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.

Let ${\displaystyle (S,{\mathcal {S}})}$, ${\displaystyle (T,{\mathcal {T}})}$ be two measurable spaces. A function

${\displaystyle \kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]}$

is called a (transition) kernel from ${\displaystyle S}$ to ${\displaystyle T}$ if the following two conditions hold:^[1]

• For any fixed ${\displaystyle B\in {\mathcal {T}}}$, the mapping

${\displaystyle s\mapsto \kappa (s,B)}$

is ${\displaystyle {\mathcal {S}}/{\mathcal {B}}([0,+\infty ])}$-measurable;

• For every fixed ${\displaystyle s\in S}$, the mapping

${\displaystyle B\mapsto \kappa (s,B)}$

is a measure on ${\displaystyle (T,{\mathcal {T}})}$.

Transition kernels are usually classified by the measures they define. Those measures are defined as

${\displaystyle \kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]}$

with

${\displaystyle \kappa _{s}(B)=\kappa (s,B)}$

for all ${\displaystyle B\in {\mathcal {T}}}$ and all ${\displaystyle s\in S}$. With this notation, the kernel ${\displaystyle \kappa }$ is called^[1][2]

• a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all ${\displaystyle \kappa _{s}}$ are sub-probability measures
• a Markov kernel, stochastic kernel or probability kernel if all ${\displaystyle \kappa _{s}}$ are probability measures
• a finite kernel if all ${\displaystyle \kappa _{s}}$ are finite measures
• a ${\displaystyle \sigma }$-finite kernel if all ${\displaystyle \kappa _{s}}$ are ${\displaystyle \sigma }$-finite measures
• a ${\displaystyle s}$-finite kernel if ${\displaystyle \kappa }$ can be written as a countable sum of finite kernels (so that in particular, all ${\displaystyle \kappa _{s}}$ are ${\displaystyle s}$-finite measures).
• a uniformly ${\displaystyle \sigma }$-finite kernel if there are at most countably many measurable sets ${\displaystyle B_{1},B_{2},\dots }$ in ${\displaystyle T}$ with ${\displaystyle \kappa _{s}(B_{i})<\infty }$ for all ${\displaystyle s\in S}$ and all ${\displaystyle i\in \mathbb {N} }$.

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

Let ${\displaystyle \kappa ^{1}}$ be a s-finite kernel from ${\displaystyle S}$ to ${\displaystyle T}$ and ${\displaystyle \kappa ^{2}}$ be a s-finite kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$. Then the product ${\displaystyle \kappa ^{1}\otimes \kappa ^{2}}$ of the two kernels is defined as^[3][4]

${\displaystyle \kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]}$

${\displaystyle \kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)}$

for all ${\displaystyle A\in {\mathcal {T}}\otimes {\mathcal {U}}}$.

The product of two kernels is a kernel from ${\displaystyle S}$ to ${\displaystyle T\times U}$. It is again a s-finite kernel and is a ${\displaystyle \sigma }$-finite kernel if ${\displaystyle \kappa ^{1}}$ and ${\displaystyle \kappa ^{2}}$ are ${\displaystyle \sigma }$-finite kernels. The product of kernels is also associative, meaning it satisfies

${\displaystyle (\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})}$

for any three suitable s-finite kernels ${\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}$.

The product is also well-defined if ${\displaystyle \kappa ^{2}}$ is a kernel from ${\displaystyle T}$ to ${\displaystyle U}$. In this case, it is treated like a kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$ that is independent of ${\displaystyle S}$. This is equivalent to setting

${\displaystyle \kappa ((s,t),A):=\kappa (t,A)}$

for all ${\displaystyle A\in {\mathcal {U}}}$ and all ${\displaystyle s\in S}$.^[4][3]

Let ${\displaystyle \kappa ^{1}}$ be a s-finite kernel from ${\displaystyle S}$ to ${\displaystyle T}$ and ${\displaystyle \kappa ^{2}}$ a s-finite kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$. Then the composition ${\displaystyle \kappa ^{1}\cdot \kappa ^{2}}$ of the two kernels is defined as^[5][3]

${\displaystyle \kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]}$

${\displaystyle (s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)}$

for all ${\displaystyle s\in S}$ and all ${\displaystyle B\in {\mathcal {U}}}$.

The composition is a kernel from ${\displaystyle S}$ to ${\displaystyle U}$ that is again s-finite. The composition of kernels is associative, meaning it satisfies

${\displaystyle (\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})}$

for any three suitable s-finite kernels ${\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}$. Just like the product of kernels, the composition is also well-defined if ${\displaystyle \kappa ^{2}}$ is a kernel from ${\displaystyle T}$ to ${\displaystyle U}$.

An alternative notation is for the composition is ${\displaystyle \kappa ^{1}\kappa ^{2}}$^[3]

Let ${\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}}$ be the set of positive measurable functions on ${\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})}$.

Every kernel ${\displaystyle \kappa }$ from ${\displaystyle S}$ to ${\displaystyle T}$ can be associated with a linear operator

${\displaystyle A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}}$

given by^[6]

${\displaystyle (A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).}$

The composition of these operators is compatible with the composition of kernels, meaning^[3]

${\displaystyle A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}}$