Precisely quantifying the flow of radiation such as neutrons or photons through a structural material or a living body represents a long-standing problem in statistical physics. A fundamental question concerns the occupation statistics of the transported particles within the body when entering from the outer surface, ie the distribution of the travelled length l and the number of collisions n performed by the stochastic process in the body.

Stochastic radiation transport is modeled by Pearson random walks, which can be coupled to a birth-death mechanism (neutron multiplication in fissile materials for instance): a random number of particles emerge from a collision, which leads to branching particle trajectories. In particular, branching Pearson random walks with exponentially distributed lengths stem from assuming that the traversed medium is homogeneous. In this case, the Markovian nature of the transport process leads to remarkably simple Cauchy-like formulas, relating the surface to the volume averages of l and n.

In many important applications of transport theory, the hypothesis of uncorrelated scattering centers is however deemed to fail, which
thus calls for models based on non-exponential random walks. During this talk, we will see how such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, and have thus a universal character.

Our results are key to such technological issues as the analysis of radiation flow for nuclear reactor design and medical diagnosis and apply more broadly to physical and biological systems with diffusion, reproduction and death. The proposed formalism may also apply to animal search strategies in the presence of non-exponential displacements.