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The boundary of the state space of a [[Markov process|Markov process]] or of its image in some compact space, constructed by a scheme similar to the Martin scheme (see [[#References|[1]]]).

A probabilistic interpretation of Martin's construction was first proposed by J.L. Doob (see [[#References|[4]]]), who discussed the case of discrete Markov chains.

Let  $  P ( t , x , B ) $
be the [[Transition function|transition function]] of a homogeneous Markov process  $  X = ( x _ {t} , \zeta , F _ {t} , {\mathsf P} _ {x} ) $, 
given on a separable, locally compact space  $  E $, 
where  $  t \geq  0 $, 
$  x \in E $, 
$  B \in {\mathcal B} $, 
and  $  {\mathcal B} $
is the family of Borel sets in  $  E $. 
A function  $  g _  \alpha  ( x , y ) \geq  0 $
defined for  $  \alpha \geq  0 $, 
$  x \in E $, 
$  y \in E $, 
which is  $  ( {\mathcal B} \times {\mathcal B} ) $-
measurable for fixed  $  \alpha $
is called a Green's function if for each  $  B \in {\mathcal B} $,

$$ 
\int\limits _ { B } g _  \alpha  ( x , y ) m ( d y )  \equiv \ 
\int\limits _ { 0 } ^  \infty   e ^ {- \alpha t } P ( t , x , B )  d t ,
$$

where  $  m $
is a measure on  $  {\mathcal B} $. 
To avoid ambiguity in the definition of a Green's function, it can be required in addition that for any continuous function  $  f ( x) $
with compact support, the function

$$ 
g _  \alpha  ( \cdot )  = \ 
\int\limits _ { E } f ( x)
g _  \alpha  ( x , \cdot )
m ( d x )
$$

is  $  \Lambda $-
continuous (meaning that there exists a function  $  F ( t , \omega ) $
which is left continuous in  $  t $
and such that

$$ 
{\mathsf P} _ {x} \{ F ( t , \omega ) \neq
g _  \alpha  ( x _ {t} ( \omega ) ) \}  \equiv  0 ,\ \ 
x \in E ,\  t > 0 \textrm{ )  } .
$$

Fixing a measure  $  \gamma $
in  $  {\mathcal B} $
and postulating the existence of a Green's function, one defines the Martin kernel

$$ 
K _ {y}  ^  \alpha  ( x)  = \ 

\frac{g _  \alpha  ( x , y ) }{q ( y ) }
 ,
$$

where

$$ 
q ( y)  =  \int\limits _ { E }
g _  \alpha  ( x , y )
\gamma ( d x )
$$

(here some restrictions must be introduced to ensure, in particular, the positivity and  $  \Lambda $-
continuity of  $  q ( y) $). 
If  $  \gamma $
is the unit measure concentrated at some point and  $  X $
is a [[Wiener process|Wiener process]] terminating at the first exit time for some domain, then the definition of  $  K _ {y}  ^ {0} ( x) $
reduces to an analogous form [[#References|[1]]]. Under broad conditions one can establish the existence of a compact set  $  {\mathcal E} $(
the  "Martin compactum" ), a measure  $  K _ {y}  ^  \alpha  ( d x ) $
on  $  {\mathcal B} $(
$  \alpha \geq  0 $, 
$  y \in {\mathcal E} $) 
and a mapping  $  i :  E \rightarrow {\mathcal E} $
for which: a)  $  i ( E) $
is dense in  $  {\mathcal E} $; 
b) the function

$$ 
K _ {y}  ^  \alpha  ( f  )  = \ 
\int\limits _  {\mathcal B}  f ( x) K _ {y}  ^  \alpha  ( d x )
$$

separates points and is continuous on  $  {\mathcal E} $
as  $  f $
runs through all continuous function in  $  E $
with compact support; and c) the measure  $  K _ {i ( y) }   ^  \alpha  ( d x ) $
coincides with  $  K _ {y}  ^  \alpha  ( x) m ( d x ) $
if  $  y \in E $. 
The boundary of the set  $  i ( E) $
in  $  {\mathcal E} $
is called the Martin boundary or exit-boundary (in the study of decompositions of excessive measures the dual object, the entrance-boundary, arises; see [[#References|[3]]], [[#References|[4]]]).

In order to describe the properties of  $  {\mathcal E} $
it is convenient to invoke  $  h $-
processes in the sense of Doob: to each [[Excessive function|excessive function]]  $  h $
is associated the transition function

$$ 
P  ^ {h} ( t , x , B )  = \ 
h  ^ {-} 1 ( x) \int\limits _ { E }
h ( y) P ( t , x , d y )
$$

on  $  ( E  ^ {h} , {\mathcal B}  ^ {h} ) $, 
where  $  E  ^ {h} = \{ {x \in E } : {0 < h ( x) < \infty } \} $
and  $  {\mathcal B}  ^ {h} = \{ {A \in B } : {A \subset  E  ^ {h} } \} $; 
the corresponding Markov process is an  $  h $-
process. All  $  h $-
processes can be realized, together with  $  X $, 
on the space of elementary events, so that they are distinguished only by the families of measures  $  \{ {\mathsf P} _ {x}  ^ {h} \} $. 
One constructs in  $  {\mathcal E} $
a modification of  $  x _ {t} $, 
a left-continuous process  $  z _ {t} $(
$  0 < t \leq  \zeta $) 
for which  $  {\mathsf P} _ {x}  ^ {h} \{ z _ {t} \neq i ( x _ {t} ) \} \equiv 0 $
if  $  h \in L _ {1} ( \gamma ) $. 
In the topology of  $  {\mathcal E} $
the limit  $  z _  \zeta  = \lim\limits _ {t \uparrow \zeta }   z _ {t} $
exists almost certainly.

There is a set  $  U \subset  {\mathcal E} $(
the  "exit space" ) such that: first,  $  {\mathsf P} _ {x}  ^ {h} \{ z _  \zeta  \in U \} \equiv 1 $
for all  $  h ( x) $
of the above form; secondly, the measure  $  K _ {y}  ^  \alpha  $
for  $  y \in U $
has a density  $  k _ {y}  ^  \alpha  ( \cdot ) $
with respect to  $  m $, 
where one can take for  $  k _ {y}  ^ {0} ( \cdot ) $
an excessive function whose spectral measure is the unit measure concentrated at  $  y $; 
and thirdly,  $  h ( x) $
admits a unique integral decomposition of the form

$$ 
h ( x)  =  \int\limits _ { U } k _ {y}  ^ {0} ( x) \mu ( d x ) .
$$

The measure  $  \mu $
in the decomposition is called the spectral measure of the function  $  h $; 
it is given by the formula

$$ 
\mu ( B)  =  \int\limits _ { E }
h ( x) {\mathsf P} _ {x}  ^ {h}
\{ z _  \zeta  \in B \}
\gamma ( d x ) ,
$$

where  $  B $
is a Borel set in  $  {\mathcal E} $.

In the theory of Markov processes other types of compactifications are also used, particularly those in which any function of the form

$$ 
\int\limits _ { 0 } ^  \infty  
e ^ {- \alpha t } \ 
d t \int\limits _ { E }
f ( y) P ( t , x , d y ) ,\ \ 
\alpha > 0 ,\  x \in E ,
$$

has a continuous extension for a sufficiently general set of functions  $  f $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.S. Martin,   "Minimal positive harmonic functions"  ''Trans. Amer. Math. Soc.'' , '''49'''  (1941)  pp. 137–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Motoo,   "Application of additive functionals to the boundary problem of Markov processes. Lévy's system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062560/m06256090.png" />-processes" , ''Proc. 5-th Berkeley Symp. Math. Stat. Probab.'' , '''2''' :  2  (1967)  pp. 75–110</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Kunita,   T. Watanabe,   "Some theorems concerning resolvents over locally compact spaces" , ''Proc. 5-th Berkeley Symp. Math. Stat. Probab.'' , '''2''' :  2  (1967)  pp. 131–164</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Doob,   "Discrete potential theory and boundaries"  ''J. Math. and Mech.'' , '''8''' :  3  (1959)  pp. 433–458; 993</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T. Watanabe,   "On the theory of Martin boundaries induced by countable Markov processes"  ''Mem. Coll. Sci. Kyoto Univ. Ser. A'' , '''33''' :  1  (1960)  pp. 39–108</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.A. Hunt,   "Markov chains and Markov boundaries"  ''Illinois J. Math.'' , '''4'''  (1960)  pp. 313–340</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.L. Hennequin,   A. Tortrat,   "Théorie des probabilites et quelques applications" , Masson  (1965)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Kunita,   T. Watanabe,   "Markov processes and Martin boundaries I"  ''Illinois J. Math.'' , '''9''' :  3  (1965)  pp. 485–526</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.G. Shur,   ''Trudy Moskov. Inst. Elektron. Mashinostr.'' , '''5'''  (1970)  pp. 192–251</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  T. Jeulin,   "Compactification de Martin d'un processus droit"  ''Z. Wahrsch. Verw. Gebiete'' , '''42''' :  3  (1978)  pp. 229–260</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  E.B. Dynkin,   "Boundary theory of Markov processes (the discrete case)"  ''Russian Math. Surveys'' , '''24''' :  2  (1969)  pp. 1–42  ''Uspekhi Mat. Nauk'' , '''24''' :  4  (1969)  pp. 89–152</TD></TR></table>

====Comments====
One of the other types of compactifications used in the theory of Markov processes is the [[Ray–Knight compactification|Ray–Knight compactification]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)</TD></TR></table>