Transition from ergodic to explosive behavior in a family of stochastic   differential equations
  We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $\alpha_{1}=\alpha_{2}$ such that when $\alpha_{2}>\alpha_{1}$ the system is ergodic and when $\alpha_{2}<\alpha_{1}$ solutions are not defined for all times. H\"{o}rmander's hypoellipticity theorem and geometric control theory are also utilized. 
Jeremiah Birrell, David P. Herzog, Jan Wehr
math.PR math-ph math.DS math.MP