  We prove the existence of uncountably many positive harmonic functions for
random walks on the euclidean lattice with non-zero drift, killed when leaving
two dimensional convex cones with vertex in 0. Our proof is an adaption of the
proof for the positive quadrant from [Ignatiouk-Robert, Loree]. We also make
the natural conjecture about the Martin boundary for general convex cones in
two dimensions. This is still an open problem and here we only indicate where
the proof technique for the positive quadrant breaks down.

Probability