View source for Transition-operator semi-group

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" /> [[#References|[1]]]. 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>

and in the third

<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/t09377016.png" /></td> </tr></table>

(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377018.png" /> belong to the corresponding spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377020.png" />). In all these cases the semi-group property holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377022.png" />, and any of the three semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377023.png" /> is called a transition-operator semi-group.

In what follows, only the first case is considered. The usual definition of the infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377024.png" /> of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377025.png" /> (this is also the infinitesimal generator of the process) is as follows:

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377027.png" /> for which this limit exists as a limit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377028.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377030.png" /> is a measurable function of the pair of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377031.png" />, and one introduces the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377032.png" /> of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377034.png" />, by:

<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/t09377035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377036.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377039.png" />. Under certain assumptions the integral (*) exists also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377041.png" /> satisfies the "Poisson equation" 

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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" />).

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 [[#References|[2]]], [[#References|[3]]]. Also, using the infinitesimal generator one can find the mean values of various functionals. For example, under certain assumptions the function

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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" />.

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" /> [[#References|[2]]]. 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

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if the limit exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377059.png" /> runs through a system of neighbourhoods of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377060.png" /> contracting towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377061.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377062.png" /> is the moment of first exit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377063.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377064.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377065.png" />, the fraction in the limit is set equal to zero). In many cases the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377066.png" /> amounts to calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093770/t09377067.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, "The parabolic differential equations and the associated semi-groups of transformations" ''Ann. of Math.'' , '''55''' (1952) pp. 468–519 {{MR|0047886}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) {{MR|0131898}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) {{MR|0264757}} {{ZBL|0169.49204}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1966) {{MR|0210154}} {{ZBL|0138.10207}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Loève, "Probability theory" , '''II''' , Springer (1978) {{MR|0651017}} {{MR|0651018}} {{ZBL|0385.60001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) {{MR|0958914}} {{ZBL|0649.60079}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> S. Albeverio, Zh.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400</TD></TR></table>