  Excursion reflected Brownian motion (ERBM) is a strong Markov process defined
in a finitely connected domain $D \subset \mathbb{C}$ that behaves like a
Brownian motion away from the boundary of $D$ and picks a point according to
harmonic measure from infinity to reflect from every time it hits a boundary
component. We give a construction of ERBM using its conformal invariance and
develop the basic theory of its harmonic functions. One important reason for
studying ERBM is the hope that it will be a useful tool in the study of SLE in
multiply connected domains. To this end, we develop the basic theory of the
Poisson kernel and Green's function for ERBM and show how it can be used to
construct conformal maps into certain classes of multiply connected domains.
