Excessive function

for a Markov process

The analogue of a non-negative superharmonic function.

Suppose that in a measurable space $ ( E , {\mathcal B} ) $ a homogeneous Markov chain is given with single-step transition probabilities $ {\mathsf P} ( x , B ) $( $ x \in E $, $ B \in {\mathcal B} $). A measurable function $ f : E \rightarrow [ 0 , \infty ] $ relative to $ {\mathcal B} $ is said to be excessive for this chain if

$$ \int\limits _ { E } f ( y) {\mathsf P} ( x , d y ) \leq f ( x) $$

everywhere in $ E $. 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 $ X = ( x _ {t} , \zeta , {\mathcal F} _ {t} , {\mathsf P} _ {x} ) $ in $ ( E , {\mathcal B} ) $ with transition function $ P ( t , x , B ) $, the definition of an excessive function is somewhat more complicated. A set $ B $ belongs to the $ \sigma $- algebra $ \overline {\mathcal B} \; $ if for any finite measure $ \mu $ on $ {\mathcal B} $ one can find sets $ B _ \mu ^ {1} $ and $ B _ \mu ^ {2} $ such that $ B _ \mu ^ {1} \subset B \subset B _ \mu ^ {2} $ and $ \mu ( B _ \mu ^ {2} \setminus B _ \mu ^ {1} ) = 0 $. A function $ f : E \rightarrow [ 0 , \infty ] $ is said to be excessive if it is $ \overline {\mathcal B} \; $- measurable and if for $ t \geq 0 $ everywhere in $ E $:

$$ P ^ {t} f ( x) = \int\limits f ( y) P ( t , x , d y ) \leq f ( x ) , $$

and

$$ f ( x) = \lim\limits _ {s \downarrow 0 } P ^ {s} f ( x) . $$

For the part of a Wiener process in a certain domain $ E \subset \mathbf R ^ {n} $( see Functional of a Markov process) the class of excessive functions is the same as that of superharmonic functions supplemented by $ f ( x) \equiv \infty $.

In the case of a standard process $ X $ in a locally compact separable space $ E $ the inequality

$$ M _ {x} f ( x _ \tau ) \leq f ( x ) , $$

for an excessive function $ f ( x) $, is satisfied throughout $ E $, where $ \tau $ is the Markov moment, $ M _ {x} $ is the mathematical expectation corresponding to the measure $ {\mathsf P} _ {x} $ and $ f ( x _ \tau ) = 0 $ for $ \tau \geq \zeta $. Another frequently used property of an excessive function is that $ {\mathsf P} _ {x} $- almost certainly the function $ f ( x _ {t} ) $ is right continuous on the interval $ [ 0 , \zeta ] $( see [3]).

An excessive function $ f ( x) < \infty $ is called harmonic if $ f ( x) \equiv M _ {x} f ( x _ \tau ) $, where $ \tau $ is the first exit time of $ X $ from $ K $, $ K \subset E $ being any given compact set. A potential is, by definition, any excessive function $ f ( x) < \infty $ for which

$$ \lim\limits _ {n \rightarrow \infty } M _ {x} f ( x _ {\tau _ {n} } ) = 0 $$

for any choice of Markov moments $ \tau _ {n} $, $ n \geq 1 $, such that $ \tau _ {n} \rightarrow \zeta $, as $ n \rightarrow \infty $. For the part of a Wiener process in a domain $ E \subset \mathbf R ^ {n} $ harmonic functions and potentials are, respectively, non-negative harmonic functions on $ E $ in the classical sense and Green potentials of Borel measures concentrated on $ E $.

An example of a potential is the potential $ M _ {x} \gamma _ \zeta $ of an additive functional $ \gamma _ {t} \geq 0 $ in $ X $, provided that $ M _ {x} \gamma _ \zeta < \infty $. An excessive function $ f ( x) < \infty $ is the potential of an additive functional if and only if

$$ \lim\limits _ {n \rightarrow \infty } M _ {x} f ( x _ {\tau _ {n} } ) = 0 , $$

where $ \tau _ {n} $ is the first entry time of the set $ \{ {x } : {f ( x) \geq n } \} $.

Within the framework of Brélot's axiomatic theory of harmonic spaces all non-negative superharmonic functions are excessive for some standard process.

References

[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
[2]	A.N. Shiryaev, "Statistical sequential analysis" , Amer. Math. Soc. (1973) (Translated from Russian)
[3]	E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian)
[4]	R.K. Getoor, "Markov processes: Ray processes and right processes" , Lect. notes in math. , 440 , Springer (1975)
[5]	M.G. Shur, "Functions harmonic for a Markov process" Math. Notes , 13 (1973) pp. 355–359 Mat. Zametki , 13 : 4 (1973) pp. 587–596
[6]	P.A. Meyer, "Fonctionelles multiplicatives et additives de Markov" Ann. Inst. Fourier , 12 (1962) pp. 125–230
[7]	P.A. Meyer, "Brélot's axiomatic theory of the Dirichlet problem and Hunt's theory" Ann. Inst. Fourier , 13 (1963) pp. 357–372

Comments

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].

References

[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)