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 $ ( E, {\mathcal B}) $ be such that the $ \sigma $- algebra $ {\mathcal B} $ contains all one-point subsets from $ E $, and let $ T $ be a subset of the real line $ \mathbf R $. A function $ P( s, x; t, B) $ given for $ s, t \in T $, $ s \leq t $, $ x \in E $ and $ B \in {\mathcal B} $ is called a transition function for $ ( E, {\mathcal B}) $ if: a) for given $ s $, $ x $ and $ t $, it is a measure on $ {\mathcal B} $, with $ P( s, x; t, B) \leq 1 $; b) for given $ s $, $ t $ and $ B $, it is a $ {\mathcal B} $- measurable function of the point $ x $; c) $ P( s, x; s, \{ x \} ) \equiv 1 $ and for all limit points $ s $ of $ T $ from the right in the topology of $ \mathbf R $,

$$ \lim\limits _ {\begin{array}{c} t\downarrow s \\ t \in T \end{array} } P( s, x; t, E) = 1; $$

and d) for all $ x \in E $, $ B \in {\mathcal B} $ and $ s \leq t \leq u $ from $ T $, the Kolmogorov–Chapman equation is fulfilled:

$$ \tag{* } P( s, x; u , B) = \int\limits _ { E } P( s, x; t, dy) P( t, y; u , B) $$

(in some cases, requirement c) may be omitted or weakened). A transition function is called a Markov transition function if $ P( s, x; t, E) \equiv 1 $, and a subMarkov transition function otherwise. If $ E $ is at most countable, then the transition function is specified by means of the matrix of transition probabilities

$$ P ^ {st} = \| P _ {xy} ( s, t) \| $$

(see Transition probabilities; Matrix of transition probabilities). It often happens that for any admissible $ s $, $ x $ and $ t $ the measure $ P( s, x; t, \cdot ) $ has a density $ p( s, x; t, \cdot ) $ with respect to a certain measure. If in this case the following form of equation (*) is satisfied:

$$ p( s, x; u , z) = \int\limits _ { E } p( s, x; t, y) p( t, y; u , z) dy $$

for any $ x $ and $ z $ from $ E $ and $ s \leq t \leq u $ from $ T $, then $ p( s, x; t, y) $ is called a transition density.

Under very general conditions (cf. [N], [GS]), the transition function $ P( s, x; t, B) $ can be related to a Markov process $ X = ( x _ {t} , \zeta , {\mathcal F} _ {t} ^ {s} , {\mathsf P} _ {s,x} ) $ for which $ {\mathsf P} _ {s,x} \{ x _ {t} \in B \} = P( s, x; t, B) $( in the case of a Markov transition function, this process does not terminate, i.e. $ \zeta = \infty $ $ P _ {s,x} $- 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 $ T $ be homogeneous in the sense that the set of values of $ t- s $ for $ s \leq t $ from $ T $ forms a semi-group $ \widetilde{T} $ in $ \mathbf R $ under addition (for example, $ T = \mathbf R $, $ T = \{ {t \in \mathbf R } : {t \geq 0 } \} $, $ T = \{ 0, 1 ,\dots \} $). If, moreover, the transition function $ P( s, x; t, B) $ depends only on the difference $ t- s $, i.e. if $ P( s, x; t, B) = P( t- s, x, B) $, where $ P( t, x, B) $ is a function of $ t \in \widetilde{T} $, $ x \in E $, $ B \in {\mathcal B} $ satisfying the corresponding form of conditions a)–d), then $ P( s, x; t, B) $ is called a homogeneous transition function. The latter name is also given to a function $ P( t, x, B) $ for which (*) takes the form

$$ P( t+ s, x, B) = \int\limits _ { E } P( t, x, dy) P( s, y, B), $$

$$ s, t \in \widetilde{T} ,\ x \in E ,\ B \in {\mathcal B} . $$

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 $ ( E _ {t} , {\mathcal B} _ {t} ) $, $ t \in T $, while a transition function with respect to this family is defined as a function $ P( s, x; t, B) $, where $ s, t \in T $, $ s \leq t $, $ x \in E _ {s} $, $ B \in {\mathcal B} _ {t} $, 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