%0 Journal Article
%T Transition from ergodic to explosive behavior in a family of stochastic differential equations
%A Jeremiah Birrell
%A David P. Herzog
%A Jan Wehr
%J Mathematics
%D 2011
%I arXiv
%R 10.1016/j.spa.2011.12.014
%X 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.
%U http://arxiv.org/abs/1105.2378v1