Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

Question

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?

Answer 1

For example, let $(A_t)_{t \geq 0} \subset {\rm SO}(n)$ be any continuous family which is eventually constant, and consider the map $T: {\bf R}^n \to {\bf R}^n$ defined by $Tx = A_{\|x\|}(x)$. (I made the family eventually constant to ensure that $T$ is Lipschitz; this can be weakened.)

Answer 2

Consider the one parameter transformation $g_t$ generated by a vector field $X$. If $g_t$ preserves the measure, we get a first order equation on $X$, which I think could have many solutions.

Answer 3

As noted above there are many more mappings which leave the Gaussian invariant. In a certain sense they can all be written down explicitly (when one interprets the latter expression with a litte goodwill---one must allow for operations like integration and taking inverse functions, all of which can, in principal, be done numerically in concrete situations).

We start with some historical remarks which will put the solutions in perspective---the prototype problem is for the case of functions which leave ordinary Lebesgue measure invariant in two dimensions. This is clearly a significant problem with applications to mathematical cartography (description of all equal area map projections) and thermodynamics (the basic physical principles of Gibbsian thermodynamics can be coded in the fact that the natural map from $(p,V)$ space to $(S,T)$ space given by the equations of state is area preserving So it is no surprise that this question was already tackled and solved be Gauß who essentially discovered the concept of generating functions which is so central to symplectic geometry (in two dimensions, of course, the concepts of areaa preserving mappings and symplectic mappings coincide).
A result which me quote in order to motivate the gaussian case is as follows: each area-preserving mapping of the plane which leaves the family of parallels to the $y$-axis invariant (nota bene, the family not the invidual curves) has the form $$u(x)=a(x) , \quad v(x,y)=\frac y {a'(x)} + b(x)$$ for suitable functions $a$ and $b$ of one variable. (Health warning: since we are concerned to show the plethora of possible solutions we are opting for a rather informal presentations---statements such as the above would require a more careful formulation--- which wouldn't be appropriate for this forum---to be precise. In particular, many of the statements are local in nature).

The corrresponding result in the Gaussian case is a bit more complicated. Consider a function of the form $$\phi_{a,b}(x,y) = ((a(x),\frac{e^{- \frac{x^2}2}\text{erf}(y) }{a'(x)}+ b(x)).$$ (Another health warning: I have ignored the finger-breaking constants associated with the Gaussian here).

Then a function $\phi_{A,B}^{-1}\circ \phi_{a,b}$ (where $A$ and $B$ are two further functions of one variable) leaves the Gaussian invariant and preserves the above family of curves (the parallels to the $y$-axis)---indeed this is a characterisation of such functions. One can remove the restriction on the parallels but I will leave that for another day.

The point that I am making is that one has the free choice of functions of one variable in this construction and so a great deal of latitude. In fact, one can show that if one has two suitable families of curves then one can always find an infinite family of mappings which satisfy the conditions of the OP and map the one family into the other one. One can also specify how much further information is required to force uniqueness of the solution. (This version is important for applications---in thermodynamics the families of curves which are the images of the coordinate network under the mappings are the isotherms and adiabats, in cartography, the parallels and meridians).