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Propagating lyapunov functions to prove noise-induced stabilization

Publication ,  Journal Article
Athreyaz, A; Kolba, T; Mattingly, JC
Published in: Electronic Journal of Probability
2012

We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.

Duke Scholars

Published In

Electronic Journal of Probability

DOI

ISSN

1083-6489

Publication Date

2012

Volume

17

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 0105 Mathematical Physics
  • 0104 Statistics
 

Citation

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ICMJE
MLA
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Athreyaz, A., Kolba, T., & Mattingly, J. C. (2012). Propagating lyapunov functions to prove noise-induced stabilization. Electronic Journal of Probability, 17. https://doi.org/10.1214/EJP.v17-2410
Athreyaz, A., T. Kolba, and J. C. Mattingly. “Propagating lyapunov functions to prove noise-induced stabilization.” Electronic Journal of Probability 17 (2012). https://doi.org/10.1214/EJP.v17-2410.
Athreyaz A, Kolba T, Mattingly JC. Propagating lyapunov functions to prove noise-induced stabilization. Electronic Journal of Probability. 2012;17.
Athreyaz, A., et al. “Propagating lyapunov functions to prove noise-induced stabilization.” Electronic Journal of Probability, vol. 17, 2012. Scival, doi:10.1214/EJP.v17-2410.
Athreyaz A, Kolba T, Mattingly JC. Propagating lyapunov functions to prove noise-induced stabilization. Electronic Journal of Probability. 2012;17.

Published In

Electronic Journal of Probability

DOI

ISSN

1083-6489

Publication Date

2012

Volume

17

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 0105 Mathematical Physics
  • 0104 Statistics