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 ,
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
(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
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).
|[N]||J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901|
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|[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|
|[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://www.encyclopediaofmath.org/index.php?title=Transition_function&oldid=26963