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We construct obliquely reflected Brownian motions in all bounded simplyconnected planar domains, including non-smooth domains, with general reflectionvector fields on the boundary. Conformal mappings and excursion theory are ourmain technical tools. A key intermediate step, which may be of independentinterest, is an alternative characterization of reflected Brownian motions insmooth bounded planar domains with a given field of angles of obliquereflection on the boundary in terms of a pair of quantities, namely anintegrable positive harmonic function, which represents the stationarydistribution of the process, and a real number that represents, in a suitablesense, the asymptotic rate of rotation of the process around a reference pointin the domain. Furthermore, we also show that any obliquely reflected Brownianmotion in a simply connected Jordan domain can be obtained as a suitable limitof obliquely reflected Brownian motions in smooth domains.

Obliquely reflected Brownian motion in non-smooth planar domains