View source for Feller process

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[[Category:Markov processes]]

A homogeneous [[Markov process|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383803.png" /> is an additive sub-semi-group of the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383804.png" />, with values in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383805.png" /> with a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383806.png" /> and a Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383807.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383808.png" />, the [[Transition function|transition function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838012.png" />, of which has a certain property of smoothness, namely that for a continuous bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838013.png" /> 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/f/f038/f038380/f03838014.png" /></td> </tr></table>

is continuous. This requirement on the transition function is natural because the transition operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838016.png" />, acting on the space of bounded Borel functions, leave invariant the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838017.png" /> of continuous bounded functions, that is, the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838018.png" /> of transition operators can be considered as acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838019.png" />. The first semi-groups of this type were studied by W. Feller (1952, see [[#References|[1]]]).

As a rule, one imposes additional conditions on the topological space; usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838020.png" /> is a locally compact metrizable space. In this case, a Feller process that satisfies the condition of stochastic continuity admits a modification that is a standard Markov process (see [[Markov process|Markov process]], the strong Markov property). Conversely, a standard Markov process is a Feller process for a natural topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838021.png" />; a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838022.png" /> is constituted by the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838023.png" /> such that the first exit moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838024.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838025.png" /> almost-surely satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838026.png" /> if the process starts in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838027.png" /> (see [[#References|[1]]]).

An important subclass of Feller processes is formed by the strong Feller processes [[#References|[2]]]; in this case a stricter smoothness condition is imposed on the transition function: The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838028.png" /> must be continuous for every bounded Borel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838029.png" />. If, moreover, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838030.png" /> is continuous in the variation norm in the space of bounded measures, then the Markov process corresponding to this transition function is called a strong Feller process in the narrow sense. If the transition functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838032.png" /> correspond to strong Feller processes, then their composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838033.png" /> corresponds to a strong Feller process in the narrow sense under the usual assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838034.png" />. Non-degenerate diffusion processes (cf. [[Diffusion process|Diffusion process]]) are strong Feller processes (see [[#References|[3]]]). A natural generalization of strong Feller processes are Markov processes with a continuous component (see [[#References|[4]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838035.png" /> is a subset of the natural numbers, then a Feller process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838037.png" />, is called a Feller chain. An example of a Feller chain is provided by a [[Random walk|random walk]] on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838038.png" />: a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838042.png" /> is a sequence of independent identically-distributed random variables. Here the random walk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838043.png" /> is a strong Feller chain if and only if the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838044.png" /> has a density.

There is a natural generalization for Feller processes of the classification of the states of a Markov chain with a countable number of states (see [[Markov chain|Markov chain]]). Two states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838047.png" /> are in communication if for any neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838051.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838054.png" /> (chains with a countable set of states are Feller chains with the discrete topology). Ergodic properties and methods for investigating them have a definite character for Feller processes in comparison to classical [[Ergodic theory|ergodic theory]]. The "most-regular" behaviour is found with irreducible (topologically-indecomposable) Feller processes; these are Feller processes all states of which are in communication (see [[#References|[7]]]). Here the ergodic properties of a Feller process are of a comparatively weak nature.

As an example one can compare properties such as recurrence for a Markov chain with a general space of states. Suppose that for any initial state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838055.png" /> and any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838057.png" /> it is almost-surely true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838058.png" /> for an infinite set of values of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838060.png" /> takes values in the natural numbers). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838061.png" /> is a system of sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838063.png" /> is some measure, then one obtains the recurrence property of a chain in the sense of Harris (see [[#References|[8]]]), and if for the Feller process one chooses as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838064.png" /> the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838066.png" />, the diffusion (topological recurrence) property is obtained (see [[#References|[7]]]). A random walk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838067.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838068.png" /> has finite expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838069.png" /> is a diffusion Feller chain if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838070.png" />, and if the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838071.png" /> is not arithmetic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838072.png" /> is moreover recurrent in the sense of Harris only if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838073.png" /> the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838074.png" /> has an absolutely-continuous component.

From the formal point of view, the theory of Markov chains with a general state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838075.png" /> can be reduced to the study of Feller chains with a compact state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838076.png" /> — the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838077.png" /> obtained by means of the Gel'fand–Naimark theorem (see [[Banach algebra|Banach algebra]] and [[#References|[9]]]). This extension, however, is "too large" ; other constructions of Feller extensions are also possible for Markov chains (see [[#References|[10]]]).

The theory of Feller processes and Feller chains is also a probabilistic generalization of [[Topological dynamics|topological dynamics]], since a deterministic (degenerate) Feller process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838079.png" />, corresponds to the dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838080.png" />, where the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838081.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838082.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838083.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838084.png" /> (almost-surely).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.V. Girsanov, "On transforming a certain class of stochastic processes by absolutely continuous substitution of measures" ''Theor. Probab. Appl.'' , '''5''' : 3 (1960) pp. 285–301 ''Teor. Veroyatnost. i Primenen.'' , '''5''' : 3 (1960) pp. 314–330 {{MR|133152}} {{ZBL|0100.34004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.A. Molchanov, "Strong Feller property of diffusion processes on smooth manifolds" ''Theor. Probab. Appl.'' , '''13''' : 3 (1968) pp. 471–475 ''Teor. Veroyatnost. i Primenen.'' , '''13''' : 3 (1968) pp. 493–498 {{MR|}} {{ZBL|0177.21805}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P. Tuominen, R. Tweedie, "Markov chains with continuous components" ''Proc. London Math. Soc.'' , '''38''' (1979) pp. 89–114 {{MR|0520974}} {{ZBL|0396.60059}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Foguel, "The ergodic theory of positive operators on continuous functions" ''Ann. Scuola Norm. Sup. Pisa'' , '''27''' : 1 (1973) pp. 19–51 {{MR|0372154}} {{ZBL|0258.47010}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Sine, "Sample path convergence of stable Markov processes II" ''Indiana Univ. Math. J.'' , '''25''' : 1 (1976) pp. 23–43 {{MR|0391261}} {{ZBL|0329.60021}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S.N. Smirnov, "On the asymptotic behavior of Feller chains" ''Soviet Math. Dokl.'' , '''25''' : 2 (1982) pp. 399–403 ''Dokl. Akad. Nauk SSSR'' , '''263''' : 3 (1982) pp. 554–558 {{MR|0650363}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.I. Zhdanok, "Ergodic theorems for nonsmooth Markov processes" , ''Topological spaces and their mappings'' , Riga (1981) pp. 18–33 (In Russian) (English summary) {{MR|0630418}} {{ZBL|0477.60061}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.G. Shur, "Invariant measures for Markov chains and Feller extensions of chains" ''Theory Probab. Appl.'' , '''26''' : 3 (1981) pp. 485–497 ''Teor. Veroyatnost. i Primenen.'' , '''26''' : 3 (1981) pp. 496–509 {{MR|0627857}} {{ZBL|0499.60074}} </TD></TR></table>



====Comments====
In the West a Feller process is usually indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838085.png" /> (and not by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838086.png" />). Feller processes are important for three main reasons:

a) numerous natural (homogeneous) Markov processes are Feller; e.g., a [[Diffusion process|diffusion process]], a [[Stochastic process with stationary increments|stochastic process with stationary increments]], among them a [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]];

b) the notion of a Feller semi-group (i.e. a [[Transition-operator semi-group|transition-operator semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838087.png" /> as defined in the main article) lies at the interface between the stochastic and the analytic study of semi-groups of linear operators (see also [[Semi-group of operators|Semi-group of operators]]);

c) by way of the so-called Ray–Knight compactification it is possible to look at a strong Markov process as if it were "almost" a Feller process (with a nice topology on the state space), and so the make use of the smoothness of the latter.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''C''' , North-Holland (1988) (Translated from French) {{MR|0939365}} {{ZBL|0716.60001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1966) pp. Chapt. X {{MR|0210154}} {{ZBL|0138.10207}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV {{MR|0203748}} {{ZBL|0108.14202}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K.L. Chung, "Lectures from Markov processes to Brownian motion" , Springer (1982) {{MR|0648601}} {{ZBL|0503.60073}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) {{MR|0781738}} {{MR|0614594}} {{ZBL|0502.60001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> T.G. Kurtz, "Markov processes" , Wiley (1986) {{MR|0838085}} {{ZBL|0592.60049}} </TD></TR></table>