Abstract
We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete—in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.
Similar content being viewed by others
References
Herzog, D.P., Mattingly, J.C.: Noise-induced stabilization of planar flows I. ArXiv e-prints: arXiv:1404.0957 (2014)
Herzog, D.P., Mattingly, J.C.: Noise-induced stabilization of planar flows II. ArXiv e-prints: arXiv:1404.0955 (2014)
Scheutzow, M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 11(1), 97–113 (1993)
Birrell, J., Herzog, D.P., Wehr, J.: Transition from ergodic to explosive behavior in a family of stochastic differential equations. Stoch. Process. Appl. 122(4), 1519–1539 (2012)
Athreya, A., Kolba, T., Mattingly, J.C.: Propagating Lyapunov functions to prove noise-induced stabilization. Electron. J. Probab. 17, 1–38 (2012)
Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100(3), 365–393 (1994)
Elworthy, K.D.: Stochastic dynamical systems and their flows. In: Proceedings of the International Conference on Stochastic Analysis, Northwestern University, Evanston, pp. 79–95. Academic Press, New York (1978)
Li, X., Scheutzow, M.: Lack of strong completeness for stochastic flows. Ann. Probab. 39(4), 1407–1421 (2011)
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)
Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7(1), 81–94 (1956)
Acknowledgments
We would like to acknowledge the DFG Research training group 1845 Stochastic Analysis with applications in biology, finance and physics for its financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leimbach, M., Scheutzow, M. Blow-up of a Stable Stochastic Differential Equation. J Dyn Diff Equat 29, 345–353 (2017). https://doi.org/10.1007/s10884-015-9467-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-015-9467-5