Markov kernel

In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Let and be measurable spaces. A Markov kernel with source and target is a map with the following properties: For every (fixed) , the map is -measurable For every (fixed) , the map is a probability measure on 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 , and (the power set of ). Then a Markov kernel is fully determined by the probability it assigns to singletons for each : Now the random walk that goes to the right with probability and to the left with probability is defined by where is the Kronecker delta. The transition probabilities for the random walk are equivalent to the Markov kernel. More generally take and both countable and . Again a Markov kernel is defined by the probability it assigns to singleton sets for each We define a Markov process by defining a transition probability where the numbers define a (countable) stochastic matrix i.e. We then define Again the transition probability, the stochastic matrix and the Markov kernel are equivalent reformulations. Let be a measure on , and a measurable function with respect to the product -algebra such that then i.e. the mapping defines a Markov kernel. This example generalises the countable Markov process example where was the counting measure. Moreover it encompasses other important examples such as the convolution kernels, in particular the Markov kernels defined by the heat equation. The latter example includes the Gaussian kernel on with standard Lebesgue measure and Take and arbitrary measurable spaces, and let be a measurable function. Now define i.e. for all . Note that the indicator function is -measurable for all iff is measurable.