Martin boundary in the theory of Markov processes

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

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

Let $ P ( t , x , B ) $ be the 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 terminating at the first exit time for some domain, then the definition of $ K _ {y} ^ {0} ( x) $ reduces to an analogous form [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 [3], [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 $ 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

[1]	R.S. Martin, "Minimal positive harmonic functions" Trans. Amer. Math. Soc. , 49 (1941) pp. 137–172
[2]	M. Motoo, "Application of additive functionals to the boundary problem of Markov processes. Lévy's system of -processes" , Proc. 5-th Berkeley Symp. Math. Stat. Probab. , 2 : 2 (1967) pp. 75–110
[3]	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
[4]	J.L. Doob, "Discrete potential theory and boundaries" J. Math. and Mech. , 8 : 3 (1959) pp. 433–458; 993
[5]	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
[6]	G.A. Hunt, "Markov chains and Markov boundaries" Illinois J. Math. , 4 (1960) pp. 313–340
[7]	P.L. Hennequin, A. Tortrat, "Théorie des probabilites et quelques applications" , Masson (1965)
[8]	H. Kunita, T. Watanabe, "Markov processes and Martin boundaries I" Illinois J. Math. , 9 : 3 (1965) pp. 485–526
[9]	M.G. Shur, Trudy Moskov. Inst. Elektron. Mashinostr. , 5 (1970) pp. 192–251
[10]	T. Jeulin, "Compactification de Martin d'un processus droit" Z. Wahrsch. Verw. Gebiete , 42 : 3 (1978) pp. 229–260
[11]	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

Comments

One of the other types of compactifications used in the theory of Markov processes is the Ray–Knight compactification.

References

[a1]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984)