Collision densities and mean residence times for d-dimensional exponential flights

In this paper we analyze some aspects of exponential flights, a stochastic process that governs the evolution of many random transport phenomena, such as neutron propagation, chemical or biological species migration, and electron motion. We introduce a general framework for d-dimensional setups and emphasize that exponential flights represent a deceivingly simple system, where in most cases closed-form formulas can hardly be obtained. We derive a number of exact (where possible) or asymptotic results, among which are the stationary probability density for two-dimensional systems, a long-standing issue in physics, and the mean residence time in a given volume. Bounded or unbounded domains as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings.