# Transition function

transition probability

2010 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