The mean-square dichotomy spectrum and a bifurcation to a mean-square   attractor
  The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor. 
Thai Son Doan, Martin Rasmussen, Peter E. Kloeden
math.DS