An equation of the form
that is, a condition imposed on the transition function (, , , being a measurable space), enabling one (under certain conditions on ) to construct a Markov process for which the conditional probability is the same as . Conversely, for a Markov process its transition function , which by definition is equal to , 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]).
|[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)|
In Western literature this equation is usually referred to as the Chapman–Kolmogorov equation.
See also (the editorial comments to) Einstein–Smoluchowski equation.
|[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|
Kolmogorov–Chapman equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kolmogorov%E2%80%93Chapman_equation&oldid=22656