ell(z)|\le2L|s|\le2Lt_{0}<1$. Thus, for $s=0$ trivially and for $s\ne0$ by the triangle inequality,

The map $f(s,\cdot)=\ell\circ G_{s}^{(2)}$ is Borel, $\ell$ being Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel. As $\mu\in\mathcal{D}_{\log}$, $\ell$ is $\mu\boxtimes\mu$-integrable (The Logarithmic Energy of a Probability Measure on the Real Line §energy); by (6) and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, together with $\int_{\mathbb{R}^{2}}1\,d(\mu\boxtimes\mu)=1$ (claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space),

so $f(s,\cdot)$ is $\mu\boxtimes\mu$-integrable by Integrable Function and the Lebesgue Integral. By Step 2(b) with $T=G_{s}$ and $\Phi=\ell$, $\ell$ is $\nu_{s}\boxtimes\nu_{s}$-integrable and

Next, let $a\in\mathbb{R}$. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, $\nu_{s}(\{a\})=\mu(G_{s}^{-1}(\{a\}))$. Since $G_{s}$ is injective (Step 1), $G_{s}^{-1}(\{a\})$ is either empty or a one-point set $\{x\}$; in the first case its measure is $0$, and in the second $\mu(\{x\})=0$ because $\mu\in\mathcal{D}_{\log}$. So $\nu_{s}(\{a\})=0$ for every $a$. As $\nu_{s}\in\mathcal{P}_{2}(\mathbb{R})$, The Logarithmic Energy of a Probability Measure on the Real Line §energy gives $\nu_{s}\in\mathcal{D}_{\log}$, with, by (7),

This proves the first assertion of clause 1 with the positive number $t_{0}$.

Step 5 (differentiation at $0$). Apply Differentiation under the Integral Sign to the measure space $(\mathbb{R}^{2},\mathcal{B}(\mathbb{R}^{2}),\mu\boxtimes\mu)$, the open interval $U=(-t_{0},t_{0})$ and the function $f$ of Step 3. Its condition (i) holds by Step 4, condition (ii) by Step 3, and condition (iii) with the constant function $2L$, which is $\mu\boxtimes\mu$-integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space (as $\mu\boxtimes\mu$ is a Borel measure of total mass $1$, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), by (5). Hence $z\mapsto D_{1}f(0,z)$ is measurable and integrable, and the function $E:U\to\mathbb{R}$, $E(s)=\int_{\mathbb{R}^{2}}f(s,\cdot)\,d(\mu\boxtimes\mu)$, is differentiable at $0$ with

By (8), $E$ is the function $t\mapsto\mathcal{E}_{\log}((\mathrm{id}+t\psi')_{\#}\mu)$ of clause 1. Since $\mu(\{x\})=0$ for every $x$, the diagonal $\Delta$, a Borel set by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel, satisfies $(\mu\boxtimes\mu)(\Delta)=0$ by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §diagonal. By (5), $D_{1}f(0,\cdot)$ and $-F_{\psi}$ agree off $\Delta$, hence almost everywhere; $-F_{\psi}$ is integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral; so by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

This proves clause 1.

Clause 2. If moreover $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, the free score $\Xi_{\mu}$ satisfies $\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}=\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)$ by Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line §score, so the derivative of clause 1 equals $-\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}$.