Existence of non-volume preserving automorphisms of normal distribution

Question

Let $X=\mathbb{R}^d$, let $p$ the standard normal distribution on $X$, with zero mean and identity covariance, and let $f:X \to X$ be a diffeomorphism that preserves this normal distribution. For simplicity, let the Jacobian $df$ of $f$ have positive determinant. Then we have that:

$$\log p(f(x))-\log p(x)=- ||f(x)||^2+||x||^2 = -\log \det d f(x) $$

It's easy to construct volume-preserving diffeomorphisms $f$, with $\det df=1$. These preserve the distance to the origin and have for each radius $r > 0$ a uniform measure preserving diffeomorphism $h_r:S^{d-1} \to S^{d-1}$. There diffeomorphisms $h_r$ can be orthogonal transformations or be the flow of a divergence free vector field on the sphere.

Now my question is: do there also exist diffeomorphisms $f$ that preserve the standard normal distribution, but that do not preserve the distance to the origin, so that they are not volume preserving?

What I tried so far (please let me know it I made mistakes here):

• in the case $X=\mathbb{R}^1$, then only the identity is a positive normal distribution preserving diffeomorphism (still assuming positive $\det df$).
• we have that $\det df(0) \ge 1$ and $\det df(f^{-1}(0)) \le 1$, so either $f(0)=0$ with $\det df(0) = 1$, or there is a set at which $\det df=1$ which separate $0$ and $f^{-1}(0)$.
• if $f$ preserves the origin then there can not exist an open ball $U$ around the origin in which $\det df < 1$ everywhere but at the origin, because then $||f(x)||^2 < ||x||^2 \implies f(U) \subsetneq U \implies p(f(U)) < p(U)$, contradicting the preservation of the standard normal distribution.
• any standard normal preserving $f$ that factorizes into a radial diffeomorphism $g: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ and for each radius $r > 0$ a diffeomorphism $h_r:S^{d-1} \to S^{d-1}$ is volume preserving, because $h_r$ should preserve the uniform measure and $g$ should preserve the marginal distribution over the radius, implying it is the identity.
• if $f$ has a triangular Jacobian, so that $f(x)_i$ is constant in $x_j$ for all $j > i$, then you can solve these recursively and find that $f(x)=x$.

Thanks!

Answer 1

This is probably not the most direct route, but it is one which I think is geometrically illustrative: Since the normal distribution can be pushed forward to a uniform distribution on the unit ball, the transformations you're interested in correspond to the volume-preserving maps on the unit ball.

Let $\nu$ be the standard normal distribution in $\mathbb{R}^n$. Consider a "radial" transformation $\varphi(\vec{x})=\rho(\|\vec{x}\|)\vec{x}/\|\vec{x}\|$ where $\rho:[0,\infty)\to[0,\infty)$ is a strictly increasing differentiable function with $\rho(0)=0$. For a suitable choice of $\rho$ we, can ensure that the pushforward of the normal distribution $\varphi_*\nu$ is the uniform distribution on the unit ball $\mathbb{B}^n$. To compute the appropriate form of $\rho$, we can consider the spherical shell $U=\{\vec{x}\in\mathbb{R}^n:r<\|\vec{x}\|<r+dr\}$ and note that $\nu(U)$ must be proportional to the volume of $\varphi(U)$: $$ r^{n-1}e^{-r^2/2}dr\propto\rho^{n-1}(r)d\rho(r) \\ $$ With some computation, this yields $$ \rho(r)\propto\left(\int_0^rt^{n-1}e^{-t^2/2}dt\right)^{1/n} $$ Finding the appropriate choice of constant so that $\lim_{r\to\infty}\rho(r)=1$, we have $$ \rho(r)=\left(\frac{2^{1-n/2}}{\Gamma(n/2)}\int_0^rt^{n-1}e^{-t^2/2}dt\right)^{1/n} $$ One can show that for this $\rho$, $\varphi:\mathbb{R}^n\to\mathbb{B}^n$ is a diffeomorphism, and $\varphi_*\nu$ is uniform.

Thus, given any volume preserving map $f:\mathbb{B}^n\to\mathbb{B}^n$, $\varphi^{-1}\circ f\circ\varphi$ is a $\nu$-preserving automorphism of $\mathbb{R}^n$. In dimension $n>1$ there are many such automorphisms (think, for instance, of the ways in which water can flow in a spherical container). As a simple example in $n=2$, consider any smooth function $H:\overline{\mathbb{B}^2}\to\mathbb{R}$ which vanishes on the boundary. The flow of the corresponding Hamiltonian vector field $V_H=(-\partial_yH,\partial_x H)$ is a volume preserving diffeomorphism of $\mathbb{B}^n$.