# Transition-operator semi-group

2010 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