Martin representation theorem

Remarks

Idea of the proof: for an exhaustion of $\Omega$ by smooth subdomains $\Omega_j$, the reduced function of $h$ on $\overline{\Omega_j}$ is a Green potential $\int G(\cdot, y)\, d\nu_j(y)$. Rewriting it as $\int M(\cdot, y)\, G(x_0, y)\, d\nu_j(y)$ gives measures of mass at most $h(x_0)$ on the compact space $\Omega \cup \Delta$, and a weak-star limit gives a representing measure on $\Delta$. Choquet's theory of extreme points, or a direct argument of Martin, shows that the measure can be taken on $\Delta_1$, where it is unique. Example: for the unit ball with $x_0 = 0$, $\Delta = \Delta_1$ is the unit sphere and $M(x, \zeta)$ is the Poisson kernel. So the theorem becomes the Herglotz representation of positive harmonic functions as Poisson integrals of positive measures. Uniqueness needs the condition $\mu(\Delta \setminus \Delta_1) = 0$: a non-minimal point $\xi$ gives a function $M(\cdot, \xi)$ that has two representations, by $\delta_\xi$ and by a measure on $\Delta_1$. The theorem is due to R. S. Martin (1941).

Going deeper

For a student

The theorem reduces questions about all positive harmonic functions to questions about the minimal ones and about measures on the Martin boundary. It is the starting point for fine boundary limits, where Naïm's thin sets and Doob's conditioned Brownian motion describe how $h / M(\cdot, \xi)$ behaves near $\xi$. For Lipschitz domains, the boundary Harnack principle shows that $\Delta = \Delta_1$ is the Euclidean boundary. The same scheme applies to Markov chains, where it describes all positive harmonic functions of a transition operator.