## The Annals of Probability

### Obliquely reflected Brownian motion in nonsmooth planar domains

#### 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
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

#### 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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