Transition-operator semi-group
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
[1]. 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
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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 [2], [3]. 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
[2]. 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
.
References
[1] | W. Feller, "The parabolic differential equations and the associated semi-groups of transformations" Ann. of Math. , 55 (1952) pp. 468–519 |
[2] | E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) |
[3] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) |
Comments
References
[a1] | R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) |
[a2] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 |
[a3] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) |
[a4] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966) |
[a5] | M. Loève, "Probability theory" , II , Springer (1978) |
[a6] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) |
[a7] | M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) |
[a8] | S. Albeverio, Zh.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400 |
Transition-operator semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition-operator_semi-group&oldid=17099