Poisson Process with States Congruent (Mod 2)

Abstract

IN connexion with the examination of stochastic processes in chemistry we have been led to consider a modification of the Poisson process which we feel may be of independent interest. As is well known¹, the simple Poisson process is a Markov process {X(t), t≥0} with denumerable state space E = {0, 1, 2, …}, satisfying the following conditions: For xɛE, (a). The probability of the transition x→x + 1 in the interval (t, t + Δt) is λΔt + o(Δt), where λ is a positive constant; (b) The probability of the transition x→x + n(n > 1) is o(Δt); (c) The probability of the transition x→x (no change of state) is 1 − λΔt + o(Δt). The modification we consider can be expressed as follows: (a′) The probability of the transition x→x + 1 in (t, t + Δt) is λ₁Δt + o(Δt) if x is an even integer, and is λ₂Δt + o(Δt) if x is an odd integer, where λ₁ and λ₂ are positive constants. We call a process satisfying (a′), (b) and (c) a Poisson process with states congruent (mod 2). In view of the foregoing definition a simple Poisson process has states congruent (mod 1).