The transition from ergodic to explosive behavior in a family of stochastic differential equations

We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hörmander’s hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 1 = α 2 such that when α 2 > α 1 the system is ergodic and when α 2 < α 1 solutions are not defined for all times.