In probability theory, a Markov kernel (also known as a stochastic kernel or probability 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 -measurable for every
- The map is a probability measure on for every .
In other words it associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra 
Take (the power set of ), then the Markov kernel with
where is the indicator function, describes the transition rule for the random walk on
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
In the convention of Markov kernels we write
Construction of a Markov kernel
If is a finite measure on and is a measurable function with respect to the product -algebra and has the property
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