Discontinuous Markoff processes

Abstract

The present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as point x in an abstract space; phases are points (x, t) in the product state×time space; sets of states are denoted by X, sets of phases by S. It is shown in § 2 that such a process is specified by two functions:the probability χ0 (X, t|x0, t0) of a transition x0→X without “jumps” in the time interval [t0, t), and the probability distribution Ψ (S|x0, t0) of the first jump time and the consequent state, given an initial phase (x0, t0). The total transition probability χ (X, t|x0, t0) is required to satisfy the integral equation $\chi \left( {\left. {X,t} \right|x_0 ,t_0 } \right) = \chi _0 \left( {\left. {X,t} \right|x_0 ,t_0 } \right) + \int {\chi \left( {\left. {X,t} \right|\xi ,\tau } \right)} \psi \left( {\left. {d\xi ,d\tau } \right|x_0 ,t_0 } \right).$ The main problem is to study the existence and uniqueness of the solutions of I.E. which also satisfy the conditions (stated in § 1) for being transition probabilities of a Markoff process. Previous work (cf. § 4) on this subject relates to special cases, mainly to processes where transitions occur only by jumps. In § 5, two auxiliary sets of functions are introduced: the distributions ψn(|x0, t0) of the nth jump time and consequent state (which form a Markoff chain), and the transition probabilities χ0 (X, t|x0, t0)