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 \cite{Kol1997} 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.

翻译：在知道相关多变量空间等级为$R_P美元的情况下,我们解决了回收概率大于$美元(例如,如果存在的话,其密度为$_P美元)的问题。在\cite{Kol1997}中显示,多变量空间等级是概率测量的特征。我们通过以(可能的分数)部分差价方方(ff_P=\LLL_n(R_P))的形式,以(可能的分数)部分差价方(f_P=美元)的形式,以部分差价方(例如,如果有的话,其密度为美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元/美元