The First Variation of the Entropy Along a Gradient Perturbation of the Identity

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ as in that definition, and with the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, their gradient maps $\nabla\psi$ and Laplacians $\Delta\psi$ as fixed there. For a test function $\psi$, $\partial_{j}\partial_{i}\psi$ are its second partial derivatives and $D^{2}\psi(x)\in\mathcal{S}(d)$ its Hessian matrix at $x$, as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives; for $t\in\mathbb{R}$, $\mathrm{id}+t\nabla\psi$ is the map $x\mapsto x+t\,\nabla\psi(x)$. The determinant is written $\det$, the natural logarithm $\log$, and $I_{d}$ is the identity matrix; $c_{d}$ and $K_{d}$ are the constants of Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §expansion, and the natural number $d$ is read in $\mathbb{R}$ where a real number is required. Differentiability at $0$ of a real function on an open interval $(-t_{0},t_{0})$ is that of Derivative at an Interior Point. Let $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, and let $b$ be a nonnegative real number with $|\partial_{j}\partial_{i}\psi(x)|\le b$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$.

1. (Push-forward)¶ Let $t\in\mathbb{R}$ satisfy $2d\,|t|\,b\le1$. Then $0<\det(I_{d}+tD^{2}\psi(x))$ for every $x\in\mathbb{R}^{d}$, the map $\mathrm{id}+t\nabla\psi$ is Borel, the function $x\mapsto\log\det(I_{d}+tD^{2}\psi(x))$ is Borel and bounded, $(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, and

2. (Expansion)¶ Let $t\in\mathbb{R}$ satisfy $2d\,|t|\,b\le1$ and $|t|\,b\le c_{d}$. Then

3. (First variation)¶ There is a positive real number $t_{0}$ such that $(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ for every $t\in(-t_{0},t_{0})$, and the function

is differentiable at $0$ with derivative $-\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu$.