Transition-operator semi-group

2020 Mathematics Subject Classification: Primary: 60J35 Secondary: 47D07 [MSN][ZBL]

The semi-group of operators generated by the transition function of a Markov process. From the transition function $ P( t, x, A) $ of a homogeneous Markov process $ X = ( x _ {t} , \zeta , {\mathcal F} _ {t} , {\mathsf P} _ {x} ) $ in a state space $ ( E, {\mathcal B}) $ one can construct certain semi-groups of linear operators $ P ^ {t} $ acting in some Banach space $ B $[F]. Very often, $ B $ is the space $ B ( E) $ of bounded real-valued measurable functions $ f $ in $ E $ with the uniform norm (or for a Feller process $ X $, the space $ C ( E) $ of continuous functions with the same norm) or else the space $ V( E) $ of finite countably-additive functions $ \phi $ on $ {\mathcal B} $ with the complete variation as norm. In the first two cases one puts

$$ P ^ {t} f( x) = \int\limits _ { E } f( y) {\mathsf P} ( t, x, dy); $$

and in the third

$$ P ^ {t} \phi ( A) = \int\limits _ { E } {\mathsf P} ( t, y, A) \phi ( dy) $$

(here $ f $ and $ \phi $ belong to the corresponding spaces, $ x \in E $, $ A \in {\mathcal B} $). In all these cases the semi-group property holds: $ P ^ {t} P ^ {s} = P ^ {t+} s $, $ s, t \geq 0 $, and any of the three semi-groups $ \{ P ^ {t} \} $ is called a transition-operator semi-group.

In what follows, only the first case is considered. The usual definition of the infinitesimal generator $ A $ of the semi-group $ \{ P ^ {t} \} $( this is also the infinitesimal generator of the process) is as follows:

$$ Af = \lim\limits _ {t \downarrow 0 } \frac{1}{t} ( P ^ {t} f - f ) $$

for all $ f \in B ( E) $ for which this limit exists as a limit in $ B ( E) $. It is assumed that $ P( t, x, A) $ for $ A \in {\mathcal B} $ is a measurable function of the pair of variables $ ( t, x) $, and one introduces the resolvent $ R ^ \alpha $ of the process $ X $, $ \alpha > 0 $, by:

$$ \tag{* } R ^ \alpha f = \int\limits _ { 0 } ^ \infty e ^ {- \alpha t } P ^ {t} f dt ,\ \ f \in B ( E). $$

If $ \| P ^ {t} f- f \| \rightarrow 0 $ as $ t \downarrow 0 $, then $ Ag = \alpha g - f $, where $ g = R ^ \alpha f $. Under certain assumptions the integral (*) exists also for $ \alpha = 0 $, and $ g = R ^ {0} f $ satisfies the "Poisson equation"

$$ Ag = - f $$

(for this reason, in particular, $ R ^ {0} f $ is called the potential of $ f $).

Knowledge of the infinitesimal generator enables one to derive important characteristics of the initial process; the classification of Markov processes amounts to the description of their corresponding infinitesimal generators [Dy], [GS]. Also, using the infinitesimal generator one can find the mean values of various functionals. For example, under certain assumptions the function

$$ v( t, x) = {\mathsf E} _ {x} \left [ \mathop{\rm exp} \left \{ \int\limits _ { 0 } ^ { {t } \wedge \zeta } c( x _ {s} ) ds \right \} f( x _ {t \wedge \zeta } ) \right ] ,\ \ t \geq 0,\ x \in E, $$

is a unique solution to $ v _ {t} ^ \prime = Av + cv $, $ v( 0, x) = f( x) $, which is a not-too-rapidly-increasing function of $ t $. Here $ {\mathsf E} _ {x} $ is the mathematical expectation corresponding to $ {\mathsf P} _ {x} $, while $ t \wedge \zeta = \min ( t, \zeta ) $.

The operator $ A $ is related to the characteristic operator $ \mathfrak A $[Dy]. Let $ X $ be a Markov process that is right continuous in a topological space $ E $. For a Borel function $ f $ one puts

$$ \mathfrak A f( x) = \lim\limits _ {U \downarrow x } \left [ \frac{ {\mathsf E} _ {x} f( x _ \tau ) - f( x) }{ {\mathsf E} _ {x} \tau } \right ] , $$

if the limit exists for all $ x \in E $, where $ U $ runs through a system of neighbourhoods of the point $ x $ contracting towards $ x $ and where $ \tau $ is the moment of first exit of $ X $ from $ U $( if $ {\mathsf E} _ {x} \tau = \infty $, the fraction in the limit is set equal to zero). In many cases the calculation of $ Af $ amounts to calculating $ \mathfrak A f $.

References

[F]	W. Feller, "The parabolic differential equations and the associated semi-groups of transformations" Ann. of Math. , 55 (1952) pp. 468–519 MR0047886
[Dy]	E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) MR0131898
[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027

References

[BG]	R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) MR0264757 Zbl 0169.49204
[Do]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[Dy2]	E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[F2]	W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966) MR0210154 Zbl 0138.10207
[L]	M. Loève, "Probability theory" , II , Springer (1978) MR0651017 MR0651018 Zbl 0385.60001
[DM]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001
[S]	M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) MR0958914 Zbl 0649.60079
[AM]	S. Albeverio, Zh.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400