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 |
Martin boundary in potential theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Martin_boundary_in_potential_theory&oldid=14144