View source for Martin boundary in potential theory

<!--
m0625501.png
$#A+1 = 79 n = 0
$#C+1 = 79 : ~/encyclopedia/old_files/data/M062/M.0602550 Martin boundary in potential theory
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The ideal boundary of a [[Green space|Green space]]  $  \Omega $(
see also [[Boundary (in the theory of uniform algebras)|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 [[#References|[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|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 [[#References|[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|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|Integral representation of an analytic function]]; [[Poisson integral|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.S. Martin,   "Minimal positive harmonic functions"  ''Trans. Amer. Math. Soc.'' , '''49'''  (1941)  pp. 137–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Ideale Ränder Riemannscher Flächen" , Springer  pp. 1963</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brelot,   "On topologies and boundaries in potential theory" , Springer  (1971)</TD></TR></table>

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

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