Kolmogorov-Chapman equation

2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]

An equation of the form

$$ P ( s , x ; u , \Gamma ) = \ \int\limits _ { E } P ( s , x ; t , d y ) P ( t , y ; u , \Gamma ) ,\ \ s < t < u , $$

that is, a condition imposed on the transition function $ P ( s , x ; t , \Gamma ) $( $ 0 \leq s \leq t < \infty $, $ x \in E $, $ \Gamma \in \mathfrak B $, $ ( E , \mathfrak B ) $ being a measurable space), enabling one (under certain conditions on $ ( E , \mathfrak B ) $) to construct a Markov process for which the conditional probability $ {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $ is the same as $ P ( s , x ; t , \Gamma ) $. Conversely, for a Markov process its transition function $ P ( s , x ; t , \Gamma ) $, which by definition is equal to $ {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $, satisfies the Kolmogorov–Chapman equation, as follows immediately from general properties of conditional probabilities. This was pointed out by S. Chapman [C] and investigated by A.N. Kolmogorov in 1931 (see [K]).

References

[C]	S. Chapman, "?", Proc. Roy. Soc. Ser. A , 119 (1928) pp. 34–54
[K]	A. Kolmogoroff, "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" Math. Ann. , 104 (1931) pp. 415–458
[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)

Comments

In Western literature this equation is usually referred to as the Chapman–Kolmogorov equation.

See also (the editorial comments to) Einstein–Smoluchowski equation.

References

[L]	P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965)
[D]	E.B. Dynkin, "Markov processes", 1, Springer (1965) pp. Sect. 5.26 (Translated from Russian)
[F]	W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1966) pp. Chapt. XV.13