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

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  • Birrell, Jeremiah
  • Herzog, David P.
  • Wehr, Jan

Abstract

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 α1=α2 such that when α2>α1 the system is ergodic and when α2<α1 solutions are not defined for all times.

Suggested Citation

  • Birrell, Jeremiah & Herzog, David P. & Wehr, Jan, 2012. "The transition from ergodic to explosive behavior in a family of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1519-1539.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1519-1539
    DOI: 10.1016/j.spa.2011.12.014
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    References listed on IDEAS

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    1. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
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    1. Carfagnini, Marco & Földes, Juraj & Herzog, David P., 2022. "A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 188-223.

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