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).
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|[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|
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|[TT]||P. Tuominen, R. Tweedie, "Markov chains with continuous components" Proc. London Math. Soc. , 38 (1979) pp. 89–114 MR0520974 Zbl 0396.60059|
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In the West a Feller process is usually indexed by (and not by ). Feller processes are important for three main reasons:
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.
|[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://www.encyclopediaofmath.org/index.php?title=Feller_process&oldid=26519