When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.

When the stochastic process under consideration is Markovian, the Chapman-Kolmogorov equation is equivalent to an identity on transition densities.

In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman-Kolmogorov equation.

Chapman-Kolmogorov equation / (F:DC)

There, he wrote down his latest work on the Chapman-Kolmogorov equation, and sent this as a "pli cacheté" (sealed envelope) to the French Academy of Sciences.

In his study of Markovian stochastic processes and their generalizations, Chapman and the Russian Andrey Kolmogorov independently developed the pivotal set of equations in the field, the 'Chapman-Kolmogorov equations'.

The original derivation of the equations by Kolmogorov starts with the Chapman-Kolmogorov equation (Kolmogorov called it Fundamental equation) for time-continuous and differentiable Markov processes on a finite, discrete state space.

In mathematics, specifically in probability theory and in particular the theory of Markovian stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.

When the probability distribution on the state space of a Markov chain is discrete and the Markov chain is homogeneous, the Chapman-Kolmogorov equations can be expressed in terms of (possibly infinite-dimensional) matrix multiplication, thus:

Writing in 1931, Andrei Kolmogorov started from the theory of discrete time Markov processes, which are described by the Chapman-Kolmogorov equation, and sought to derive a theory of continuous time Markov processes by extending this equation.

In the theory of stochastic processes in probability theory and statistics, a nuisance variable is a random variable that is fundamental to the probabilistic model, but that is of no particular interest in itself or is no longer of interest: one such usage arises for the Chapman-Kolmogorov equation.