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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, Hormander's
hypoellipticity theorem, and geometric control theory, we find a critical
parameter value @a"1=@a"2 such that when @a"2>@a"1 the system is ergodic
and when @a"2<@a"1 solutions are not defined for all times.
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