View source for Potential theory, abstract

The theory of potentials on abstract topological spaces. Abstract potential theory arose in the middle of the 20th century from the efforts to create a unified axiomatic method for treating a vast diversity of properties of the different potentials that are applied to solve problems of the theory of partial differential equations. The first sufficiently complete description of the axiomatics of  "harmonic"  functions (i.e. solutions of an admissible class of partial differential equations) and the corresponding potentials was given by M. Brelot (1957–1958, see [[#References|[1]]]), but it was concerned only with elliptic equations. The extension of the theory to a wide class of parabolic equations was obtained by H. Bauer (1960–1963, see [[#References|[3]]]). The probabilistic approach to abstract potential theory, the origins of which could be found already in the works of P. Lévy, J. Doob, G. Hunt, and others, turned out to be very fruitful.

To expose abstract potential theory, the notion of a [[Harmonic space|harmonic space]] is of great help. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741501.png" /> be a locally compact topological space. A sheaf of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741502.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741503.png" /> defined on the family of all open sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741504.png" /> such that

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741505.png" />, for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741506.png" />, is a family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741507.png" />;

2) if two open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741508.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p0741509.png" />, then the restriction of any function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415011.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415012.png" />;

3) if for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415014.png" />, of open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415015.png" /> the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415016.png" /> of some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415017.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415018.png" /> belong, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415019.png" />, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415021.png" />.

A sheaf of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415023.png" /> is called a harmonic sheaf if for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415024.png" /> the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415025.png" /> is a real vector space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415026.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415027.png" /> defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415028.png" /> containing the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415029.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415031.png" />-function if the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415032.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415033.png" />. A harmonic sheaf is non-degenerate at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415034.png" /> if in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415035.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415036.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415038.png" />.

The real distinctions between the axiomatics of Bauer, Brelot and Doob can be characterized by the convergence properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415039.png" />-functions.

a) Bauer's convergence property states that if an increasing sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415040.png" />-functions is locally bounded on some open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415041.png" />, then the limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415042.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415043.png" />-function.

b) Doob's convergence property states that if a limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415044.png" /> is finite on some dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415046.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415047.png" />-function.

c) Brelot's convergence property states that if the limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415048.png" /> of an increasing sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415049.png" />-functions on some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415050.png" /> is finite at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415052.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415053.png" />-function.

If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415054.png" /> is locally connected, the implications c)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415055.png" />b)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415056.png" />a) hold.

A sheaf of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415058.png" /> is called a hyperharmonic sheaf if for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415059.png" /> the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415060.png" /> is a convex cone of lower semi-continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415061.png" />; a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415063.png" />-function is defined in a similar way as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415064.png" />-function. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415065.png" /> is a harmonic sheaf, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415066.png" /> and generated by the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415067.png" />; hereafter, only this harmonic sheaf will be used.

Let on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415068.png" /> of an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415069.png" /> a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415070.png" /> with compact support be given. The hyperharmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415071.png" /> allows one to construct a generalized solution of the [[Dirichlet problem|Dirichlet problem]] for certain open sets in the class of corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415072.png" />-functions by the [[Perron method|Perron method]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415073.png" /> be a family of lower semi-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415074.png" />-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415075.png" />, bounded from below on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415076.png" />, positive outside some compact set and such that

<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/p/p074/p074150/p07415077.png" /></td> </tr></table>

define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415078.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415079.png" />. Now, let,

<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/p/p074/p074150/p07415080.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415081.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415082.png" />. Similarly,

<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/p/p074/p074150/p07415083.png" /></td> </tr></table>

or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415084.png" />. A function is called resolutive if for this function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415086.png" /> coincide, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415087.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415088.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415089.png" />-function; this function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415090.png" /> is a generalized solution of the Dirichlet problem in the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415091.png" />-functions. An open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415092.png" /> is resolutive with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415093.png" /> if every finite continuous function with compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415094.png" /> is resolutive. For a resolutive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415095.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415096.png" /> is a positive linear functional, hence it determines a positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415098.png" />, which is called the harmonic measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p07415099.png" /> (or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150100.png" />) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150101.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150102.png" />).

A locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150103.png" /> with a hyperharmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150104.png" /> turns into a harmonic space if the four corresponding axioms (see [[Harmonic space|Harmonic space]]) hold; moreover, in the convergence axiom the property is understood in the sense of Bauer.

Often (it is like this in classical examples) one takes as a basis the harmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150105.png" />, and the axiom of completeness serves then as a definition of a hyperharmonic sheaf. For instance, the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150107.png" />, together with the sheaf of classical solutions of the [[Laplace equation|Laplace equation]] or of the [[Heat equation|heat equation]] as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150108.png" />, is a harmonic space. A harmonic space is locally connected, does not contain isolated points and has a basis consisting of connected resolutive sets (resolutive domains).

An open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150109.png" /> in a harmonic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150110.png" /> with the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150111.png" /> as hyperharmonic sheaf is a harmonic subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150112.png" />. A hyperharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150113.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150114.png" /> is called a superharmonic function if for any relatively compact resolutive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150115.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150116.png" />, the greatest minorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150117.png" /> is harmonic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150118.png" />. Many properties of classical superharmonic functions (see [[Subharmonic function|Subharmonic function]]) also hold in this case. A potential is a positive superharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150119.png" /> such that its greatest harmonic minorant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150120.png" /> is identically equal to zero. A harmonic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150121.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150123.png" />-harmonic (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150125.png" />-harmonic) space if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150128.png" /> there exists a positive superharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150129.png" /> (a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150130.png" />, respectively) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150131.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150132.png" />. Any open set in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150133.png" />-harmonic space is resolutive.

Taking a harmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150134.png" /> as basis and defining the corresponding hyperharmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150135.png" /> by the axiom of completeness, one obtains the Bauer space, which coincides with the harmonic space for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150136.png" />. If the harmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150137.png" />, for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150138.png" />, consists of the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150139.png" /> of the heat equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150140.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150141.png" /> has the Doob convergence property and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150142.png" /> together with this sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150143.png" /> is a (Bauer) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150144.png" />-space. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150145.png" /> is a superharmonic function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150146.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150147.png" />.

A Brelot space is characterized by the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150148.png" /> does not have isolated points and is locally connected; the regular sets with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150149.png" /> form a base of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150150.png" /> (regularity means resolutivity of the classical Dirichlet problem in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150151.png" />); and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150152.png" /> has the Brelot convergence property. The Brelot spaces form a proper subclass of the so-called elliptic harmonic spaces (see [[#References|[4]]]), i.e. the elliptic Bauer spaces. If the harmonic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150153.png" />, for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150155.png" />, consists of the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150156.png" /> of the Laplace equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150157.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150158.png" /> together with this sheaf is a Brelot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150159.png" />-space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150160.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150161.png" /> is a Brelot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150162.png" />-space. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150163.png" /> is a hyperharmonic function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150164.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150165.png" />.

A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150166.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150167.png" /> of a resolutive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150168.png" /> is called a regular boundary point if for any finite continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150169.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150170.png" /> the following limit relation holds:

<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/p/p074/p074150/p074150171.png" /></td> </tr></table>

otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150172.png" /> is called an irregular boundary point. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150173.png" /> be a [[Filter|filter]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150174.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150175.png" />. A strictly-positive hyperharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150176.png" /> defined on the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150177.png" /> with some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150178.png" /> and converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150179.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150180.png" /> is called a barrier of the filter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150181.png" />. If for a relatively compact resolutive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150182.png" /> in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150183.png" />-harmonic space all filters that converge to points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150184.png" /> have barriers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150185.png" /> is a regular set, i.e. all its boundary points are regular. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150186.png" /> is a relatively compact open set in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150187.png" />-harmonic space on which there exists a strictly-positive hyperharmonic function converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150188.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150189.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150190.png" /> is a regular set.

Besides studies concerning resolutivity and regularity in the Dirichlet problem, the following problems are of major interest in abstract potential theory: the theory of [[Capacity|capacity]] of point sets in harmonic spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150191.png" />; the theory of balayage (see [[Balayage method|Balayage method]]) for functions and measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150192.png" />; and the theory of integral representations of positive superharmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150193.png" /> generalizing the Martin representations (see [[Martin boundary in potential theory|Martin boundary in potential theory]]).

Already at the beginning of the 20th century it became evident that potential theory is closely related to certain concepts of probability theory such as [[Brownian motion|Brownian motion]]; [[Wiener process|Wiener process]]; and [[Markov process|Markov process]]. For instance, the probability that the trajectory of a Brownian motion in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150194.png" /> starting at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150195.png" /> will hit for the first time the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150196.png" /> on a (Borel) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150197.png" /> is exactly the [[Harmonic measure|harmonic measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150198.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150199.png" />; the polar sets (cf. [[Polar set|Polar set]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150200.png" /> are the sets that are almost-certainly not hit by the trajectory. Later on, probabilistic methods contributed to a more profound understanding of certain ideas from potential theory and led to a series of new results; on the other hand, the potential-theoretic approach led to a better understanding of the results of probability theory and also leads to new results in it.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150201.png" /> be a locally compact space with a countable base, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150203.png" /> be the classes of finite continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150204.png" />, respectively, with compact support and convergent to zero at infinity. A measure kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150205.png" /> is a (Borel) function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150206.png" /> for every relatively compact (Borel) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150207.png" />. Using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150208.png" />, to each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150210.png" />, corresponds a potential function

<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/p/p074/p074150/p074150211.png" /></td> </tr></table>

and to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150212.png" /> corresponds a potential measure

<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/p/p074/p074150/p074150213.png" /></td> </tr></table>

The identity kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150214.png" /> vanishes when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150215.png" /> and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150216.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150217.png" />, it changes neither <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150218.png" /> nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150219.png" />. For instance, in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150220.png" /> the kernel

<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/p/p074/p074150/p074150221.png" /></td> </tr></table>

determines the [[Newton potential|Newton potential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150222.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150223.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150224.png" /> is the measure with density equal to the density of the Newton potential of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150225.png" /> (see [[Potential theory|Potential theory]]).

A product kernel has the form

<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/p/p074/p074150/p074150226.png" /></td> </tr></table>

A family of kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150227.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150228.png" />, with the composition law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150229.png" /> is a one-parameter semi-group. A kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150230.png" /> satisfies the complete maximum principle if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150231.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150233.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150234.png" /> on the set where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150235.png" /> leads to this inequality everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150236.png" />. The principal theorem in this theory is Hunt's theorem, which in its simplest version is the following: If the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150237.png" /> under a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150238.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150239.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150240.png" /> satisfies the complete maximum principle, then there exists a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150242.png" />, such that

<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/p/p074/p074150/p074150243.png" /></td> </tr></table>

(a Feller semi-group); moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150244.png" /> transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150245.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150246.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150247.png" /> is the identity kernel; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150248.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150249.png" />, locally uniform; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150250.png" />. A measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150251.png" /> is called an excessive function with respect to the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150252.png" /> if always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150253.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150254.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150255.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150256.png" /> is called an invariant function. The corresponding formulas are also valid for the potential measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150257.png" />.

The theory of Hunt (1957–1958) outlined above has a direct probabilistic sense. Let on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150258.png" /> be given some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150259.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150260.png" /> of Borel sets and a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150261.png" />. A random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150262.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150263.png" />-measurable mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150264.png" /> into the state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150265.png" />. The family of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150266.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150267.png" />, is a [[Markov process|Markov process]] (for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150268.png" /> is the trajectory of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150269.png" />) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150270.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150271.png" />, there exists a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150272.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150273.png" /> such that a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150274.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150275.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150276.png" />, is a Borel function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150277.png" />; and c) the form of a trajectory passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150278.png" /> at a moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150279.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150280.png" />, is independent of the positions of the points preceding it. In such Markov processes the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150281.png" /> are interpreted as semi-groups of measures

<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/p/p074/p074150/p074150282.png" /></td> </tr></table>

Studies of excessive and invariant functions with respect to the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150283.png" /> are of great importance.

On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150284.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150285.png" />-harmonic space with a countable base, then one can always choose on it a potential kernel to meet the requirements of Hunt's theorem; in this case the excessive functions of the associated semi-group are precisely the non-negative hyperharmonic functions. Hunt's theorem can be also generalized for some types of Bauer spaces (see [[#References|[4]]], [[#References|[7]]]).

Other concepts from abstract potential theory, such as, for example, balayage, polar and thin sets, also have their probabilistic interpretation within the framework of the general theory of random processes; this facilitates studies of the latter. On the other hand, the potential-theoretic approach to a series of concepts, such as, for example, martingales, which are beyond the limits of Markov processes, turned out to be of great importance.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,   "Lectures on potential theory" , Tata Inst.  (1960)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brelot,   "Les étapes et les aspects multiples de la théorie du potentiel"  ''Enseign. Math.'' , '''18''' :  1  (1972)  pp. 1–36</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bauer,   "Harmonische Räume und ihre Potentialtheorie" , Springer  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Meyer,   "Probability and potentials" , Blaisdell  (1966)</TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top">  G.A. Hunt,   "Markov processes and potentials, I"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 44–93</TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top">  G.A. Hunt,   "Markov processes and potentials, II"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 316–362</TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top">  G.A. Hunt,   "Markov processes and potentials, III"  ''Illinois J. Math.'' , '''2'''  (1958)  pp. 151–213</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.M. Blumenthal,   R.K. Getoor,   "Markov processes and potential theory" , Acad. Press  (1968)</TD></TR></table>



====Comments====
Abstract potential theory is also called axiomatic potential theory.

A measure kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150286.png" /> is also defined as a (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150287.png" />-additive, non-negative) measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150288.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150289.png" /> can be defined whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150290.png" /> is Borel and non-negative; in the article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150291.png" /> is assumed finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150292.png" />. By definition, a Feller semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150293.png" /> transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150294.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150295.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150296.png" /> is the identity kernel and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150297.png" /> the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150298.png" /> is continuous at 0 for the uniform topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150299.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150300.png" /> are related by Hunt's theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150301.png" /> is called the potential kernel of the Feller semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150302.png" />, and for any non-negative Borel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150303.png" /> the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150305.png" /> is an excessive function with respect to the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150306.png" />. One can define analogously excessive measures and potential measures, and their study is also of great importance. The passage in the article on the definition of a Markov process is somewhat misleading: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150307.png" /> equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150308.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150309.png" /> has to be replaced by an extraneous measurable space, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150310.png" />, and the state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150311.png" /> by the locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150312.png" /> where the Hunt kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150313.png" /> is defined, so that in the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150314.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150315.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150316.png" /> (and Borel). See [[Markov process|Markov process]].

Around 1959, A. Beurling and J. Deny introduced another branch of abstract potential theory: the notion of Dirichlet space, an axiomatization of the theory of the [[Dirichlet integral|Dirichlet integral]]. See [[#References|[a3]]].

Several abstract theories have been introduced, aimed at a unification of different branches of potential theory, e.g. the theory of balayage spaces, cf. [[#References|[a2]]], and the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150318.png" />-cones, cf. [[#References|[a1]]]. Both concepts use a convex cone of functions (the positive hyperharmonic functions, e.g.), satisfying some convergence properties, a Riesz property and a separation property as their main tool. See also [[#References|[a4]]] for short surveys.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Boboc,   Gh. Bucur,   A. Cornea,   "Order and convexity in potential theory: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150319.png" />-cones" , Springer  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bliedtner,   W. Hansen,   "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Fukushima,   "Dirichlet forms and Markov processes" , North-Holland  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Král (ed.)  J. Lukeš (ed.)  J. Veselý (ed.) , ''Potential theory. Survey and problems (Prague, 1987)'' , ''Lect. notes in math.'' , '''1344''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''C''' , North-Holland  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F.Y. Maeda,   "Dirichlet integrals on harmonic spaces" , ''Lect. notes in math.'' , '''803''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Röckner,   "Markov property of generalized fields and axiomatic potential theory"  ''Math. Ann.'' , '''264'''  (1983)  pp. 153–177</TD></TR></table>