Feller process

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 [1]).

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 [1]).

An important subclass of Feller processes is formed by the strong Feller processes [2]; 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 [3]). A natural generalization of strong Feller processes are Markov processes with a continuous component (see [4]).

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 [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 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 [8]), and if for the Feller process one chooses as the topology on , the diffusion (topological recurrence) property is obtained (see [7]). 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 [9]). This extension, however, is "too large" ; other constructions of Feller extensions are also possible for Markov chains (see [10]).

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

[1]	E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian)
[2]	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
[3]	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
[4]	P. Tuominen, R. Tweedie, "Markov chains with continuous components" Proc. London Math. Soc. , 38 (1979) pp. 89–114
[5]	S. Foguel, "The ergodic theory of positive operators on continuous functions" Ann. Scuola Norm. Sup. Pisa , 27 : 1 (1973) pp. 19–51
[6]	R. Sine, "Sample path convergence of stable Markov processes II" Indiana Univ. Math. J. , 25 : 1 (1976) pp. 23–43
[7]	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
[8]	D. Revuz, "Markov chains" , North-Holland (1975)
[9]	A.I. Zhdanok, "Ergodic theorems for nonsmooth Markov processes" , Topological spaces and their mappings , Riga (1981) pp. 18–33 (In Russian) (English summary)
[10]	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

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

[a1]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French)
[a2]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966) pp. Chapt. X
[a3]	M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV
[a4]	K.L. Chung, "Lectures from Markov processes to Brownian motion" , Springer (1982)
[a5]	A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian)
[a6]	T.G. Kurtz, "Markov processes" , Wiley (1986)