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September 2017 Obliquely reflected Brownian motion in nonsmooth planar domains
Krzysztof Burdzy, Zhen-Qing Chen, Donald Marshall, Kavita Ramanan
Ann. Probab. 45(5): 2971-3037 (September 2017). DOI: 10.1214/16-AOP1130

Abstract

We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including nonsmooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.

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Krzysztof Burdzy. Zhen-Qing Chen. Donald Marshall. Kavita Ramanan. "Obliquely reflected Brownian motion in nonsmooth planar domains." Ann. Probab. 45 (5) 2971 - 3037, September 2017. https://doi.org/10.1214/16-AOP1130

Information

Received: 1 December 2015; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812199
MathSciNet: MR3706737
Digital Object Identifier: 10.1214/16-AOP1130

Subjects:
Primary: 58J65 , 60H20 , 60J65
Secondary: 30C20 , 30J99

Keywords: Brownian motion with darning , conformal mapping , excursion reflected Brownian motion , excursion reflected Brownian motion , Oblique reflection , rate of rotation of obliquely reflected Brownian motion , reflected Brownian motion , simply connected domains , stationary distribution

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • September 2017
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