View source for Markov process, stationary

A [[Markov process|Markov process]] which is a [[Stationary stochastic process|stationary stochastic process]]. There is a stationary Markov process associated with a homogeneous Markov [[Transition function|transition function]] if and only if there is a stationary initial distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625001.png" /> corresponding to this function, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625002.png" /> satisfies

<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/m/m062/m062500/m0625003.png" /></td> </tr></table>

If the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625004.png" /> is finite, then a stationary initial distribution always exists, independent of whether the process has discrete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625005.png" /> or continuous time. For a process in discrete time and for a countable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625006.png" />, a condition for existence of a stationary distribution has been found by A.N. Kolmogorov [[#References|[1]]]: It is necessary and sufficient that there is class of communicating states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625007.png" /> such that the mathematical expectation of the time for reaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625008.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m0625009.png" /> is finite for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250010.png" />. This criterion has been generalized to strong Markov processes with an arbitrary phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250011.png" />: For the existence of a stationary process it is sufficient that there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250012.png" /> such that the expectation of the time of reaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250014.png" /> is finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250015.png" />. There is the following sufficient condition for the existence of a stationary Markov process in terms of Lyapunov stochastic functions (cf. [[Lyapunov stochastic function|Lyapunov stochastic function]]): If there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250018.png" />, then there is a stationary Markov process associated with the Markov transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250019.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250020.png" /> is the infinitesimal generator of the process.

When the stationary initial distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250021.png" /> is unique, the corresponding stationary process is ergodic. In this case the Cesàro mean of the transition probabilities converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250022.png" />. Under certain additional conditions,

<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/m/m062/m062500/m06250023.png" /></td> </tr></table>

A stationary initial distribution satisfies the Fokker–Planck(–Kolmogorov) equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250025.png" /> is the adjoint operator to the infinitesimal operator of the process. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250026.png" /> is the adjoint operator to the generating differential operator of the process for diffusion processes. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250027.png" /> has a density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250028.png" /> with respect to the Lebesgue measure which satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062500/m06250029.png" />. In the one-dimensional case this equation can be solved by quadrature.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   "Markov chains with a countable number of states" , Moscow  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Doob,   "Stochastic processes" , Wiley  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.B. Sevast'yanov,   "An ergodic theorem for Markov processes and its application to telephone systems with refusals"  ''Theor. Probab. Appl.'' , '''2'''  (1957)  pp. 104–112  ''Teor. Veroyatnost. i Primenen.'' , '''2''' :  1  (1957)  pp. 106–116</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities", Springer (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''', Wiley (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Parzen, "Stochastic processes", Holden-Day (1962)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Yu.A. Rozanov, "Stationary random processes", Holden-Day (1967) (Translated from Russian)</TD></TR></table>