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We define a family of stochastic Loewner evolution-type processes in finitelyconnected domains, which are called continuous LERW (loop-erased random walk).A continuous LERW describes a random curve in a finitely connected domain thatstarts from a prime end and ends at a certain target set, which could be aninterior point, or a prime end, or a side arc. It is defined using the usualchordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus adrift term. The distributions of continuous LERW are conformally invariant. Acontinuous LERW preserves a family of local martingales, which are composed ofgeneralized Poisson kernels, normalized by their behaviors near the target set.These local martingales resemble the discrete martingales preserved by thecorresponding LERW on the discrete approximation of the domain. For all kindsof targets, if the domain satisfies certain boundary conditions, we use thesemartingales to prove that when the mesh of the discrete approximation is smallenough, the continuous LERW and the corresponding discrete LERW can be coupledtogether, such that after suitable reparametrization, with probability close to1, the two curves are uniformly close to each other.

The scaling limits of planar LERW in finitely connected domains