Kolmogorov equations
equations characterizing continuous-time Markov processes
In probability theory, Kolmogorov equations characterize continuous-time Markov processes. In particular, they describe how the probability of a continuous-time Markov process in a certain state changes over time. There are four distinct equations: the Kolmogorov forward equation for continuous processes, now understood to be identical to the Fokker–Planck equation, the Kolmogorov forward equation for jump processes, and two Kolmogorov backward equations for processes with and without discontinuous jumps.
Diffusion processes vs. jump processes
Writing in 1931, Andrey 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. He found that there are two kinds of continuous time Markov processes, depending on the assumed behavior over small intervals of time:
If you assume that "in a small time interval there is an overwhelming probability that the state will remain unchanged; however, if it changes, the change may be radical", then you are led to what are called jump processes.
The other case leads to processes such as those "represented by diffusion and by Brownian motion; there it is certain that some change will occur in any time interval, however small; only, here it is certain that the changes during small time intervals will be also small".
For each of these two kinds of processes, Kolmogorov derived a forward and a backward system of equations (four in all).
History
The equations are named after Andrey Kolmogorov since they were highlighted in his 1931 foundational work.
William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair,
in both jump and diffusion processes.
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