  We define a family of stochastic Loewner evolution-type processes in finitely
connected domains, which are called continuous LERW (loop-erased random walk).
A continuous LERW describes a random curve in a finitely connected domain that
starts from a prime end and ends at a certain target set, which could be an
interior point, or a prime end, or a side arc. It is defined using the usual
chordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus a
drift term. The distributions of continuous LERW are conformally invariant. A
continuous LERW preserves a family of local martingales, which are composed of
generalized Poisson kernels, normalized by their behaviors near the target set.
These local martingales resemble the discrete martingales preserved by the
corresponding LERW on the discrete approximation of the domain. For all kinds
of targets, if the domain satisfies certain boundary conditions, we use these
martingales to prove that when the mesh of the discrete approximation is small
enough, the continuous LERW and the corresponding discrete LERW can be coupled
together, such that after suitable reparametrization, with probability close to
1, the two curves are uniformly close to each other.
