Difference between revisions of "Transition-operator semi-group"
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− | The [[Semi-group of operators|semi-group of operators]] generated by the [[Transition function|transition function]] of a [[Markov process|Markov process]]. From the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937701.png" /> of a homogeneous Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937702.png" /> in a state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937703.png" /> one can construct certain semi-groups of linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937704.png" /> acting in some Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937705.png" /> | + | The [[Semi-group of operators|semi-group of operators]] generated by the [[Transition function|transition function]] of a [[Markov process|Markov process]]. From the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937701.png" /> of a homogeneous Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937702.png" /> in a state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937703.png" /> one can construct certain semi-groups of linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937704.png" /> acting in some Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937705.png" /> {{Cite|F}}. Very often, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937706.png" /> is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937707.png" /> of bounded real-valued measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937708.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t0937709.png" /> with the uniform norm (or for a [[Feller process|Feller process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377010.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377011.png" /> of continuous functions with the same norm) or else the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377012.png" /> of finite countably-additive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377014.png" /> with the complete variation as norm. In the first two cases one puts |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377015.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377015.png" /></td> </tr></table> | ||
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(for this reason, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377043.png" /> is called the potential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377044.png" />). | (for this reason, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377043.png" /> is called the potential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377044.png" />). | ||
− | 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 | + | 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 {{Cite|Dy}}, {{Cite|GS}}. Also, using the infinitesimal generator one can find the mean values of various functionals. For example, under certain assumptions the function |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377045.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377045.png" /></td> </tr></table> | ||
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is a unique solution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377047.png" />, which is a not-too-rapidly-increasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377048.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377049.png" /> is the mathematical expectation corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377050.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377051.png" />. | is a unique solution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377047.png" />, which is a not-too-rapidly-increasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377048.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377049.png" /> is the mathematical expectation corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377050.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377051.png" />. | ||
− | The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377052.png" /> is related to the characteristic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377053.png" /> | + | The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377052.png" /> is related to the characteristic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377053.png" /> {{Cite|Dy}}. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377054.png" /> be a Markov process that is right continuous in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377055.png" />. For a Borel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377056.png" /> one puts |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377057.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377057.png" /></td> </tr></table> | ||
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====References==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|F}}|| W. Feller, "The parabolic differential equations and the associated semi-groups of transformations" ''Ann. of Math.'' , '''55''' (1952) pp. 468–519 {{MR|0047886}} {{ZBL|}} | |
− | + | |- | |
+ | |valign="top"|{{Ref|Dy}}|| E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) {{MR|0131898}} {{ZBL|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} | ||
+ | |} | ||
====Comments==== | ====Comments==== | ||
Line 48: | Line 52: | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|BG}}|| R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) {{MR|0264757}} {{ZBL|0169.49204}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Do}}|| J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 {{MR|0731258}} {{ZBL|0549.31001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Dy2}}|| E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|F2}}|| W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1966) {{MR|0210154}} {{ZBL|0138.10207}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , '''II''' , Springer (1978) {{MR|0651017}} {{MR|0651018}} {{ZBL|0385.60001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|DM}}|| C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''1–3''' , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|S}}|| M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) {{MR|0958914}} {{ZBL|0649.60079}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|AM}}|| S. Albeverio, Zh.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400 | ||
+ | |} |
Revision as of 14:16, 31 May 2012
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 of a homogeneous Markov process
in a state space
one can construct certain semi-groups of linear operators
acting in some Banach space
[F]. 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 [Dy], [GS]. 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
[Dy]. 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
[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 |
Comments
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 |
Transition-operator semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition-operator_semi-group&oldid=24665