for a Markov process
The analogue of a non-negative superharmonic function.
Suppose that in a measurable space a homogeneous Markov chain is given with single-step transition probabilities (, ). A measurable function relative to is said to be excessive for this chain if
everywhere in . For an indecomposable chain with an at most countable set of states, among the excessive functions there exist non-constant ones if and only if at least one of the states is non-recurrent.
For a given homogeneous Markov process in with transition function , the definition of an excessive function is somewhat more complicated. A set belongs to the -algebra if for any finite measure on one can find sets and such that and . A function is said to be excessive if it is -measurable and if for everywhere in :
For the part of a Wiener process in a certain domain (see Functional of a Markov process) the class of excessive functions is the same as that of superharmonic functions supplemented by .
In the case of a standard process in a locally compact separable space the inequality
for an excessive function , is satisfied throughout , where is the Markov moment, is the mathematical expectation corresponding to the measure and for . Another frequently used property of an excessive function is that -almost certainly the function is right continuous on the interval (see ).
An excessive function is called harmonic if , where is the first exit time of from , being any given compact set. A potential is, by definition, any excessive function for which
for any choice of Markov moments , , such that , as . For the part of a Wiener process in a domain harmonic functions and potentials are, respectively, non-negative harmonic functions on in the classical sense and Green potentials of Borel measures concentrated on .
An example of a potential is the potential of an additive functional in , provided that . An excessive function is the potential of an additive functional if and only if
where is the first entry time of the set .
Within the framework of Brélot's axiomatic theory of harmonic spaces all non-negative superharmonic functions are excessive for some standard process.
|[1a]||G.A. Hunt, "Markov processes and potentials I" Illinois J. Math. , 1 : 1 (1957) pp. 44–93|
|[1b]||G.A. Hunt, "Markov processes and potentials II" Illinois J. Math. , 1 : 3 (1957) pp. 316–369|
|[1c]||G.A. Hunt, "Markov processes and potentials III" Illinois J. Math. , 2 : 2 (1958) pp. 151–213|
|||A.N. Shiryaev, "Statistical sequential analysis" , Amer. Math. Soc. (1973) (Translated from Russian)|
|||E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian)|
|||R.K. Getoor, "Markov processes: Ray processes and right processes" , Lect. notes in math. , 440 , Springer (1975)|
|||M.G. Shur, "Functions harmonic for a Markov process" Math. Notes , 13 (1973) pp. 355–359 Mat. Zametki , 13 : 4 (1973) pp. 587–596|
|||P.A. Meyer, "Fonctionelles multiplicatives et additives de Markov" Ann. Inst. Fourier , 12 (1962) pp. 125–230|
|||P.A. Meyer, "Brélot's axiomatic theory of the Dirichlet problem and Hunt's theory" Ann. Inst. Fourier , 13 (1963) pp. 357–372|
The definition of an excessive function and its properties are due to G.A. Hunt . Another definition via resolvents is used in R.M. Blumenthal and R.K. Getoor [a1]. More recent references are [a2]–[a4]. For Brélot's theory of harmonic spaces, see [a5].
|[a1]||R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968)|
|[a2]||M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980)|
|[a3]||K.L. Chung, "Lectures from Markov processes to Brownian motion" , Springer (1982)|
|[a4]||J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390|
|[a5]||M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)|
Excessive function. M.G. Shur (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Excessive_function&oldid=15609