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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.
 
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
+
To expose abstract potential theory, the notion of a [[Harmonic space|harmonic space]] is of great help. Let $  X $
 +
be a locally compact topological space. A sheaf of functions on $  X $
 +
is a mapping $  \mathfrak F $
 +
defined on the family of all open sets of $  X $
 +
such that
 +
 
 +
1)  $  \mathfrak F ( U) $,
 +
for any open set  $  U \subset  X $,
 +
is a family of functions  $  u :  U \rightarrow \overline{\mathbf R}\; = [ - \infty , \infty ] $;
 +
 
 +
2) if two open sets  $  U , V $
 +
are such that $  U \subset  V \subset  X $,
 +
then the restriction of any function from  $  \mathfrak F ( V) $
 +
to  $  U $
 +
belongs to  $  \mathfrak F ( U) $;
  
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" />;
+
3) if for any family  $  \{ U _ {i} \} $,
 +
$  i \in I $,
 +
of open sets  $  U _ {i} \subset  X $
 +
the restrictions to  $  U _ {i} $
 +
of some function  $  u $
 +
defined on  $  \cup _ {i \in I }  U _ {i} $
 +
belong, for any $  i \in I $,
 +
to  $  \mathfrak F ( U _ {i} ) $,  
 +
then  $  u \in \mathfrak F ( \cup _ {i \in I }  U _ {i} ) $.
  
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" />;
+
A sheaf of functions  $  \mathfrak H $
 +
on  $  X $
 +
is called a harmonic sheaf if for any open set  $  U \subset  X $
 +
the family  $  \mathfrak H ( U) $
 +
is a real vector space of continuous functions on  $  U $.  
 +
A function  $  u $
 +
defined on some set  $  S \subset  X $
 +
containing the open set  $  U $
 +
is called an  $  \mathfrak H $-
 +
function if the restriction $  u \mid  _ {U} $
 +
belongs to  $  \mathfrak H ( U) $.
 +
A harmonic sheaf is non-degenerate at a point  $  x \in X $
 +
if in a neighbourhood of $  x $
 +
there exists an  $  \mathfrak H $-
 +
function $  u $
 +
such that  $  u ( x) \neq 0 $.
  
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" />.
+
The real distinctions between the axiomatics of Bauer, Brelot and Doob can be characterized by the convergence properties of $  \mathfrak H $-
 +
functions.
  
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" />.
+
a) Bauer's convergence property states that if an increasing sequence of $  \mathfrak H $-
 +
functions is locally bounded on some open set $  U \subset  X $,
 +
then the limit function $  v $
 +
is an $  \mathfrak H $-
 +
function.
  
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.
+
b) Doob's convergence property states that if a limit function  $  v $
 +
is finite on some dense set  $  U \subset  X $,
 +
then  $  v $
 +
is an  $  \mathfrak H $-
 +
function.
  
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.
+
c) Brelot's convergence property states that if the limit function  $  v $
 +
of an increasing sequence of $  \mathfrak H $-
 +
functions on some domain  $  U \subset  X $
 +
is finite at a point  $  x \in U $,  
 +
then $  v $
 +
is an $  \mathfrak H $-
 +
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.
+
If the space  $  X $
 +
is locally connected, the implications c) $  \Rightarrow $
 +
b) $  \Rightarrow $
 +
a) hold.
  
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.
+
A sheaf of functions  $  \mathfrak U $
 +
on  $  X $
 +
is called a hyperharmonic sheaf if for any open set  $  U \subset  X $
 +
the family  $  \mathfrak U ( U) $
 +
is a convex cone of lower semi-continuous functions $  u : U \rightarrow ( - \infty , \infty ] $;
 +
a  $  \mathfrak U $-
 +
function is defined in a similar way as an  $  \mathfrak H $-
 +
function. The mapping  $  U \rightarrow \mathfrak U ( U) \cap ( - \mathfrak U ( U) ) $
 +
is a harmonic sheaf, denoted by  $  \mathfrak H = \mathfrak H _ {\mathfrak U} $
 +
and generated by the sheaf  $  \mathfrak U $;
 +
hereafter, only this harmonic sheaf will be used.
  
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.
+
Let on the boundary  $  \partial  U $
 +
of an open set  $  U \subset  X $
 +
a continuous function  $  \phi : \partial  U \rightarrow ( - \infty , \infty ) $
 +
with compact support be given. The hyperharmonic sheaf  $  \mathfrak U $
 +
allows one to construct a generalized solution of the [[Dirichlet problem|Dirichlet problem]] for certain open sets in the class of corresponding  $  \mathfrak H $-
 +
functions by the [[Perron method|Perron method]]. Let  $  \overline{\mathfrak U}\; _  \phi  $
 +
be a family of lower semi-continuous  $  \mathfrak U $-
 +
functions  $  u $,
 +
bounded from below on  $  U $,
 +
positive outside some compact set and such that
  
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.
+
$$
 +
\lim\limits _ {x \rightarrow y }  \inf  u ( x)  \geq  \phi ( y) ,\ \
 +
y \in \partial  U ;
 +
$$
  
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
+
define  $  \underline{\mathfrak U} {} _  \phi  $
 +
by $  \underline{\mathfrak U} {} _  \phi  = - \overline{\mathfrak U}\; _  \phi  $.  
 +
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/p07415077.png" /></td> </tr></table>
+
$$
 +
\overline{H}\; _  \phi  ( x)  = \inf \{ {u ( x) } : {
 +
u \in \overline{\mathfrak U}\; _  \phi  } \}
 +
,\ \
 +
x \in U ,
 +
$$
  
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,
+
and let  $  \overline{H}\; _  \phi  = \infty $
 +
if  $  \overline{\mathfrak U}\; _  \phi  = \emptyset $.  
 +
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/p07415080.png" /></td> </tr></table>
+
$$
 +
\underline{H} {} _  \phi  ( x)  = \sup \{ {u ( x) } : {
 +
u \in \underline{\mathfrak U} {} _  \phi  } \}
 +
,\ \
 +
x \in U ,
 +
$$
  
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,
+
or  $  \underline{H} {} _  \phi  = - \infty $.  
 +
A function is called resolutive if for this function  $  \overline{H}\; _  \phi  $
 +
and  $  \underline{H} {} _  \phi  $
 +
coincide,  $  \overline{H}\; _  \phi  = \underline{H} {} _  \phi  = H _  \phi  $,
 +
and if  $  H _  \phi  $
 +
is an  $  \mathfrak H $-
 +
function; this function  $  H _  \phi  $
 +
is a generalized solution of the Dirichlet problem in the class of  $  \mathfrak H $-
 +
functions. An open set  $  U \subset  X $
 +
is resolutive with respect to  $  \mathfrak U $
 +
if every finite continuous function with compact support on  $  \partial  U $
 +
is resolutive. For a resolutive set  $  U $
 +
the mapping  $  H _  \phi  :  C _ {c} ( \partial  U ) \rightarrow \mathbf R $
 +
is a positive linear functional, hence it determines a positive measure  $  \mu _ {x} $,
 +
$  x \in U $,
 +
which is called the harmonic measure on  $  \partial  U $(
 +
or on  $  U $)
 +
at the point  $  x $(
 +
with respect to  $  \mathfrak U $).
  
<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>
+
A locally compact space  $  X $
 +
with a hyperharmonic sheaf  $  \mathfrak U $
 +
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.
  
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" />).
+
Often (it is like this in classical examples) one takes as a basis the harmonic sheaf  $  \mathfrak H $,
 +
and the axiom of completeness serves then as a definition of a hyperharmonic sheaf. For instance, the Euclidean space  $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
together with the sheaf of classical solutions of the [[Laplace equation|Laplace equation]] or of the [[Heat equation|heat equation]] as  $  \mathfrak H $,
 +
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).
  
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.
+
An open set  $  U $
 +
in a harmonic space $  X $
 +
with the restriction  $  \mathfrak U \mid  _ {U} $
 +
as hyperharmonic sheaf is a harmonic subspace of  $  X $.  
 +
A hyperharmonic function  $  u $
 +
on  $  U \subset  X $
 +
is called a superharmonic function if for any relatively compact resolutive set  $  V $
 +
with  $  \overline{V}\; \subset  U $,
 +
the greatest minorant  $  \mu  ^ {V} u $
 +
is harmonic,  $  \mu  ^ {V} u \in \mathfrak H ( V) $.
 +
Many properties of classical superharmonic functions (see [[Subharmonic function|Subharmonic function]]) also hold in this case. A potential is a positive superharmonic function  $  u $
 +
such that its greatest harmonic minorant on  $  X $
 +
is identically equal to zero. A harmonic space  $  X $
 +
is called a  $  \mathfrak C $-
 +
harmonic (or  $  \mathfrak P $-
 +
harmonic) space if for any point  $  x \in X $
 +
there exists a positive superharmonic function  $  u $(
 +
a potential  $  u $,  
 +
respectively) on  $  X $
 +
such that  $  u > 0 $.
 +
Any open set in a  $  \mathfrak P $-
 +
harmonic space is resolutive.
  
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).
+
Taking a harmonic sheaf  $  \mathfrak H $
 +
as basis and defining the corresponding hyperharmonic sheaf $  \mathfrak H  ^ {*} $
 +
by the axiom of completeness, one obtains the Bauer space, which coincides with the harmonic space for  $  \mathfrak H  ^ {*} $.  
 +
If the harmonic sheaf $  \mathfrak H $,
 +
for any open set  $  U \subset  \mathbf R  ^ {n} \times \mathbf R $,
 +
consists of the solutions $  h $
 +
of the heat equation $  \Delta h - \partial  h / \partial  t = 0 $,  
 +
then  $  \mathfrak H $
 +
has the Doob convergence property and  $  \mathbf R  ^ {n} \times \mathbf R $
 +
together with this sheaf  $  \mathfrak H $
 +
is a (Bauer)  $  \mathfrak P $-
 +
space. Here,  $  v $
 +
is a superharmonic function of class  $  C  ^ {2} $
 +
if and only if  $  \Delta v - \partial  v / \partial  t \leq  0 $.
  
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.
+
A Brelot space is characterized by the following conditions: $  X $
 +
does not have isolated points and is locally connected; the regular sets with respect to  $  \mathfrak H $
 +
form a base of  $  X $(
 +
regularity means resolutivity of the classical Dirichlet problem in the class  $  \mathfrak H $);
 +
and  $  \mathfrak H $
 +
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  $  \mathfrak H $,
 +
for any open set  $  U \subset  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
consists of the solutions  $  u $
 +
of the Laplace equation  $  \Delta u = 0 $,
 +
then  $  \mathbf R  ^ {2} $
 +
together with this sheaf is a Brelot  $  \mathfrak C $-
 +
space, and  $  \mathbf R  ^ {n} $
 +
for  $  n \geq  3 $
 +
is a Brelot  $  \mathfrak P $-
 +
space. Here,  $  v $
 +
is a hyperharmonic function of class  $  C  ^ {2} $
 +
if and only if  $  \Delta v \leq  0 $.
  
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 point  $  y $
 +
of the boundary  $  \partial  U $
 +
of a resolutive set  $  U $
 +
is called a regular boundary point if for any finite continuous function $  \phi $
 +
on  $  \partial  U $
 +
the following limit relation holds:
  
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" />.
+
$$
 +
\lim\limits _ {x \rightarrow y }  H _  \phi  ( x) = \phi ( y) ,\ \
 +
x \in U ;
 +
$$
  
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:
+
otherwise  $  y $
 +
is called an irregular boundary point. Let  $  F $
 +
be a [[Filter|filter]] on  $  U $
 +
converging to  $  y $.  
 +
A strictly-positive hyperharmonic function  $  v $
 +
defined on the intersection of  $  U $
 +
with some neighbourhood of  $  y $
 +
and converging to  $  0 $
 +
along  $  F $
 +
is called a barrier of the filter  $  F $.  
 +
If for a relatively compact resolutive set $  U $
 +
in a  $  \mathfrak C $-
 +
harmonic space all filters that converge to points  $  y \in \partial  U $
 +
have barriers, then  $  U $
 +
is a regular set, i.e. all its boundary points are regular. If  $  U $
 +
is a relatively compact open set in a  $  \mathfrak P $-
 +
harmonic space on which there exists a strictly-positive hyperharmonic function converging to  $  0 $
 +
at each point  $  y \in \partial  U $,
 +
then  $  U $
 +
is a regular set.
  
<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>
+
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  $  X $;  
 +
the theory of balayage (see [[Balayage method|Balayage method]]) for functions and measures on  $  X $;  
 +
and the theory of integral representations of positive superharmonic functions on  $  X $
 +
generalizing the Martin representations (see [[Martin boundary in potential theory|Martin boundary in potential theory]]).
  
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.
+
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  $  G \subset  \mathbf R  ^ {2} $
 +
starting at the point  $  x _ {0} \in G $
 +
will hit for the first time the boundary  $  \partial  G $
 +
on a (Borel) set $  E \subset  \partial  G $
 +
is exactly the [[Harmonic measure|harmonic measure]] of  $  E $
 +
at  $  x _ {0} $;
 +
the polar sets (cf. [[Polar set|Polar set]]) on  $  \partial  G $
 +
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.
  
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]]).
+
Let  $  X $
 +
be a locally compact space with a countable base, let  $  C _ {c} $
 +
and $  C _ {0} $
 +
be the classes of finite continuous functions on  $  X $,
 +
respectively, with compact support and convergent to zero at infinity. A measure kernel  $  N ( x , E ) \geq  0 $
 +
is a (Borel) function in $  x \in X $
 +
for every relatively compact (Borel) set  $  E \subset  X $.  
 +
Using  $  N $,
 +
to each function  $  f \geq  0 $,
 +
$  f \in C _ {c} $,
 +
corresponds a potential function
  
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.
+
$$
 +
N f ( x) = \int\limits f ( y) N ( x , d y ) ,\ \
 +
x \in X ,
 +
$$
  
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
+
and to a measure $  \theta \geq  0 $
 +
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/p074150211.png" /></td> </tr></table>
+
$$
 +
\theta N ( E)  = \int\limits N ( x , E )  d \theta ( x) .
 +
$$
  
and to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074150/p074150212.png" /> corresponds a potential measure
+
The identity kernel  $  I ( x , E ) $
 +
vanishes when  $  x \notin E $
 +
and is equal to $  1 $
 +
when  $  x \in E $,
 +
it changes neither  $  f ( x) $
 +
nor  $  \theta ( E) $.  
 +
For instance, in the Euclidean space  $  \mathbf R  ^ {3} $
 +
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/p074150213.png" /></td> </tr></table>
+
$$
 +
N ( x , E )  = \int\limits _ { E }
  
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
+
\frac{dy}{| x - y | }
  
<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]]).
+
determines the [[Newton potential|Newton potential]] $  N f $
 +
with density $  f $,  
 +
and $  \theta N $
 +
is the measure with density equal to the density of the Newton potential of the measure $  \theta $(
 +
see [[Potential theory|Potential theory]]).
  
 
A product kernel has the form
 
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>
+
$$
 +
M N ( x , E )  = \int\limits N ( y , E )
 +
M ( x , d y ) .
 +
$$
  
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
+
A family of kernels $  \{ N _ {t} \} $,  
 +
$  t \geq  0 $,  
 +
with the composition law $  N _ {t+} s = N _ {t} N _ {s} $
 +
is a one-parameter semi-group. A kernel $  N $
 +
satisfies the complete maximum principle if for any $  f , g \geq  0 $
 +
from $  C _ {c} $
 +
and  $  a > 0 $
 +
the inequality $  N f \leq  N g + a $
 +
on the set where $  f > 0 $
 +
leads to this inequality everywhere on $  X $.  
 +
The principal theorem in this theory is Hunt's theorem, which in its simplest version is the following: If the image of $  C _ {c} $
 +
under a transformation $  N $
 +
is dense in $  C _ {0} $
 +
and if $  N $
 +
satisfies the complete maximum principle, then there exists a semi-group $  \{ P _ {t} \} $,  
 +
$  t \geq  0 $,  
 +
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>
+
$$
 +
N f ( x)  = \int\limits _ { 0 } ^  \infty  P _ {t} f ( x)  d t ,\ \
 +
f \geq  0
 +
$$
  
(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" />.
+
(a Feller semi-group); moreover, $  P _ {t} $
 +
transforms $  C _ {c} $
 +
into $  C _ {0} $;  
 +
$  P _ {0} $
 +
is the identity kernel; $  \lim\limits _ {t \rightarrow 0 }  P _ {t} f = f $,  
 +
$  f \in C _ {0} $,  
 +
locally uniform; and $  P _ {t} ( 1) \leq  1 $.  
 +
A measurable function $  f \geq  0 $
 +
is called an excessive function with respect to the semi-group $  \{ P _ {t} \} $
 +
if always $  P _ {t} f \leq  f $
 +
and if $  \lim\limits _ {t \rightarrow 0 }  P _ {t} f = f $;  
 +
if $  P _ {t} f = f $,  
 +
then $  f $
 +
is called an invariant function. The corresponding formulas are also valid for the potential measure $  \theta N $.
  
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
+
The theory of Hunt (1957–1958) outlined above has a direct probabilistic sense. Let on $  X $
 +
be given some $  \sigma $-
 +
algebra $  \mathfrak U $
 +
of Borel sets and a probability measure $  {\mathsf P} $.  
 +
A random variable $  S = S ( x) $
 +
is a $  \mathfrak U $-
 +
measurable mapping of $  X $
 +
into the state space $  \overline{\mathbf R}\; = [ - \infty , \infty ] $.  
 +
The family of random variables $  \{ S _ {t} \} $,  
 +
$  t \geq  0 $,  
 +
is a [[Markov process|Markov process]] (for which $  S _ {t} ( x) $
 +
is the trajectory of a point $  x \in X $)  
 +
if for any $  y $,  
 +
$  - \infty \leq  y < \infty $,  
 +
there exists a probability measure $  {\mathsf P}  ^ {y} $
 +
on $  \mathfrak U $
 +
such that a) $  {\mathsf P}  ^ {y} ( \{ S _ {0} = y \} ) = 1 $;  
 +
b) $  {\mathsf P}  ^ {y} ( A) $,  
 +
$  A \in \mathfrak U $,  
 +
is a Borel function of $  y $;  
 +
and c) the form of a trajectory passing through $  y $
 +
at a moment $  r $,  
 +
for $  t \geq  r $,  
 +
is independent of the positions of the points preceding it. In such Markov processes the semi-groups $  \{ {\mathsf P} _ {t} \} $
 +
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>
+
$$
 +
{\mathsf P} _ {t} ( y , B )  = {\mathsf P}  ^ {y}
 +
( \{ S _ {t} \in B \} ) .
 +
$$
  
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.
+
Studies of excessive and invariant functions with respect to the semi-groups $  \{ {\mathsf P} _ {t} \} $
 +
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]]]).
+
On the other hand, if $  X $
 +
is a $  \mathfrak P $-
 +
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.
 
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.
Line 93: Line 401:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
Abstract potential theory is also called axiomatic potential theory.
 
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]].
+
A measure kernel $  N( x, E) $
 +
is also defined as a ( $  \sigma $-
 +
additive, non-negative) measure in $  E $,  
 +
and $  Nf: x \rightarrow Nf( x) $
 +
can be defined whenever $  f $
 +
is Borel and non-negative; in the article above, $  Nf $
 +
is assumed finite if $  f \in C _ {c} $.  
 +
By definition, a Feller semi-group $  ( P _ {t} ) $
 +
transforms $  C _ {0} $
 +
to $  C _ {0} $,  
 +
$  P _ {0} $
 +
is the identity kernel and for $  f \in C _ {0} $
 +
the transformation $  t \rightarrow P _ {t} f $
 +
is continuous at 0 for the uniform topology. If $  N $
 +
and $  ( P _ {t} ) $
 +
are related by Hunt's theorem, $  N $
 +
is called the potential kernel of the Feller semi-group $  ( P _ {t} ) $,  
 +
and for any non-negative Borel function $  f $
 +
the potential $  Nf $
 +
of $  f $
 +
is an excessive function with respect to the semi-group $  ( P _ {t} ) $.  
 +
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 $  X $
 +
equipped with the $  \sigma $-
 +
algebra $  \mathfrak U $
 +
has to be replaced by an extraneous measurable space, say $  ( \Omega , \mathfrak U ) $,  
 +
and the state space $  \overline{\mathbf R}\; $
 +
by the locally compact space $  X $
 +
where the Hunt kernel $  N $
 +
is defined, so that in the equality $  {\mathsf P} _ {t} ( y, B) = {\mathsf P}  ^ {y} ( S _ {t} \in B ) $
 +
one has $  y \in X $
 +
and $  B \subseteq X $(
 +
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]]].
 
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.
+
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 $  H $-
 +
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====
 
====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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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 [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 [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 is of great help. Let $ X $ be a locally compact topological space. A sheaf of functions on $ X $ is a mapping $ \mathfrak F $ defined on the family of all open sets of $ X $ such that

1) $ \mathfrak F ( U) $, for any open set $ U \subset X $, is a family of functions $ u : U \rightarrow \overline{\mathbf R}\; = [ - \infty , \infty ] $;

2) if two open sets $ U , V $ are such that $ U \subset V \subset X $, then the restriction of any function from $ \mathfrak F ( V) $ to $ U $ belongs to $ \mathfrak F ( U) $;

3) if for any family $ \{ U _ {i} \} $, $ i \in I $, of open sets $ U _ {i} \subset X $ the restrictions to $ U _ {i} $ of some function $ u $ defined on $ \cup _ {i \in I } U _ {i} $ belong, for any $ i \in I $, to $ \mathfrak F ( U _ {i} ) $, then $ u \in \mathfrak F ( \cup _ {i \in I } U _ {i} ) $.

A sheaf of functions $ \mathfrak H $ on $ X $ is called a harmonic sheaf if for any open set $ U \subset X $ the family $ \mathfrak H ( U) $ is a real vector space of continuous functions on $ U $. A function $ u $ defined on some set $ S \subset X $ containing the open set $ U $ is called an $ \mathfrak H $- function if the restriction $ u \mid _ {U} $ belongs to $ \mathfrak H ( U) $. A harmonic sheaf is non-degenerate at a point $ x \in X $ if in a neighbourhood of $ x $ there exists an $ \mathfrak H $- function $ u $ such that $ u ( x) \neq 0 $.

The real distinctions between the axiomatics of Bauer, Brelot and Doob can be characterized by the convergence properties of $ \mathfrak H $- functions.

a) Bauer's convergence property states that if an increasing sequence of $ \mathfrak H $- functions is locally bounded on some open set $ U \subset X $, then the limit function $ v $ is an $ \mathfrak H $- function.

b) Doob's convergence property states that if a limit function $ v $ is finite on some dense set $ U \subset X $, then $ v $ is an $ \mathfrak H $- function.

c) Brelot's convergence property states that if the limit function $ v $ of an increasing sequence of $ \mathfrak H $- functions on some domain $ U \subset X $ is finite at a point $ x \in U $, then $ v $ is an $ \mathfrak H $- function.

If the space $ X $ is locally connected, the implications c) $ \Rightarrow $ b) $ \Rightarrow $ a) hold.

A sheaf of functions $ \mathfrak U $ on $ X $ is called a hyperharmonic sheaf if for any open set $ U \subset X $ the family $ \mathfrak U ( U) $ is a convex cone of lower semi-continuous functions $ u : U \rightarrow ( - \infty , \infty ] $; a $ \mathfrak U $- function is defined in a similar way as an $ \mathfrak H $- function. The mapping $ U \rightarrow \mathfrak U ( U) \cap ( - \mathfrak U ( U) ) $ is a harmonic sheaf, denoted by $ \mathfrak H = \mathfrak H _ {\mathfrak U} $ and generated by the sheaf $ \mathfrak U $; hereafter, only this harmonic sheaf will be used.

Let on the boundary $ \partial U $ of an open set $ U \subset X $ a continuous function $ \phi : \partial U \rightarrow ( - \infty , \infty ) $ with compact support be given. The hyperharmonic sheaf $ \mathfrak U $ allows one to construct a generalized solution of the Dirichlet problem for certain open sets in the class of corresponding $ \mathfrak H $- functions by the Perron method. Let $ \overline{\mathfrak U}\; _ \phi $ be a family of lower semi-continuous $ \mathfrak U $- functions $ u $, bounded from below on $ U $, positive outside some compact set and such that

$$ \lim\limits _ {x \rightarrow y } \inf u ( x) \geq \phi ( y) ,\ \ y \in \partial U ; $$

define $ \underline{\mathfrak U} {} _ \phi $ by $ \underline{\mathfrak U} {} _ \phi = - \overline{\mathfrak U}\; _ \phi $. Now, let,

$$ \overline{H}\; _ \phi ( x) = \inf \{ {u ( x) } : { u \in \overline{\mathfrak U}\; _ \phi } \} ,\ \ x \in U , $$

and let $ \overline{H}\; _ \phi = \infty $ if $ \overline{\mathfrak U}\; _ \phi = \emptyset $. Similarly,

$$ \underline{H} {} _ \phi ( x) = \sup \{ {u ( x) } : { u \in \underline{\mathfrak U} {} _ \phi } \} ,\ \ x \in U , $$

or $ \underline{H} {} _ \phi = - \infty $. A function is called resolutive if for this function $ \overline{H}\; _ \phi $ and $ \underline{H} {} _ \phi $ coincide, $ \overline{H}\; _ \phi = \underline{H} {} _ \phi = H _ \phi $, and if $ H _ \phi $ is an $ \mathfrak H $- function; this function $ H _ \phi $ is a generalized solution of the Dirichlet problem in the class of $ \mathfrak H $- functions. An open set $ U \subset X $ is resolutive with respect to $ \mathfrak U $ if every finite continuous function with compact support on $ \partial U $ is resolutive. For a resolutive set $ U $ the mapping $ H _ \phi : C _ {c} ( \partial U ) \rightarrow \mathbf R $ is a positive linear functional, hence it determines a positive measure $ \mu _ {x} $, $ x \in U $, which is called the harmonic measure on $ \partial U $( or on $ U $) at the point $ x $( with respect to $ \mathfrak U $).

A locally compact space $ X $ with a hyperharmonic sheaf $ \mathfrak U $ turns into a harmonic space if the four corresponding axioms (see 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 $ \mathfrak H $, and the axiom of completeness serves then as a definition of a hyperharmonic sheaf. For instance, the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, together with the sheaf of classical solutions of the Laplace equation or of the heat equation as $ \mathfrak H $, 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 $ U $ in a harmonic space $ X $ with the restriction $ \mathfrak U \mid _ {U} $ as hyperharmonic sheaf is a harmonic subspace of $ X $. A hyperharmonic function $ u $ on $ U \subset X $ is called a superharmonic function if for any relatively compact resolutive set $ V $ with $ \overline{V}\; \subset U $, the greatest minorant $ \mu ^ {V} u $ is harmonic, $ \mu ^ {V} u \in \mathfrak H ( V) $. Many properties of classical superharmonic functions (see Subharmonic function) also hold in this case. A potential is a positive superharmonic function $ u $ such that its greatest harmonic minorant on $ X $ is identically equal to zero. A harmonic space $ X $ is called a $ \mathfrak C $- harmonic (or $ \mathfrak P $- harmonic) space if for any point $ x \in X $ there exists a positive superharmonic function $ u $( a potential $ u $, respectively) on $ X $ such that $ u > 0 $. Any open set in a $ \mathfrak P $- harmonic space is resolutive.

Taking a harmonic sheaf $ \mathfrak H $ as basis and defining the corresponding hyperharmonic sheaf $ \mathfrak H ^ {*} $ by the axiom of completeness, one obtains the Bauer space, which coincides with the harmonic space for $ \mathfrak H ^ {*} $. If the harmonic sheaf $ \mathfrak H $, for any open set $ U \subset \mathbf R ^ {n} \times \mathbf R $, consists of the solutions $ h $ of the heat equation $ \Delta h - \partial h / \partial t = 0 $, then $ \mathfrak H $ has the Doob convergence property and $ \mathbf R ^ {n} \times \mathbf R $ together with this sheaf $ \mathfrak H $ is a (Bauer) $ \mathfrak P $- space. Here, $ v $ is a superharmonic function of class $ C ^ {2} $ if and only if $ \Delta v - \partial v / \partial t \leq 0 $.

A Brelot space is characterized by the following conditions: $ X $ does not have isolated points and is locally connected; the regular sets with respect to $ \mathfrak H $ form a base of $ X $( regularity means resolutivity of the classical Dirichlet problem in the class $ \mathfrak H $); and $ \mathfrak H $ has the Brelot convergence property. The Brelot spaces form a proper subclass of the so-called elliptic harmonic spaces (see [4]), i.e. the elliptic Bauer spaces. If the harmonic sheaf $ \mathfrak H $, for any open set $ U \subset \mathbf R ^ {n} $, $ n \geq 2 $, consists of the solutions $ u $ of the Laplace equation $ \Delta u = 0 $, then $ \mathbf R ^ {2} $ together with this sheaf is a Brelot $ \mathfrak C $- space, and $ \mathbf R ^ {n} $ for $ n \geq 3 $ is a Brelot $ \mathfrak P $- space. Here, $ v $ is a hyperharmonic function of class $ C ^ {2} $ if and only if $ \Delta v \leq 0 $.

A point $ y $ of the boundary $ \partial U $ of a resolutive set $ U $ is called a regular boundary point if for any finite continuous function $ \phi $ on $ \partial U $ the following limit relation holds:

$$ \lim\limits _ {x \rightarrow y } H _ \phi ( x) = \phi ( y) ,\ \ x \in U ; $$

otherwise $ y $ is called an irregular boundary point. Let $ F $ be a filter on $ U $ converging to $ y $. A strictly-positive hyperharmonic function $ v $ defined on the intersection of $ U $ with some neighbourhood of $ y $ and converging to $ 0 $ along $ F $ is called a barrier of the filter $ F $. If for a relatively compact resolutive set $ U $ in a $ \mathfrak C $- harmonic space all filters that converge to points $ y \in \partial U $ have barriers, then $ U $ is a regular set, i.e. all its boundary points are regular. If $ U $ is a relatively compact open set in a $ \mathfrak P $- harmonic space on which there exists a strictly-positive hyperharmonic function converging to $ 0 $ at each point $ y \in \partial U $, then $ U $ 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 of point sets in harmonic spaces $ X $; the theory of balayage (see Balayage method) for functions and measures on $ X $; and the theory of integral representations of positive superharmonic functions on $ X $ generalizing the Martin representations (see 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; Wiener process; and Markov process. For instance, the probability that the trajectory of a Brownian motion in a domain $ G \subset \mathbf R ^ {2} $ starting at the point $ x _ {0} \in G $ will hit for the first time the boundary $ \partial G $ on a (Borel) set $ E \subset \partial G $ is exactly the harmonic measure of $ E $ at $ x _ {0} $; the polar sets (cf. Polar set) on $ \partial G $ 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 $ X $ be a locally compact space with a countable base, let $ C _ {c} $ and $ C _ {0} $ be the classes of finite continuous functions on $ X $, respectively, with compact support and convergent to zero at infinity. A measure kernel $ N ( x , E ) \geq 0 $ is a (Borel) function in $ x \in X $ for every relatively compact (Borel) set $ E \subset X $. Using $ N $, to each function $ f \geq 0 $, $ f \in C _ {c} $, corresponds a potential function

$$ N f ( x) = \int\limits f ( y) N ( x , d y ) ,\ \ x \in X , $$

and to a measure $ \theta \geq 0 $ corresponds a potential measure

$$ \theta N ( E) = \int\limits N ( x , E ) d \theta ( x) . $$

The identity kernel $ I ( x , E ) $ vanishes when $ x \notin E $ and is equal to $ 1 $ when $ x \in E $, it changes neither $ f ( x) $ nor $ \theta ( E) $. For instance, in the Euclidean space $ \mathbf R ^ {3} $ the kernel

$$ N ( x , E ) = \int\limits _ { E } \frac{dy}{| x - y | } $$

determines the Newton potential $ N f $ with density $ f $, and $ \theta N $ is the measure with density equal to the density of the Newton potential of the measure $ \theta $( see Potential theory).

A product kernel has the form

$$ M N ( x , E ) = \int\limits N ( y , E ) M ( x , d y ) . $$

A family of kernels $ \{ N _ {t} \} $, $ t \geq 0 $, with the composition law $ N _ {t+} s = N _ {t} N _ {s} $ is a one-parameter semi-group. A kernel $ N $ satisfies the complete maximum principle if for any $ f , g \geq 0 $ from $ C _ {c} $ and $ a > 0 $ the inequality $ N f \leq N g + a $ on the set where $ f > 0 $ leads to this inequality everywhere on $ X $. The principal theorem in this theory is Hunt's theorem, which in its simplest version is the following: If the image of $ C _ {c} $ under a transformation $ N $ is dense in $ C _ {0} $ and if $ N $ satisfies the complete maximum principle, then there exists a semi-group $ \{ P _ {t} \} $, $ t \geq 0 $, such that

$$ N f ( x) = \int\limits _ { 0 } ^ \infty P _ {t} f ( x) d t ,\ \ f \geq 0 $$

(a Feller semi-group); moreover, $ P _ {t} $ transforms $ C _ {c} $ into $ C _ {0} $; $ P _ {0} $ is the identity kernel; $ \lim\limits _ {t \rightarrow 0 } P _ {t} f = f $, $ f \in C _ {0} $, locally uniform; and $ P _ {t} ( 1) \leq 1 $. A measurable function $ f \geq 0 $ is called an excessive function with respect to the semi-group $ \{ P _ {t} \} $ if always $ P _ {t} f \leq f $ and if $ \lim\limits _ {t \rightarrow 0 } P _ {t} f = f $; if $ P _ {t} f = f $, then $ f $ is called an invariant function. The corresponding formulas are also valid for the potential measure $ \theta N $.

The theory of Hunt (1957–1958) outlined above has a direct probabilistic sense. Let on $ X $ be given some $ \sigma $- algebra $ \mathfrak U $ of Borel sets and a probability measure $ {\mathsf P} $. A random variable $ S = S ( x) $ is a $ \mathfrak U $- measurable mapping of $ X $ into the state space $ \overline{\mathbf R}\; = [ - \infty , \infty ] $. The family of random variables $ \{ S _ {t} \} $, $ t \geq 0 $, is a Markov process (for which $ S _ {t} ( x) $ is the trajectory of a point $ x \in X $) if for any $ y $, $ - \infty \leq y < \infty $, there exists a probability measure $ {\mathsf P} ^ {y} $ on $ \mathfrak U $ such that a) $ {\mathsf P} ^ {y} ( \{ S _ {0} = y \} ) = 1 $; b) $ {\mathsf P} ^ {y} ( A) $, $ A \in \mathfrak U $, is a Borel function of $ y $; and c) the form of a trajectory passing through $ y $ at a moment $ r $, for $ t \geq r $, is independent of the positions of the points preceding it. In such Markov processes the semi-groups $ \{ {\mathsf P} _ {t} \} $ are interpreted as semi-groups of measures

$$ {\mathsf P} _ {t} ( y , B ) = {\mathsf P} ^ {y} ( \{ S _ {t} \in B \} ) . $$

Studies of excessive and invariant functions with respect to the semi-groups $ \{ {\mathsf P} _ {t} \} $ are of great importance.

On the other hand, if $ X $ is a $ \mathfrak P $- 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 [4], [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

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

Comments

Abstract potential theory is also called axiomatic potential theory.

A measure kernel $ N( x, E) $ is also defined as a ( $ \sigma $- additive, non-negative) measure in $ E $, and $ Nf: x \rightarrow Nf( x) $ can be defined whenever $ f $ is Borel and non-negative; in the article above, $ Nf $ is assumed finite if $ f \in C _ {c} $. By definition, a Feller semi-group $ ( P _ {t} ) $ transforms $ C _ {0} $ to $ C _ {0} $, $ P _ {0} $ is the identity kernel and for $ f \in C _ {0} $ the transformation $ t \rightarrow P _ {t} f $ is continuous at 0 for the uniform topology. If $ N $ and $ ( P _ {t} ) $ are related by Hunt's theorem, $ N $ is called the potential kernel of the Feller semi-group $ ( P _ {t} ) $, and for any non-negative Borel function $ f $ the potential $ Nf $ of $ f $ is an excessive function with respect to the semi-group $ ( P _ {t} ) $. 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 $ X $ equipped with the $ \sigma $- algebra $ \mathfrak U $ has to be replaced by an extraneous measurable space, say $ ( \Omega , \mathfrak U ) $, and the state space $ \overline{\mathbf R}\; $ by the locally compact space $ X $ where the Hunt kernel $ N $ is defined, so that in the equality $ {\mathsf P} _ {t} ( y, B) = {\mathsf P} ^ {y} ( S _ {t} \in B ) $ one has $ y \in X $ and $ B \subseteq X $( and Borel). See 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. See [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. [a2], and the theory of $ H $- cones, cf. [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 [a4] for short surveys.

References

[a1] N. Boboc, Gh. Bucur, A. Cornea, "Order and convexity in potential theory: -cones" , Springer (1981)
[a2] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a3] M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980)
[a4] J. Král (ed.) J. Lukeš (ed.) J. Veselý (ed.) , Potential theory. Survey and problems (Prague, 1987) , Lect. notes in math. , 1344 , Springer (1988)
[a5] M. Brelot (ed.) H. Bauer (ed.) J.-M. Bony (ed.) J. Deny (ed.) G. Mokobodzki (ed.) , Potential theory (CIME, Stresa, 1969) , Cremonese (1970)
[a6] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French)
[a7] F.Y. Maeda, "Dirichlet integrals on harmonic spaces" , Lect. notes in math. , 803 , Springer (1980)
[a8] M. Röckner, "Markov property of generalized fields and axiomatic potential theory" Math. Ann. , 264 (1983) pp. 153–177
How to Cite This Entry:
Potential theory, abstract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_theory,_abstract&oldid=14793
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article