Minimal Noise-Induced Stabilization of One-Dimensional Diffusions

  • Tony Allen West Virginia University
  • Emily Gebhardt
  • Adam Kluball
  • Tiffany Kolba Valparaiso University


The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. In this paper, we investigate under what conditions one-dimensional, autonomous stochastic differential equations are stable, where we take the notion of stability to be that of global stochastic boundedness. Specifically, we find the minimum amount of noise necessary for noise-induced stabilization to occur when the drift and noise coefficients are power, polynomial, exponential, or logarithmic functions.


Richard Durrett. Stochastic calculus. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. A practical introduction.

Samuel Karlin and Howard M. Taylor. A second course in stochastic processes. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.

Rafail Khasminskii. Stochastic stability of differential equations, volume 66 of Stochastic Modelling and Applied Probability. Springer, Heidelberg, second edition, 2012. With contributions by G. N. Milstein and M. B. Nevelson.

Bernt Øksendal. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, sixth edition, 2003. An introduction with applications.

M. Scheutzow. Noise-induced transitions for one-dimensional diffusions. Stochastic Anal. Appl., 14(5):535–563, 1996.

James Stewart. Calculus: Early Transcendentals. Thompson Brooks/Cole, 6th edition, 2008.
How to Cite
ALLEN, Tony et al. Minimal Noise-Induced Stabilization of One-Dimensional Diffusions. Minnesota Journal of Undergraduate Mathematics, [S.l.], v. 3, n. 1, july 2017. ISSN 2378-5810. Available at: <>. Date accessed: 19 feb. 2018.