PDE characterization of geometric distribution functions and quantiles

We show that, in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric (or spatial) distribution function. The reconstruction takes the form of a (potentially fractional) linear PDE, where the differential operator is given in closed form. This result implies that, contrary to a common belief in the statistical depth community, geometric cdfs in principle provide exact control over the probability content of all depth regions. We present a comprehensive study of the regularity of the geometric distribution function, and show that, contrary to the univariate case, a continuous density in general does not give rise to a geometric cdf with enough regularity to reconstruct the density pointwise. Surprisingly, we prove that the reconstruction displays different behaviours in odd and even dimension: it is local in odd dimension and completely nonlocal in even dimension. We investigate this issue and provide a partial counterpart for even dimensions, and establish a general representation formula of the geometric cdf of spherically symmetric probability laws in odd dimensions. We finally provide explicit density reconstructions from their corresponding geometric distribution function in dimension 2 and 3.