Martin kernel

Summary

The Martin kernel of a domain is the Green's function normalised by its value at a fixed reference point: $M(x, y) = G(x, y) / G(x_0, y)$. As the pole $y$ runs to the boundary, the functions $M(\cdot, y)$ accumulate at positive harmonic functions. These limits are the points of the Martin boundary. For a ball, the limits are the Poisson kernels, so the Martin kernel generalises the Poisson kernel to arbitrary domains.