Martin boundary in potential theory

The ideal boundary of a Green space (see also Boundary (in the theory of uniform algebras)), which allows one to construct the characteristic representation of positive harmonic functions in . Let be a locally compact, non-compact, topological space, and let be a family of continuous functions . The Constantinescu–Cornea theorem [2] asserts that, up to a homeomorphism, there is a unique compact space with the following properties: 1) is an everywhere-dense subspace of ; 2) each extends continuously to a function on , separating points on the ideal boundary of relative to ; and 3) is an open set in .

Now, let be a bounded domain in a Euclidean space , , or, more generally, a Green space; let be the Green function on with pole and let be fixed. The Martin space or Martin compactification of is obtained via the Constantinescu–Cornea theorem by taking for the family

where, by definition, . The Martin boundary is the corresponding ideal boundary . The Martin topology is the topology on the Martin space . Two Martin spaces , corresponding to different points are homeomorphic. The function , the extension of , is harmonic in and jointly continuous in the variables ; is a metrizable space. Martin's fundamental theorem [1] asserts: The class of all positive harmonic functions on is characterized by the Martin representation:

(*)

where is a positive Radon measure on . The measure in (*) is not uniquely determined by the function . A harmonic function is called minimal in if each harmonic function such that in is proportional to . Minimal harmonic functions are proportional to , the corresponding points are called minimal, and the set of minimal points is called the minimal Martin boundary. If one poses the additional condition that in (*) be concentrated on , one obtains the canonical Martin representation:

in which the measure is uniquely determined by .

Examples. a) If is a ball of radius in , , then

is the Poisson kernel, is the Euclidean closure , the Martin boundary is the sphere , all points of which are minimal. The representation (*) in this case reduces to the Poisson–Herglotz formula (see Integral representation of an analytic function; Poisson integral).

b) The Martin boundary coincides with the Euclidean boundary whenever is a sufficiently smooth hypersurface in , .

c) If is a simply-connected domain in the plane, then the Martin boundary coincides with the set of limit elements, or Carathéodory prime ends. Thus, an element of the Martin boundary can be considered as a generalization of the notion of a prime end to dimension .

References

[1]	R.S. Martin, "Minimal positive harmonic functions" Trans. Amer. Math. Soc. , 49 (1941) pp. 137–172
[2]	C. Constantinescu, A. Cornea, "Ideale Ränder Riemannscher Flächen" , Springer pp. 1963
[3]	M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971)

Comments

See also [a1], Chapt. 12, for a concise treatment. For Martin boundaries for the heat equation or in probabilistic potential theory, see [a3].

References

[a1]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[a2]	M. Brelot, "Axiomatique des fonctions harmoniques" , Univ. Montréal (1966)
[a3]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390