Boundary behaviour of quotients of martin kernels

Given two intersecting domains, we investigate the boundary behaviour of the quotient of Martin kernels of each domain. To this end, we give a characterization of minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem, we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.