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 . 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.
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.
- §36. Kernels and semigroups of kernels