Transition kernel


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.

Definition

Let, be two measurable spaces. A function
is called a kernel from to iff the following two conditions hold:
Transition kernels are usually classified by the measures they define. Those measures are defined as
with
for all and all. With this notation, the kernel is called
In this section, let, and be measurable spaces and denote the product σ-algebra of and with

Product of kernels

Definition

Let be a s-finite kernel from to and be a s-finite kernel from to. Then the product of the two kernels is defined as
for all.

Properties and comments

The product of two kernels is a kernel from to. It is again a s-finite kernel and is a -finite kernel if and are -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 is a kernel from to. In this case, it is treated like a kernel from to that is independent of. This is equivalent to setting
for all and all.

Composition of kernels

Definition

Let be a s-finite kernel from to and a s-finite kernel from to. Then the composition of the two kernels is defined as
for all and all.

Properties and comments

The composition is a kernel from to that is again s-finite. 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 is a kernel from to.
An alternative notation is for the composition is

Kernels as operators

Let be the set of positive measurable functions on.
Every kernel from to can be associated with a linear operator
given by
The composition of these operators is compatible with the composition of kernels, meaning