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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
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 4 ()
    Pages: 1519-1539

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1519-1539

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    Related research

    Keywords: Ergodic property; Stochastic differential equations; Degenerate noise; Invariant (probability) measures; Geometric control theory; Lyapunov functions;

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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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