Integral Equations and Operator Theory

, Volume 69, Issue 1, pp 1–27 | Cite as

Pathwise Stability of Degenerate Stochastic Evolutions

  • Joris Bierkens
Open Access


For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.

Mathematics Subject Classification (2010)

60Hxx 93Dxx 47A62 34K50 


Stochastic evolution equation Lyapunov exponent pathwise stability stochastic delay differential equation operator inequality 



The author wishes to express his thanks to Onno van Gaans and Sjoerd Verduyn Lunel (Mathematical Institute, Leiden) and also the anonymous reviewer for their insightful remarks which added substantially to the quality of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

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