A Martin boundary in the plane
Thomas S. Salisbury · Transactions of the American Mathematical Society · 1986
Let E E be an open connected subset of Euclidean space, with a Green function, and let λ \lambda be harmonic measure on the Martin boundary Δ \Delta of E E . We will show that, except for a λ ⊗ λ \lambda \otimes \lambda -null set of ( x , y ) ∈ Δ 2 (x,y) \in {\Delta ^2} , x x is an entrance point for Brownian motion conditioned to leave E E at y y . R. S. Martin gave examples in dimension 3 3 or higher, for which there exist minimal accessible Martin boundary points x ≠ y x e y for which this condition fails. We will give a similar example in dimension 2 2 .