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 be the transition function of a homogeneous Markov process , given on a separable, locally compact space , where , , , and is the family of Borel sets in . A function defined for , , , which is -measurable for fixed is called a Green's function if for each ,

where is a measure on . To avoid ambiguity in the definition of a Green's function, it can be required in addition that for any continuous function with compact support, the function

is -continuous (meaning that there exists a function which is left continuous in and such that

Fixing a measure in and postulating the existence of a Green's function, one defines the Martin kernel

where

(here some restrictions must be introduced to ensure, in particular, the positivity and -continuity of ). If is the unit measure concentrated at some point and is a Wiener process terminating at the first exit time for some domain, then the definition of reduces to an analogous form [1]. Under broad conditions one can establish the existence of a compact set (the "Martin compactum" ), a measure on (, ) and a mapping for which: a) is dense in ; b) the function

separates points and is continuous on as runs through all continuous function in with compact support; and c) the measure coincides with if . The boundary of the set in 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 it is convenient to invoke -processes in the sense of Doob: to each excessive function is associated the transition function

on , where and ; the corresponding Markov process is an -process. All -processes can be realized, together with , on the space of elementary events, so that they are distinguished only by the families of measures . One constructs in a modification of , a left-continuous process () for which if . In the topology of the limit exists almost certainly.

There is a set (the "exit space" ) such that: first, for all of the above form; secondly, the measure for has a density with respect to , where one can take for an excessive function whose spectral measure is the unit measure concentrated at ; and thirdly, admits a unique integral decomposition of the form

The measure in the decomposition is called the spectral measure of the function ; it is given by the formula

where is a Borel set in .

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

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

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)