Fourth-Order Differential Equation for a Two-Stage Absorbing Markov Chain with a Stochastic Forward Transition Probability

Abstract

The problem of averaging the kinetics of a two-stage absorbing Markov chain over random fluctuations in its forward transition probability approximated by the symmetric dichotomous stochastic process is solved exactly. It is shown that the temporal behavior of the population of chain’s transient state obeys a fourth-order differential equation with the tetra-exponential form of a solution given the finite frequency and mean amplitude of fluctuations. In the limit of frequent fluctuations, this tetra-exponential solution reduces to a simple bi-exponential form typical of the deterministic two-stage decay process lacking fluctuations in its transition probability. Rather, in the limit of rare fluctuations, the tetra-exponential solution, while simplifying to the tri- and bi-exponential solutions, becomes specific both for the low amplitude and the resonance amplitude fluctuations, respectively. Furthermore, there is a stochastic resonance point, where the forward transition probability is in resonance with the mean fluctuation amplitude, whereas the backward transition probability, decay transition probability, and fluctuation frequency are negligibly small. In result, the stochastic immobilization of the two-stage absorbing Markov chain in its initial state occurs at this point.