Exponential almost sure synchronization of one-dimensional diffusions with nonregular coefficients

Abstract

We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge almost sure to one another at an exponential explicit rate as soon as the dissipative coefficient is large enough. A similar result in L_p is obtained.

KEYWORDS:

• Stochastic differential equation (SDE)
• singular drift
• synchronization

Acknowledgments

The authors would like to warmly thank M. Scheutzow for fruitful discussions on this topic. It is also their pleasure to thank an anonymous referee for drawing their attention to the cited article of M. Scheutzow and S. Schulze, which allowed a significant improvement of a first version of Proposition 1.

Additional information

Funding

This work was partially supported by Deutsche Forschungsgemeinschaft project “Stochastic Dynamics with Interfaces”, no. 452119141, and the Alexander von Humboldt Foundation (Research Group Linkage cooperation Singular diffusions: analytic and stochastic approaches) between the University of Potsdam and the Institute of Mathematics of the National Academy of Sciences of Ukraine.