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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. The equation was arrived at independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov.

Suppose that { fi } is an indexed collection of random variables, that is, a stochastic process. Let

\( p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)

be the joint probability density function of the values of the random variables \( f_1 \) to \( f_n \) . Then, the Chapman–Kolmogorov equation is

\( p_{i_1,\ldots,i_{n-1}}(f_1,\ldots,f_{n-1})=\int_{-\infty}^{\infty}p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)\,df_n \)

i.e. a straightforward marginalization over the nuisance variable.

(Note that we have not yet assumed anything about the temporal (or any other) ordering of the random variables—the above equation applies equally to the marginalization of any of them.)
Application to Markov chains

When the stochastic process under consideration is Markovian, the Chapman–Kolmogorov equation is equivalent to an identity on transition densities. In the Markov chain setting, one assumes that i1 < ... < in. Then, because of the Markov property,

\( p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)=p_{i_1}(f_1)p_{i_2;i_1}(f_2\mid f_1)\cdots p_{i_n;i_{n-1}}(f_n\mid f_{n-1}), \)

where the conditional probability \( p_{i;j}(f_i\mid f_j) \) is the transition probability between the times i>j. So, the Chapman–Kolmogorov equation takes the form

\( p_{i_3;i_1}(f_3\mid f_1)=\int_{-\infty}^\infty p_{i_3;i_2}(f_3\mid f_2)p_{i_2;i_1}(f_2\mid f_1) \, df_2. \)

In English, and informally, this says that the probability of going from state 1 to state 3 can be found from the probabilities of going from 1 to an intermediate state 2 and then from 2 to 3, by adding up over all the possible intermediate states 2.

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:

\( P(t+s)=P(t)P(s)\, \)

where P(t) is the transition matrix of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps.

As a corollary, it follows that to calculate the transition matrix of jump t, it is sufficient to raise the transition matrix of jump one to the power of t, that is

\( P(t)=P^t.\, \)

See also

Fokker–Planck equation (also known as Kolmogorov forward equation)
Kolmogorov backward equation
Examples of Markov chains
Master equation (physics)

References

The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
Weisstein, Eric W., "Chapman–Kolmogorov Equation" from MathWorld.

Mathematics Encyclopedia

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