Abstract
Consider reflecting Brownian motion in a bounded domain in \({\mathbb R^d}\) that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential.
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Research supported in part by NSF Grants DMS-0601783 and DMS-0600206 and by EPSRC Grant EP/D071593/.
K. Burdzy and M. Hairer gratefully acknowledge the hospitality and support of the Institut Mittag-Leffler, where part of this research was done.
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Bass, R.F., Burdzy, K., Chen, ZQ. et al. Stationary distributions for diffusions with inert drift. Probab. Theory Relat. Fields 146, 1 (2010). https://doi.org/10.1007/s00440-008-0182-6
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DOI: https://doi.org/10.1007/s00440-008-0182-6
Mathematics Subject Classification (2000)
- Primary: 60H10
- Secondary: 60J55
- 60J60