View source for Martin boundary in potential theory

The ideal boundary of a [[Green space|Green space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625501.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625502.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625503.png" /> be a locally compact, non-compact, topological space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625504.png" /> be a family of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625505.png" />. The Constantinescu–Cornea theorem [[#References|[2]]] asserts that, up to a homeomorphism, there is a unique compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625506.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625507.png" /> is an everywhere-dense subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625508.png" />; 2) each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m0625509.png" /> extends continuously to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255011.png" />, separating points on the ideal boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255013.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255014.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255015.png" /> is an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255016.png" />.

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255017.png" /> be a bounded domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255019.png" />, or, more generally, a Green space; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255020.png" /> be the [[Green function|Green function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255021.png" /> with pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255022.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255023.png" /> be fixed. The Martin space or Martin compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255025.png" /> is obtained via the Constantinescu–Cornea theorem by taking for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255026.png" /> the family

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255027.png" /></td> </tr></table>

where, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255028.png" />. The Martin boundary is the corresponding ideal boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255029.png" />. The Martin topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255030.png" /> is the topology on the Martin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255031.png" />. Two Martin spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255033.png" /> corresponding to different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255034.png" /> are homeomorphic. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255035.png" />, the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255036.png" />, is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255037.png" /> and jointly continuous in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255038.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255039.png" /> is a metrizable space. Martin's fundamental theorem [[#References|[1]]] asserts: The class of all positive harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255041.png" /> is characterized by the Martin representation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255043.png" /> is a positive [[Radon measure|Radon measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255044.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255045.png" /> in (*) is not uniquely determined by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255046.png" />. A harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255047.png" /> is called minimal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255048.png" /> if each harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255051.png" /> is proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255052.png" />. Minimal harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255053.png" /> are proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255054.png" />, the corresponding points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255055.png" /> are called minimal, and the set of minimal points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255056.png" /> is called the minimal Martin boundary. If one poses the additional condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255057.png" /> in (*) be concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255058.png" />, one obtains the canonical Martin representation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255059.png" /></td> </tr></table>

in which the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255060.png" /> is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255061.png" />.

Examples. a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255062.png" /> is a ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255065.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255066.png" /></td> </tr></table>

is the Poisson kernel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255067.png" /> is the Euclidean closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255068.png" />, the Martin boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255069.png" /> is the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255070.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255071.png" /> coincides with the Euclidean boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255072.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255073.png" /> is a sufficiently smooth hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255075.png" />.

c) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255076.png" /> is a simply-connected domain in the plane, then the Martin boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255077.png" /> coincides with the set of [[Limit elements|limit elements]], or Carathéodory prime ends. Thus, an element of the Martin boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255078.png" /> can be considered as a generalization of the notion of a prime end to dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062550/m06255079.png" />.

====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>