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 Lp is obtained.
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.