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

Jeremiah Birrell, David P. Herzog, Jan Wehr

Research output: Contribution to journalArticle

3 Scopus citations


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, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 12 such that when α 21 the system is ergodic and when α 21 solutions are not defined for all times.

Original languageEnglish (US)
Pages (from-to)1519-1539
Number of pages21
JournalStochastic Processes and their Applications
Issue number4
StatePublished - Apr 1 2012



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

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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