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{{MSC|60J35}}
 
{{MSC|60J35}}
  
 
[[Category:Markov processes]]
 
[[Category:Markov processes]]
  
A homogeneous [[Markov process|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383803.png" /> is an additive sub-semi-group of the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383804.png" />, with values in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383805.png" /> with a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383806.png" /> and a Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383807.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383808.png" />, the [[Transition function|transition function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f0383809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838012.png" />, of which has a certain property of smoothness, namely that for a continuous bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838013.png" /> the function
+
A homogeneous [[Markov process|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|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
  
<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>
+
$$
 +
x  \mapsto \
 +
P  ^ {t} f ( x)  = \
 +
\int\limits f ( y) P ( t, x, dy)
 +
$$
  
is continuous. This requirement on the transition function is natural because the transition operators <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 [[#References|[1]]]).
+
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 {{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 [[#References|[1]]]).
+
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|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 {{Cite|D}}).
  
An important subclass of Feller processes is formed by the strong Feller processes [[#References|[2]]]; 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 [[#References|[3]]]). A natural generalization of strong Feller processes are Markov processes with a continuous component (see [[#References|[4]]]).
+
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 $  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|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 $  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|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|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 [[#References|[7]]]). Here the ergodic properties of a Feller process are of a comparatively weak nature.
+
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 $  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|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 [[#References|[8]]]), 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 [[#References|[7]]]). 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.
+
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 {{Cite|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 {{Cite|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 <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 [[#References|[9]]]). This extension, however, is "too large" ; other constructions of Feller extensions are also possible for Markov chains (see [[#References|[10]]]).
+
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|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 $  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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.B. Dynkin,   "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"P. Tuominen,   R. Tweedie,   "Markov chains with continuous components" ''Proc. London Math. Soc.'' , '''38''' (1979) pp. 89–114</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"S. Foguel,   "The ergodic theory of positive operators on continuous functions" ''Ann. Scuola Norm. Sup. Pisa'' , '''27''' : 1 (1973) pp. 19–51</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"R. Sine,   "Sample path convergence of stable Markov processes II" ''Indiana Univ. Math. J.'' , '''25''' : 1 (1976) pp. 23–43</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  D. Revuz,   "Markov chains" , North-Holland (1975)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"A.I. Zhdanok,   "Ergodic theorems for nonsmooth Markov processes" , ''Topological spaces and their mappings'' , Riga (1981) pp. 18–33 (In Russian) (English summary)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"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</TD></TR></table>
+
{|
 
+
|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====
In the West a Feller process is usually indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838085.png" /> (and not by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838086.png" />). Feller processes are important for three main reasons:
+
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|diffusion process]], a [[Stochastic process with stationary increments|stochastic process with stationary increments]], among them a [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]];
 
a) numerous natural (homogeneous) Markov processes are Feller; e.g., a [[Diffusion process|diffusion process]], a [[Stochastic process with stationary increments|stochastic process with stationary increments]], among them a [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]];
  
b) the notion of a Feller semi-group (i.e. a [[Transition-operator semi-group|transition-operator semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038380/f03838087.png" /> 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|Semi-group of operators]]);
+
b) the notion of a Feller semi-group (i.e. a [[Transition-operator semi-group|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|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.
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''C''' , North-Holland (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley (1966) pp. Chapt. X</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.L. Chung,   "Lectures from Markov processes to Brownian motion" , Springer (1982)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.D. [A.D. Ventsel'] Wentzell,   "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T.G. Kurtz,   "Markov processes" , Wiley (1986)</TD></TR></table>
+
{|
 +
|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}}
 +
|}

Latest revision as of 19:38, 5 June 2020


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
How to Cite This Entry:
Feller process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feller_process&oldid=21152
This article was adapted from an original article by S.N. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article