Recovering a probability measure from its multivariate spatial rank

We address the problem of recovering a probability measure P over $\R^n$ (e.g. its density $f_P$ if one exists) knowing the associated multivariate spatial rank $R_P$ only. It has been shown in Koltchinski (1997) that multivariate spatial ranks characterize probability measures. We strengthen this result by explictly recovering $f_P$ by $R_P$ in the form of a (potentially fractional) partial differential equation $f_P = \LL_n (R_P)$, where $\LL_n$ is a differential operator given in closed form that depends on $n$. When $P$ admits no density, we further show that the equality $P=\LL_n (R_P)$ still holds in the sense of distributions (i.e. generalized functions). We throughly investigate the regularity properties of spatial ranks and use the PDE we established to give qualitative results on depths contours and regions. %We illustrate the relation between $f_P$ and $R_P$ on a few examples in dimension $2$ and $3$. We study the local properties of the operator $\LL_n$ and show that it is non-local when $n$ is even. We conclude the paper with a partial counterpart to the non-localizability in even dimensions.