  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.
