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Transition kernel

Mathematical function

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Source updated January 4, 2026

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.

01Definition

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 $(T,{\mathcal {T}})$ .

02Classification 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 $\kappa$ can be written as a countable sum of finite kernels (so that in particular, all $\kappa _{s}$ are $s$ -finite measures).
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 $i\in \mathbb {N}$ .

03Operations

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 $\kappa ^{1}\kappa ^{2}$

04Kernels 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

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

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Sources and credits

This article is adapted from the Wikipedia article “Transition kernel”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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