  Characterizing the occupation statistics of a radiation flow through confined
geometries is key to such technological issues as nuclear reactor design and
medical diagnosis. This amounts to assessing the distribution of the travelled
length $\ell$ and the number of collisions $n$ performed by the underlying
stochastic transport process, for which remarkably simple Cauchy-like formulas
were established in the case of branching Pearson random walks with
exponentially distributed jumps. In this Letter, we show that such formulas
strikingly carry over to the much broader class of branching processes with
arbitrary jumps, provided that scattering is isotropic and the average jump
size is finite.
