Noncommutative cumulants for stochastic differential equations and for generalized Dyson series

Abstract

For the stochastic equationU=VU, Kubo's ansalze for 〈U〉 in the form of differential and integrodifferential equations is investigated and a newansatz as an integral equation is added. Unique solutions in terms of noncommutative W- and K-cumulants are found by elementary functional differentiation, and expressions of van Kämpen and Terwiel are recovered. For the cumulants we find simple recursion relations and prove the important cluster property. Surprisingly, it is found that the Gaussian approximation in the differential equationansatz leads to positivity problems, while this is not the case with the integral and integrodifferential equation. The cumulant expansion technique is carried over to generalized Dyson series. In a companion paper we apply our results to quantum shot noise.