Two-point correlation properties of stochastic splitting processes

We study how the two-point density correlation properties of a point particle distribution are modified when each particle is divided, by a stochastic process, into an equal number of identical 'daughter' particles. We consider generically that there may be nontrivial correlations in the displacement fields describing the positions of the different daughters of the same 'mother' particle and then treat separately the cases in which there are, or are not, correlations also between the displacements of daughters belonging to different mothers. For both cases exact formulas are derived relating the structure factor (power spectrum) of the daughter distribution to that of the mothers. An application of these results is that they give explicit algorithms for generating, starting from regular lattice arrays, stochastic particle distributions with an arbitrarily high degree of large-scale uniformity. Such distributions are of interest, in particular, in the context of studies of self-gravitating systems in cosmology.