# Feller process

2010 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).

How to Cite This Entry:
Feller process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feller_process&oldid=46913
This article was adapted from an original article by S.N. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article