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 $α_{1}=α_{2}$ such that when $α_{2}>α_{1}$ the system is ergodic and when $α_{2}<α_{1}$ solutions are not defined for all times. Hörmander's hypoellipticity theorem and geometric control theory are also utilized.

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