Electronic Journal of Probability

Noise-induced stabilization of planar flows II

David Herzog and Jonathan Mattingly

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Abstract

We continue the work started in Part I of this article, showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir's  generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 113, 37 pp.

Dates
Accepted: 25 October 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067219

Digital Object Identifier
doi:10.1214/EJP.v20-4048

Mathematical Reviews number (MathSciNet)
MR3418545

Zentralblatt MATH identifier
1360.37012

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 37B25: Lyapunov functions and stability; attractors, repellers

Keywords
Noise-induced stabilization Lyapunov functions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Herzog, David; Mattingly, Jonathan. Noise-induced stabilization of planar flows II. Electron. J. Probab. 20 (2015), paper no. 113, 37 pp. doi:10.1214/EJP.v20-4048. https://projecteuclid.org/euclid.ejp/1465067219


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References

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  • Herzog, David P. Geometry's fundamental role in the stability of stochastic differential equations. Thesis (Ph.D.) - The University of Arizona. ProQuest LLC, Ann Arbor, MI, 2011. 108 pp. ISBN: 978-1124-60857-0.
  • David P. Herzog and Jonathan C. Mattingly. Noise–induced stabilization of planar flows I. Submitted, 2013.
  • Peskir, Goran. A change-of-variable formula with local time on surfaces. Seminaire de Probabilites XL, 69–96, Lecture Notes in Math., 1899, Springer, Berlin, 2007.
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See also

  • David Herzog, Jonathan Mattingly. Noise-induced stabilization of planar flows I.