Feller process

2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]

A homogeneous Markov process $ X ( t) $, $ t \in T $, where $ T $ is an additive sub-semi-group of the real axis $ \mathbf R $, with values in a topological space $ E $ with a topology $ {\mathcal C} $ and a Borel $ \sigma $- algebra $ {\mathcal B} $, the transition function $ P ( t, x, B) $, $ t \in T $, $ x \in E $, $ B \in {\mathcal B} $, of which has a certain property of smoothness, namely that for a continuous bounded function $ f $ the function

$$ x \mapsto \ P ^ {t} f ( x) = \ \int\limits f ( y) P ( t, x, dy) $$

is continuous. This requirement on the transition function is natural because the transition operators $ P ^ {t} $, $ t \in T $, acting on the space of bounded Borel functions, leave invariant the space $ C ( E) $ of continuous bounded functions, that is, the semi-group $ {\mathcal P} = \{ {P ^ {t} } : {t \in T } \} $ of transition operators can be considered as acting on $ C ( E) $. 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 $ ( E, {\mathcal C} ) $ 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 $ {\mathcal C} _ {0} $; a basis of $ {\mathcal C} _ {0} $ is constituted by the sets $ B \in {\mathcal B} $ such that the first exit moment $ \theta ( B) $ from $ B $ almost-surely satisfies $ \theta ( B) > 0 $ if the process starts in $ B $( 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 $ x \rightarrow P ^ {t} f ( x) $ must be continuous for every bounded Borel function $ f $. If, moreover, the function $ x \rightarrow P ( t, x, \cdot ) $ 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 $ P $ and $ Q $ correspond to strong Feller processes, then their composition $ P \cdot Q $ corresponds to a strong Feller process in the narrow sense under the usual assumptions on $ ( E, {\mathcal C} ) $. 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 $ T $ is a subset of the natural numbers, then a Feller process $ X ( t) $, $ t \in T $, is called a Feller chain. An example of a Feller chain is provided by a random walk on the line $ \mathbf R $: a sequence $ S _ {n} $, $ n \in T = \{ 0, 1 ,\dots \} $, where $ S _ {n + 1 } = S _ {n} + Y _ {n} $, and $ \{ Y _ {n} \} $ is a sequence of independent identically-distributed random variables. Here the random walk $ \{ S _ {n} \} $ is a strong Feller chain if and only if the distribution of $ Y _ {1} $ 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 $ x $ and $ y $ in $ E $ are in communication if for any neighbourhoods $ U _ {x} $ of $ x $ and $ V _ {y} $ of $ y $ there are $ t, s \in T $ such that $ P ( t, x, V _ {y} ) > 0 $ and $ P ( s, y, U _ {x} ) > 0 $( 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 $ x \in E $ and any set $ A $ in $ {\mathcal A} $ it is almost-surely true that $ X ( t) \in A $ for an infinite set of values of the time $ t $( $ t $ takes values in the natural numbers). If $ {\mathcal A} $ is a system of sets of the form $ {\mathcal A} = \{ {A } : {\mu ( A) > 0 } \} $, where $ \mu $ 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 $ {\mathcal A} $ the topology $ {\mathcal C} $ on $ E $, the diffusion (topological recurrence) property is obtained (see [Sm]). A random walk $ \{ S _ {n} \} $ for which $ Y _ {1} $ has finite expectation $ {\mathsf E} Y _ {1} $ is a diffusion Feller chain if and only if $ {\mathsf E} Y _ {1} = 0 $, and if the distribution of $ Y _ {1} $ is not arithmetic, then $ \{ S _ {n} \} $ is moreover recurrent in the sense of Harris only if for some $ n $ the distribution of $ S _ {n} $ has an absolutely-continuous component.

From the formal point of view, the theory of Markov chains with a general state space $ E $ can be reduced to the study of Feller chains with a compact state space $ \widehat{E} $— the extension of $ E $ 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 $ X ( t) $, $ t \in T $, corresponds to the dynamical system $ \{ {S _ {t} } : {t \in T } \} $, where the mapping $ ( t, x) \rightarrow S _ {t} x $ from $ T \times E $ into $ E $ is continuous and $ X ( t) = S _ {t} x $( 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 $ \mathbf R _ {+} $( and not by $ \mathbf R $). 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 $ {\mathcal P} $ 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