Martin boundary in potential theory

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

Now, let $ \Omega $ be a bounded domain in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, or, more generally, a Green space; let $ G = G ( x , y ) $ be the Green function on $ \Omega $ with pole $ y \in \Omega $ and let $ y _ {0} \in \Omega $ be fixed. The Martin space or Martin compactification $ \widehat \Omega $ of $ \Omega $ is obtained via the Constantinescu–Cornea theorem by taking for $ \Phi $ the family

$$ \Phi = \ \left \{ { x \in \Omega \rightarrow K ( x , y ) = \frac{G ( x , y ) }{G ( x , y _ {0} ) } } : { y \in \Omega } \right \} , $$

where, by definition, $ K ( x _ {0} , y _ {0} ) = 1 $. The Martin boundary is the corresponding ideal boundary $ \Delta = \widehat \Omega \setminus \Omega $. The Martin topology $ T $ is the topology on the Martin space $ \widehat \Omega $. Two Martin spaces $ \widehat \Omega {} ^ \prime $, $ \widehat \Omega {} ^ {\prime\prime} $ corresponding to different points $ y _ {0} ^ \prime , y _ {0} ^ {\prime\prime} \in \Omega $ are homeomorphic. The function $ \widehat{K} ( \xi , y) : \Delta \times \Omega \rightarrow [ 0 , + \infty ] $, the extension of $ K ( x , y ) $, is harmonic in $ y $ and jointly continuous in the variables $ ( \xi , y) $; $ \widehat \Omega $ is a metrizable space. Martin's fundamental theorem [1] asserts: The class of all positive harmonic functions $ u ( y) \geq 0 $ on $ \Omega $ is characterized by the Martin representation:

$$ \tag{* } u ( y) = \int\limits K ( \xi , y ) d \mu ( \xi ) , $$

where $ \mu $ is a positive Radon measure on $ \Delta $. The measure $ \mu $ in (*) is not uniquely determined by the function $ u $. A harmonic function $ v \geq 0 $ is called minimal in $ \Omega $ if each harmonic function $ w $ such that $ 0 \leq w \leq v $ in $ \Omega $ is proportional to $ v $. Minimal harmonic functions $ v \neq 0 $ are proportional to $ \widehat{K} ( \xi , y) $, the corresponding points $ \xi \in \Delta $ are called minimal, and the set of minimal points $ \Delta _ {1} \subset \Delta $ is called the minimal Martin boundary. If one poses the additional condition that $ \mu $ in (*) be concentrated on $ \Delta _ {1} $, one obtains the canonical Martin representation:

$$ u ( y) = \int\limits \widehat{K} ( \xi , y ) d \mu _ {1} ( \xi ) , $$

in which the measure $ \mu _ {1} \geq 0 $ is uniquely determined by $ u $.

Examples. a) If $ \Omega = \{ {x \in \mathbf R ^ {n} } : {| x | < R } \} $ is a ball of radius $ R $ in $ \mathbf R ^ {n} $, $ n \geq 2 $, then

$$ \widehat{K} ( \xi , y ) = \ \frac{R ^ {n-} 2 ( R ^ {2} - | y | ^ {2} ) }{| \xi - y | ^ {n} } $$

is the Poisson kernel, $ \widehat \Omega $ is the Euclidean closure $ \widehat \Omega = \overline \Omega \; $, the Martin boundary $ \Delta $ is the sphere $ \{ {\xi \in \mathbf R ^ {n} } : {| \xi | = R } \} $, 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 $ \Delta $ coincides with the Euclidean boundary $ \Gamma = \overline \Omega \; \setminus \Omega $ whenever $ \Gamma $ is a sufficiently smooth hypersurface in $ \mathbf R ^ {n} $, $ n \geq 2 $.

c) If $ \Omega $ is a simply-connected domain in the plane, then the Martin boundary $ \Delta $ coincides with the set of limit elements, or Carathéodory prime ends. Thus, an element of the Martin boundary $ \xi \in \Delta $ can be considered as a generalization of the notion of a prime end to dimension $ n \geq 2 $.

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