# Markov kernel

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.

## Formal definition

Let , be measurable spaces. A Markov kernel with source and target is a map with the following properties:

1. The map is - measureable for every .
2. 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 .)

## Examples

• Simple random walk: Take and , then the Markov kernel with ,

describes the transition rule for the random walk on , where is the indicator function.

• Galton-Watson process: Take , , then with i.i.d. random variables .

• 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 . for all , then the mapping  defines a Markov kernel.

## Properties

### Semidirect product

Let be a probability space and a Markov kernel from to some .

Then there exists a unique measure on , such that .

### Regular conditional distribution

Let be a Borel space, a - valued random variable on the measure space and a sub- -algebra.

Then there exists a Markov kernel from to , such that is a version of the conditional expectation for every , i.e. .

It is called regular conditional distribution of given and is not uniquely defined.

## References

1. Reiss, R. D. (1993). "A Course on Point Processes". Springer Series in Statistics. doi:10.1007/978-1-4613-9308-5. ISBN 978-1-4613-9310-8.
2. Klenke, Achim. Probability Theory: A Comprehensive Course (2 ed.). Springer. p. 180. doi:10.1007/978-1-4471-5361-0.
3. Erhan, Cinlar (2011). Probability and Stochastics. New York: Springer. pp. 37–38. ISBN 978-0-387-87858-4.
• Bauer, Heinz (1996), Probability Theory, de Gruyter, ISBN 3-11-013935-9
§36. Kernels and semigroups of kernels