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 mixing and the noise intensity. The parameter space is partitioned into three regions that correspond to qualitatively different system behaviours also known as subcritical, critical and supercritical states. We obtain a complete description of the subcritical state and, particularly, the limiting behavior of the process when we rescale the time. We develop a new technique involving fractional moments which allows us to describe partially the supercritical state. The critical state is a very difficult one and although there some open questions remain, we have obtained rigorous partial results.