The Annals of Probability
- Ann. Probab.
- Volume 45, Number 5 (2017), 2971-3037.
Obliquely reflected Brownian motion in nonsmooth planar domains
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.
Ann. Probab., Volume 45, Number 5 (2017), 2971-3037.
Received: December 2015
First available in Project Euclid: 23 September 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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
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