Failure of the Brownian Dirichlet representation on an unbounded domain
= Failure of the Brownian Dirichlet representation on an unbounded domain
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Almost-sure finiteness of the exit time does not replace boundedness of the domain. In the upper half-plane, $u(x,y)=y$ is harmonic with zero boundary values and Brownian motion reaches the boundary almost surely, but $u$ is positive inside. The stopped martingale fails to be uniformly integrable.