Transition function

transition probability

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. [1], [2]), 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 [3].

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).

References

[1]	J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)
[2]	I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)
[3]	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

Comments

For additional references see also Markov chain; Markov process.

References

[a1]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French)
[a2]	M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988)
[a3]	S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400