Transition function
transition probability
2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]
A family of measures used in the theory of Markov processes for determining the distribution at future instants from known states at previous times. Let a measurable space be such that the
-algebra
contains all one-point subsets from
, and let
be a subset of the real line
. A function
given for
,
,
and
is called a transition function for
if: a) for given
,
and
, it is a measure on
, with
; b) for given
,
and
, it is a
-measurable function of the point
; c)
and for all limit points
of
from the right in the topology of
,
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and d) for all ,
and
from
, the Kolmogorov–Chapman equation is fulfilled:
![]() | (*) |
(in some cases, requirement c) may be omitted or weakened). A transition function is called a Markov transition function if , and a subMarkov transition function otherwise. If
is at most countable, then the transition function is specified by means of the matrix of transition probabilities
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(see Transition probabilities; Matrix of transition probabilities). It often happens that for any admissible ,
and
the measure
has a density
with respect to a certain measure. If in this case the following form of equation (*) is satisfied:
![]() |
for any and
from
and
from
, then
is called a transition density.
Under very general conditions (cf. [N], [GS]), the transition function can be related to a Markov process
for which
(in the case of a Markov transition function, this process does not terminate, i.e.
-a.s.). On the other hand, the Markov property for a random process enables one, as a rule, to put the process into correspondence with a transition function [K].
Let be homogeneous in the sense that the set of values of
for
from
forms a semi-group
in
under addition (for example,
,
,
). If, moreover, the transition function
depends only on the difference
, i.e. if
, where
is a function of
,
,
satisfying the corresponding form of conditions a)–d), then
is called a homogeneous transition function. The latter name is also given to a function
for which (*) takes the form
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For certain purposes (such as regularizing transition functions) it is necessary to extend the definition. For example, one takes as given a family of measurable spaces ,
, while a transition function with respect to this family is defined as a function
, where
,
,
,
, that satisfies a suitable modification of conditions a)–d).
References
[N] | J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901 |
[GS] | I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027 |
[K] | S.E. Kuznetsov, "Any Markov process in a Borel space has a transition function" Theory Probab. Appl. , 25 : 2 (1980) pp. 384–388 Teor. Veroyatnost. i ee Primenen. , 25 : 2 (1980) pp. 389–393 MR0572574 Zbl 0456.60077 Zbl 0431.60071 |
Comments
For additional references see also Markov chain; Markov process.
References
[DM] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001 |
[S] | M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) MR0958914 Zbl 0649.60079 |
[AM] | S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400 |
Transition function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_function&oldid=26963