On Brownian paths connecting boundary points (Q1107217)

For a Greenian domain D in \(R^ n\), let \(\partial^ m_ aD\) be the set of all attainable minimal Martin boundary points; \(u\in \partial^ M_ aD\) if u is a minimal Martin boundary point such that the h-process for the Martin kernel \(h(\cdot)=K^ D_ z(u,\cdot)\) has finite lifetime. Denote by \(\mu_ x\) the harmonic measure on \(\partial^ MD\) relative to \(x\in D.\) The author proves that there exists an open connected Greenian domain D in \(R^ 2\) such that, for each set \(U\subset \partial^ M_ aD\) with \(\mu_ x(U)=0\), there are points \(u,v\in \partial^ M_ aD-U\), \(u\neq v\), which cannot be connected by an h-process starting from u and converging to v.