Difference between revisions of "Feller process"
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<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> | <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 | + | 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 {{Cite|D}}). |
− | 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 | + | 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 {{Cite|D}}). |
− | An important subclass of Feller processes is formed by the strong Feller processes | + | An important subclass of Feller processes is formed by the strong Feller processes {{Cite|G}}; 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 {{Cite|M}}). A natural generalization of strong Feller processes are Markov processes with a continuous component (see {{Cite|TT}}). |
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. | 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 | + | 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 {{Cite|Sm}}). 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 | + | 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 {{Cite|R}}), 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 {{Cite|Sm}}). 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 | + | 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 {{Cite|Z}}). This extension, however, is "too large" ; other constructions of Feller extensions are also possible for Markov chains (see {{Cite|Sh}}). |
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). | 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==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|D}}|| E.B. Dynkin, "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} | |
− | + | |- | |
+ | |valign="top"|{{Ref|G}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|M}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|TT}}|| P. Tuominen, R. Tweedie, "Markov chains with continuous components" ''Proc. London Math. Soc.'' , '''38''' (1979) pp. 89–114 {{MR|0520974}} {{ZBL|0396.60059}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Fo}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Si}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Sm}}|| 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|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|R}}|| D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Z}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Sh}}|| 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}} | ||
+ | |} | ||
====Comments==== | ====Comments==== | ||
Line 38: | Line 56: | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|DM}}|| C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''C''' , North-Holland (1988) (Translated from French) {{MR|0939365}} {{ZBL|0716.60001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|F}}|| W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1966) pp. Chapt. X {{MR|0210154}} {{ZBL|0138.10207}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV {{MR|0203748}} {{ZBL|0108.14202}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|C}}|| K.L. Chung, "Lectures from Markov processes to Brownian motion" , Springer (1982) {{MR|0648601}} {{ZBL|0503.60073}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|W}}|| 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}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|K}}|| T.G. Kurtz, "Markov processes" , Wiley (1986) {{MR|0838085}} {{ZBL|0592.60049}} | ||
+ | |} |
Revision as of 06:23, 13 May 2012
2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]
A homogeneous Markov process , , where is an additive sub-semi-group of the real axis , with values in a topological space with a topology and a Borel -algebra , the transition function , , , , of which has a certain property of smoothness, namely that for a continuous bounded function the function
is continuous. This requirement on the transition function is natural because the transition operators , , acting on the space of bounded Borel functions, leave invariant the space of continuous bounded functions, that is, the semi-group of transition operators can be considered as acting on . The first semi-groups of this type were studied by W. Feller (1952, see [D]).
As a rule, one imposes additional conditions on the topological space; usually 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, the strong Markov property). Conversely, a standard Markov process is a Feller process for a natural topology ; a basis of is constituted by the sets such that the first exit moment from almost-surely satisfies if the process starts in (see [D]).
An important subclass of Feller processes is formed by the strong Feller processes [G]; in this case a stricter smoothness condition is imposed on the transition function: The function must be continuous for every bounded Borel function . If, moreover, the function 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 and correspond to strong Feller processes, then their composition corresponds to a strong Feller process in the narrow sense under the usual assumptions on . Non-degenerate diffusion processes (cf. Diffusion process) are strong Feller processes (see [M]). A natural generalization of strong Feller processes are Markov processes with a continuous component (see [TT]).
If is a subset of the natural numbers, then a Feller process , , is called a Feller chain. An example of a Feller chain is provided by a random walk on the line : a sequence , , where , and is a sequence of independent identically-distributed random variables. Here the random walk is a strong Feller chain if and only if the distribution of 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). Two states and in are in communication if for any neighbourhoods of and of there are such that and (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. The "most-regular" behaviour is found with irreducible (topologically-indecomposable) Feller processes; these are Feller processes all states of which are in communication (see [Sm]). 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 and any set in it is almost-surely true that for an infinite set of values of the time ( takes values in the natural numbers). If is a system of sets of the form , where is some measure, then one obtains the recurrence property of a chain in the sense of Harris (see [R]), and if for the Feller process one chooses as the topology on , the diffusion (topological recurrence) property is obtained (see [Sm]). A random walk for which has finite expectation is a diffusion Feller chain if and only if , and if the distribution of is not arithmetic, then is moreover recurrent in the sense of Harris only if for some the distribution of has an absolutely-continuous component.
From the formal point of view, the theory of Markov chains with a general state space can be reduced to the study of Feller chains with a compact state space — the extension of obtained by means of the Gel'fand–Naimark theorem (see Banach algebra and [Z]). This extension, however, is "too large" ; other constructions of Feller extensions are also possible for Markov chains (see [Sh]).
The theory of Feller processes and Feller chains is also a probabilistic generalization of topological dynamics, since a deterministic (degenerate) Feller process , , corresponds to the dynamical system , where the mapping from into is continuous and (almost-surely).
References
[D] | E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901 |
[G] | 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 MR133152 Zbl 0100.34004 |
[M] | 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 Zbl 0177.21805 |
[TT] | P. Tuominen, R. Tweedie, "Markov chains with continuous components" Proc. London Math. Soc. , 38 (1979) pp. 89–114 MR0520974 Zbl 0396.60059 |
[Fo] | S. Foguel, "The ergodic theory of positive operators on continuous functions" Ann. Scuola Norm. Sup. Pisa , 27 : 1 (1973) pp. 19–51 MR0372154 Zbl 0258.47010 |
[Si] | R. Sine, "Sample path convergence of stable Markov processes II" Indiana Univ. Math. J. , 25 : 1 (1976) pp. 23–43 MR0391261 Zbl 0329.60021 |
[Sm] | 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 MR0650363 |
[R] | D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045 |
[Z] | A.I. Zhdanok, "Ergodic theorems for nonsmooth Markov processes" , Topological spaces and their mappings , Riga (1981) pp. 18–33 (In Russian) (English summary) MR0630418 Zbl 0477.60061 |
[Sh] | 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 MR0627857 Zbl 0499.60074 |
Comments
In the West a Feller process is usually indexed by (and not by ). Feller processes are important for three main reasons:
a) numerous natural (homogeneous) Markov processes are Feller; e.g., a diffusion process, a stochastic process with stationary increments, among them a Wiener process and a Poisson process;
b) the notion of a Feller semi-group (i.e. a transition-operator semi-group 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);
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
[DM] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French) MR0939365 Zbl 0716.60001 |
[F] | W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966) pp. Chapt. X MR0210154 Zbl 0138.10207 |
[L] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV MR0203748 Zbl 0108.14202 |
[C] | K.L. Chung, "Lectures from Markov processes to Brownian motion" , Springer (1982) MR0648601 Zbl 0503.60073 |
[W] | A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) MR0781738 MR0614594 Zbl 0502.60001 |
[K] | T.G. Kurtz, "Markov processes" , Wiley (1986) MR0838085 Zbl 0592.60049 |
Feller process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feller_process&oldid=23608