  We construct obliquely reflected Brownian motions in all bounded simply
connected planar domains, including non-smooth 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.
