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

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

$p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)$

be the joint probability density function of the values of the random variables f₁ to f_n. Then, the Chapman-Kolmogorov equation is

$p\_\{i\_1,\backslash ldots,i\_\{n-1\}\}(f\_1,\backslash ldots,f\_\{n-1\})=\backslash int\_\{-\backslash infty\}^\{\backslash infty\}p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)\backslash ,df\_n$

ie 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).

Particularization to Markov chains

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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

$p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)=p\_\{i\_1\}(f\_1)p\_\{i\_2;i\_1\}(f\_2\backslash mid\; f\_1)\backslash cdots\; p\_\{i\_n;i\_\{n-1\}\}(f\_n\backslash mid$

f_{n-1}), where the conditional probability $p\_\{i;j\}(f\_i\backslash mid\; f\_j)$ is the transition probability between the times $i>j$). So, the Chapman-Kolmogorov equation takes the form

$p\_\{i\_3;i\_1\}(f\_3\backslash mid\; f\_1)=\backslash int\_\{-\backslash infty\}^\backslash infty\; p\_\{i\_3;i\_2\}(f\_3\backslash mid\; f\_2)p\_\{i\_2;i\_1\}(f\_2\backslash mid\; f\_1)df\_2.$

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

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

where P(t) is the transition matrix, i.e., if X_t is the state of the process at time t, then for any two points i and j in the state space, we have

$P\_\{ij\}(t)=P(X\_t=j\backslash mid\; X\_0=i).$

See also

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• Examples of Markov chains
• Fokker-Planck equation
• Master equation (physics)

External links

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• 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.
• Mathworld

Stochastic processes | Equations