MULTILEVEL BILINEAR-SYSTEMS OF STOCHASTIC DIFFERENTIAL-EQUATIONS

A multilevel bilinear system of stochastic differential equations is a multilevel mean field system in which the drift term is also linear. Two kinds of parameters coexist in this model: the rate of spatial mix ing and the noise intensity. The parameter space is partitioned into t hree regions that correspond to qualitatively different system behavio urs also known as subcritical, critical and supercritical states. We o btain a complete description of the subcritical state and, particularl y, the limiting behavior of the process when we rescale the time. We d evelop a new technique involving fractional moments which allows us to describe partially the supercritical state. The critical state is a v ery difficult one and although there some open questions remain, we ha ve obtained rigorous partial results.