In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.
Let , be measurable spaces. A Markov kernel with source and target is a map with the following properties:
- The map is - measureable for every .
- The map is a probability measure on for every .
(i.e. It associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .)
- Simple random walk: Take and , then the Markov kernel with
describes the transition rule for the random walk on .
- Galton-Watson process: Take , , then
- General Markov processes with finite state space: Take , and , then the transition rule can be represented as a stochastic matrix with for every . In the convention of Markov kernels we write
Let be a probability space and a Markov kernel from to some . Then there exists a unique measure on , s.t.
Regular conditional distribution
It is called regular conditional distribution of given and is not uniquely defined.
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- §36. Kernels and semigroups of kernels