Proof of The Entropy of the Push-Forward of a Measure by the Gradient of a Twice Continuously Differentiable Function with Pinched Hessian

Each result cited below is universally quantified over the data in its own statement. Write $F=\nabla\Phi$, $j(x)=\det D^{2}\Phi(x)$ and $\ell(x)=\log j(x)$ for $x\in\mathbb{R}^{d}$, let $\lambda_{d}$ be Lebesgue measure, let densities with respect to $\lambda_{d}$ be those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, and let $\phi$ be the function $s\mapsto s\log s$ of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm.

Claim 1 (The map $F$). $F$ satisfies the hypotheses of Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward, with $DF(x)=D^{2}\Phi(x)$ and $\det DF(x)=j(x)>0$ for every $x$; $F$ and $F^{-1}$ are Borel, and $j$ is continuous and Borel.

By The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §bijection, $F$ is a bijection of $\mathbb{R}^{d}$ onto $\mathbb{R}^{d}$; write $F^{-1}$ for its inverse. Since $\Phi$ is of class $C^{2}$, its partial derivatives $\partial_{i}\Phi$, the components of $F$, are of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set. By The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §inverse, for every $x$ the matrix $D^{2}\Phi(x)$ is positive definite and $DF(x)=D^{2}\Phi(x)$, the components of $F^{-1}$ are of class $C^{1}$, and $DF^{-1}(F(x))$ is the inverse matrix of $D^{2}\Phi(x)=DF(x)$. As $D^{2}\Phi(x)\in\mathcal{S}(d)$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives), $DF(x)$ is symmetric and positive definite. Hence $F$ satisfies the hypotheses of Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward, $\det DF(x)=\det D^{2}\Phi(x)=j(x)$, and $0<j(x)$ by Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §positive. The remaining assertions are Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §regularity.

Claim 2 (Second moment; claim 1). By The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §bilipschitz, $\lVert F(x)-F(0_{\mathbb{R}^{d}})\rVert\le d\,L\,\lVert x\rVert$ for every $x$, so $\lVert F(x)-F(0_{\mathbb{R}^{d}})\rVert^{2}\le d^{2}L^{2}\lVert x\rVert^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and the second inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions gives

Since $F$ is Borel (Claim 1), $F_{\#}\mu\in\mathcal{P}(\mathbb{R}^{d})$ and, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward applied to the nonnegative Borel function $y\mapsto\lVert y\rVert^{2}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and by claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

because $\mu(\mathbb{R}^{d})=1$ and $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment, The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). Hence $F_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

Claim 3 (The function $\ell$). $\ell$ is Borel and bounded.

For each $x$ the matrix $D^{2}\Phi(x)$ lies in $\mathcal{S}(d)$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives) and satisfies $\varepsilon I_{d}\preceq D^{2}\Phi(x)\preceq L\,I_{d}$, so Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §pinching gives $d-d\,\varepsilon^{-1}\le\ell(x)\le d\,L-d$. With $B=|d-d\,\varepsilon^{-1}|+|d\,L-d|$, claims 1, 2 and 3 of Properties of the Absolute Value in an Ordered Field give $\ell(x)\le B$ and $-\ell(x)\le B$, hence $|\ell(x)|\le B$ by claim 6 there; so $\ell$ is bounded. The logarithm is smooth on the open set $(0,\infty)$ by The Natural Logarithm, hence continuous relative to $(0,\infty)$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, hence sequentially continuous on $(0,\infty)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, the Euclidean distance of $\mathbb{R}^{1}$ being the absolute-value metric by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §distance. Since $j$ is Borel with values in $(0,\infty)$ (Claim 1), $\ell=\log\circ j$ is Borel by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable.

Claim 4 (A pointwise identity). Let $\rho$ be a density of $\mu$ with respect to $\lambda_{d}$ and let $\tilde{\rho}(y)=\rho(F^{-1}(y))/j(F^{-1}(y))$. Then $\tilde{\rho}$ is a density of $F_{\#}\mu$ with respect to $\lambda_{d}$, and for every $x\in\mathbb{R}^{d}$

The first assertion is Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §densities, applicable by Claim 1, which also gives $\det DF=j$; in particular $\tilde{\rho}$ is Borel and nonnegative. Since $F^{-1}(F(x))=x$, $\tilde{\rho}(F(x))=\rho(x)\,j(x)^{-1}$. If $\rho(x)=0$, both sides vanish, as $\phi(0)=0$. If $0<\rho(x)$, then by the product rule of The Natural Logarithm (with $\log(j(x)^{-1})=-\log j(x)$, since $\log1=\log\exp(0)=0$), $\log\tilde{\rho}(F(x))=\log\rho(x)-\ell(x)$, and multiplying by $\tilde{\rho}(F(x))\,j(x)=\rho(x)$ gives the identity.

Claim 5 (Entropy; claim 2). By Claim 3, $\ell$ is Borel and bounded. Let $\rho$ be a density of $\mu$ with respect to $\lambda_{d}$ with $\phi\circ\rho$ integrable with respect to $\lambda_{d}$ (The Entropy of a Probability Measure on Euclidean Space §entropy). The bounded Borel function $\ell$ is integrable with respect to $\mu$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; since $\mu(A)=\int\mathbf{1}_{A}\rho\,d\lambda_{d}$ for every Borel $A$, $\mu$ is the measure with density $\rho$ with respect to $\lambda_{d}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables, and that claim shows that $\rho\,\ell$ is integrable with respect to $\lambda_{d}$ with $\int\rho\,\ell\,d\lambda_{d}=\int\ell\,d\mu$. By Claim 4 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the function $(\phi\circ\tilde{\rho}\circ F)\,j=\phi\circ\rho-\rho\,\ell$ is integrable with respect to $\lambda_{d}$, with integral $\mathrm{Ent}(\mu)-\int\ell\,d\mu$. The function $\phi\circ\tilde{\rho}$ is Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, so by Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §integrals it is integrable with respect to $\lambda_{d}$ and

Since $\tilde{\rho}$ is a density of $F_{\#}\mu$ with respect to $\lambda_{d}$ (Claim 4), $F_{\#}\mu$ has finite entropy with $\mathrm{Ent}(F_{\#}\mu)$ equal to the left-hand side, by The Entropy of a Probability Measure on Euclidean Space §entropy. Together with Claim 2, $F_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.