Initial correlations of the multiplicative process, driven by random jumps

Abstract

The authors consider a vector-valued stochastic process, which is multiplicatively driven by the Markov jump process. They obtain a closed expression for the average of the vector process by solving the Burshtein equation for the marginal average. It is shown that the solution for t>t0 requires knowledge of an initial correlation operator, due to the finite correlation time of the jump process. They derive the equation for the steady-state solution, which is applied to evaluate the stationary correlation functions. Then some specific limits of the jump process, are discussed, which arise if one takes the correlation time to be zero or infinite. For the special case L(x)=A+xB they derive from the Burshtein equation a recurrence relation between the moments of the vector process. This relation is solved and it is shown how all initial moments at t0 determine the moments for t>or=t0. The recurrence relation is applied to solve the two-state and the three-state process more explicitly. It is pointed out that the occurring initial correlations cannot be neglected in general.