Coalescence of synchronous couplings
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We consider a pair of reflected Brownian motions in a Lipschitz planar domain starting from different points but driven by the same Brownian motion. First we construct such a pair of processes in a certain weak sense, since it is not known whether a strong solution to the Skorohod equation in Lipschitz domains exists. Then we prove that the distance between the two processes converges to zero with probability one if the domain has a polygonal boundary or it is a "lip domain," i.e., a domain between the graphs of two Lipschitz functions with Lipschitz constants strictly less than 1.