# 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 $(S,{\mathcal {S}})$, $(T,{\mathcal {T}})$ be two measurable spaces. A function
    $\kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]$
is called a (transition) kernel from $S$ to $T$ if the following two conditions hold:
* For any fixed $B\in {\mathcal {T}}$, the mapping
       $s\mapsto \kappa (s,B)$
    is ${\mathcal {S}}/{\mathcal {B}}([0,+\infty ])$-measurable;
* For every fixed $s\in S$, the mapping
       $B\mapsto \kappa (s,B)$
    is a measure on $(T,{\mathcal {T}})$.

## Classification of transition kernels

Transition kernels are usually classified by the measures they define. Those measures are defined as
    $\kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]$
with
    $\kappa _{s}(B)=\kappa (s,B)$
for all $B\in {\mathcal {T}}$ and all $s\in S$. With this notation, the kernel $\kappa$ is called
* a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all $\kappa _{s}$ are sub-probability measures
* a Markov kernel, stochastic kernel or probability kernel if all $\kappa _{s}$ are probability measures
* a finite kernel if all $\kappa _{s}$ are finite measures
* a $\sigma$-finite kernel if all $\kappa _{s}$ are $\sigma$-finite measures
* a s-finite kernel if all $\kappa _{s}$ are $s$-finite measures, meaning it is a kernel that can be written as a countable sum of finite kernels
* a uniformly $\sigma$-finite kernel if there are at most countably many measurable sets $B_{1},B_{2},\dots$ in $T$ with $\kappa _{s}(B_{i})<\infty$ for all $s\in S$ and all $i\in \mathbb {N}$.

## Operations

In this section, let $(S,{\mathcal {S}})$, $(T,{\mathcal {T}})$ and $(U,{\mathcal {U}})$ be measurable spaces and denote the product σ-algebra of ${\mathcal {S}}$ and ${\mathcal {T}}$ with ${\mathcal {S}}\otimes {\mathcal {T}}$

### Product of kernels
#### Definition

Let $\kappa ^{1}$ be a s-finite kernel from $S$ to $T$ and $\kappa ^{2}$ be a s-finite kernel from $S\times T$ to $U$. Then the product $\kappa ^{1}\otimes \kappa ^{2}$ of the two kernels is defined as
    $\kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]$
    $\kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)$
for all $A\in {\mathcal {T}}\otimes {\mathcal {U}}$.

#### Properties and comments

The product of two kernels is a kernel from $S$ to $T\times U$. It is again a s-finite kernel and is a $\sigma$-finite kernel if $\kappa ^{1}$ and $\kappa ^{2}$ are $\sigma$-finite kernels. The product of kernels is also associative, meaning it satisfies
    $(\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})$
for any three suitable s-finite kernels $\kappa ^{1},\kappa ^{2},\kappa ^{3}$.
The product is also well-defined if $\kappa ^{2}$ is a kernel from $T$ to $U$. In this case, it is treated like a kernel from $S\times T$ to $U$ that is independent of $S$. This is equivalent to setting
    $\kappa ((s,t),A):=\kappa (t,A)$
for all $A\in {\mathcal {U}}$ and all $s\in S$.

### Composition of kernels
#### Definition

Let $\kappa ^{1}$ be a s-finite kernel from $S$ to $T$ and $\kappa ^{2}$ a s-finite kernel from $S\times T$ to $U$. Then the composition $\kappa ^{1}\cdot \kappa ^{2}$ of the two kernels is defined as
    $\kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]$
    $(s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)$
for all $s\in S$ and all $B\in {\mathcal {U}}$.

#### Properties and comments

The composition is a kernel from $S$ to $U$ that is again s-finite. The composition of kernels is associative, meaning it satisfies
    $(\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})$
for any three suitable s-finite kernels $\kappa ^{1},\kappa ^{2},\kappa ^{3}$. Just like the product of kernels, the composition is also well-defined if $\kappa ^{2}$ is a kernel from $T$ to $U$.
An alternative notation is for the composition is $\kappa ^{1}\kappa ^{2}$

## Kernels as operators

Let ${\mathcal {T}}^{+},{\mathcal {S}}^{+}$ be the set of positive measurable functions on $(S,{\mathcal {S}}),(T,{\mathcal {T}})$.
Every kernel $\kappa$ from $S$ to $T$ can be associated with a linear operator
    $A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}$
given by
    $(A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).$
The composition of these operators is compatible with the composition of kernels, meaning
    $A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}$