A General Theory of First-Passage Distributions in Transport and Multiplicative Processes

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Abstract
The ``Milne problem,'' expressed in probabilistic terms, is solved for general transport and multiplicative processes. If a particle initially in a given state at a given position inside a surface τ is multiply scattered while traveling through a fixed medium, then given the scattering cross sections and, if required, the probability distribution for a change of state between collisions (e.f., by diffusion or ionization), the problem is to obtain the probability that the particle eventually effects a first passage through a specified position on the surface τ and in a specified state. In the case of a multiplicative process, the problem is, given in addition the rates of creation and annihilation of particles (considering the nature of the particle as a state variable), to obtain the probability that eventually n particles will emerge for the first time through specified positions on τ and in specified states (with n = 0, 1, 2, …). A general solution is given in the form of a convergent series whose terms are obtained by iteration; this solution is unique if and only if the probability θ∞ of an infinity of atomic events before a first passage (which is the limit of a certain nonincreasing sequence) is identically zero; in the multiplicative case θ∞ ≢ 0 may be taken to mean that the process is ``supercritical.'' The mathematical theory which leads to this solution is a generalization of the corresponding theory for time‐dependent Markov processes in which the time variable is replaced by a set of surfaces ordered by inclusion of their ``insides'' and is valid for Euclidean space of any number of dimensions. Applying it to the 4‐dimensional space of special relativity with ordered sets of spacelike surfaces, one obtains a Lorentz‐invariant formulation of the theory of physical Markov processes. A few examples are given.

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