The Annals of Probability

Obliquely reflected Brownian motion in nonsmooth planar domains

Krzysztof Burdzy, Zhen-Qing Chen, Donald Marshall, and Kavita Ramanan

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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.

Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 2971-3037.

Dates
Received: December 2015
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132032

Digital Object Identifier
doi:10.1214/16-AOP1130

Mathematical Reviews number (MathSciNet)
MR3706737

Zentralblatt MATH identifier
06812199

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60H20: Stochastic integral equations
Secondary: 30C20: Conformal mappings of special domains 30J99: None of the above, but in this section

Keywords
Reflected Brownian motion oblique reflection simply connected domains conformal mapping stationary distribution excursion reflected Brownian motion Brownian motion with darning Excursion reflected Brownian motion rate of rotation of obliquely reflected Brownian motion

Citation

Burdzy, Krzysztof; Chen, Zhen-Qing; Marshall, Donald; Ramanan, Kavita. Obliquely reflected Brownian motion in nonsmooth planar domains. Ann. Probab. 45 (2017), no. 5, 2971--3037. doi:10.1214/16-AOP1130. https://projecteuclid.org/euclid.aop/1506132032


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