Transition from ergodic to explosive behavior in a family of stochastic differential equations
arXiv:1105.2378 · doi:10.1016/j.spa.2011.12.014
Abstract
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.
35 pages, 6 figures