Moving the Rescaled Head of the Diagonal Gaussian Measure by a Diffeomorphism: the Image Has an Explicit Positive Density Depending on the Head Only

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $n\in\mathbb{N}$, with $\tilde{c}^{(n)}$, $\tilde{\gamma}_{n}$, the density $\rho_{\tilde{c}^{(n)}}$ and the weighted square $|u|^{2}_{\tilde{c}^{(n)}}$ of $u\in\mathbb{R}^{n}$ as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads. Densities are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, $\det$ is the determinant and $\log$ the natural logarithm. Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ satisfy 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 $d=n$: $F$ is a bijection with inverse $F^{-1}$, the components of $F$ and of $F^{-1}$ are of class $C^{1}$ on $\mathbb{R}^{n}$, and for every $u\in\mathbb{R}^{n}$ the Jacobian matrix $DF(u)$ is symmetric and positive definite and $D(F^{-1})(F(u))$ is its inverse matrix; then $0<\det DF(u)$ for every $u$, by that theorem. For such $F$, the map $F$ is continuous 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 §regularity, hence so is the map $u\mapsto F(u)-u$, which is therefore Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; we write $T_{F}:X\to X$ for the map

$$T_{F}(x)=x+\Lambda_{n}(F-\mathrm{id})(x)\qquad(x\in X),$$

with the lift $\Lambda_{n}$ of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads, and $G_{F}:\mathbb{R}^{n}\to\mathbb{R}$ for the function

$$G_{F}(v)=\frac{\rho_{\tilde{c}^{(n)}}\bigl(F^{-1}(v)\bigr)}{\rho_{\tilde{c}^{(n)}}(v)\,\det DF\bigl(F^{-1}(v)\bigr)}\qquad(v\in\mathbb{R}^{n}).$$

1. (A bijection moving the head only)¶ The map $F^{-1}$ satisfies the hypotheses imposed on $F$, so that $T_{F^{-1}}$ is defined. $T_{F}$ and $T_{F^{-1}}$ are continuous, hence Borel, and $T_{F^{-1}}$ is the inverse of $T_{F}$. For every $x\in X$, $r_{n}(T_{F}(x))=F(r_{n}(x))$ and $Q_{n}T_{F}(x)=Q_{n}x$.

2. (Density of the image)¶ $G_{F}$ is Borel and $0<G_{F}(v)$ for every $v\in\mathbb{R}^{n}$, and $G_{F}\circ r_{n}$ is a density of $(T_{F})_{\#}\gamma_{c}$ with respect to $\gamma_{c}$.

3. (The logarithm of the density along the map)¶ For every $u\in\mathbb{R}^{n}$,

$$\log G_{F}(F(u))=\tfrac12\,|F(u)|^{2}_{\tilde{c}^{(n)}}-\tfrac12\,|u|^{2}_{\tilde{c}^{(n)}}-\log\det DF(u).$$