In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications.
Kernels can for example be used to define random measures or stochastic processes.
The most important example of kernels are the Markov kernels.
be two measurable spaces.
A function is called a (transition) kernel from
if the following two conditions hold:[1] Transition kernels are usually classified by the measures they define.
Those measures are defined as with for all
With this notation, the kernel
is called[1][2] In this section, let
be measurable spaces and denote the product σ-algebra of
be a s-finite kernel from
be a s-finite kernel from
of the two kernels is defined as[3][4] for all
The product of two kernels is a kernel from
It is again a s-finite kernel and is a
-finite kernel if
-finite kernels.
The product of kernels is also associative, meaning it satisfies for any three suitable s-finite kernels
The product is also well-defined if
In this case, it is treated like a kernel from
This is equivalent to setting for all
be a s-finite kernel from
a s-finite kernel from
of the two kernels is defined as[5][3] for all
The composition is a kernel from
The composition of kernels is associative, meaning it satisfies for any three suitable s-finite kernels
Just like the product of kernels, the composition is also well-defined if
An alternative notation is for the composition is
be the set of positive measurable functions on
can be associated with a linear operator given by[6] The composition of these operators is compatible with the composition of kernels, meaning[3]