The semi-group of operators generated by the transition function of a Markov process. From the transition function of a homogeneous Markov process in a state space one can construct certain semi-groups of linear operators acting in some Banach space . Very often, is the space of bounded real-valued measurable functions in with the uniform norm (or for a Feller process , the space of continuous functions with the same norm) or else the space of finite countably-additive functions on with the complete variation as norm. In the first two cases one puts
and in the third
(here and belong to the corresponding spaces, , ). In all these cases the semi-group property holds: , , and any of the three semi-groups is called a transition-operator semi-group.
In what follows, only the first case is considered. The usual definition of the infinitesimal generator of the semi-group (this is also the infinitesimal generator of the process) is as follows:
for all for which this limit exists as a limit in . It is assumed that for is a measurable function of the pair of variables , and one introduces the resolvent of the process , , by:
If as , then , where . Under certain assumptions the integral (*) exists also for , and satisfies the "Poisson equation"
(for this reason, in particular, is called the potential of ).
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 , . Also, using the infinitesimal generator one can find the mean values of various functionals. For example, under certain assumptions the function
is a unique solution to , , which is a not-too-rapidly-increasing function of . Here is the mathematical expectation corresponding to , while .
The operator is related to the characteristic operator . Let be a Markov process that is right continuous in a topological space . For a Borel function one puts
if the limit exists for all , where runs through a system of neighbourhoods of the point contracting towards and where is the moment of first exit of from (if , the fraction in the limit is set equal to zero). In many cases the calculation of amounts to calculating .
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Transition-operator semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition-operator_semi-group&oldid=17099