Transition kernel explained

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,lS)

(T,lT)

be two measurable spaces. A function

\kappa\colonS x lT\to[0,+infty]

is called a (transition) kernel from

if the following two conditions hold:

For any fixed

B\inlT

, the mapping

s\mapsto\kappa(s,B)

lS/lB([0,+infty])

-measurable;

For every fixed

s\inS

, the mapping

B\mapsto\kappa(s,B)

is a measure on

(T,lT)

Classification of transition kernels

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

\kappa_s\colonlT\to[0,+infty]

with

\kappa_{s(B)=\kappa(s,B)}

for all

B\inlT

and all

s\inS

. 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

\sigma

-finite kernel if all
\kappa_s

are
\sigma

-finite measures

-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,...

in
T

with
\kappa_s(B_i)<infty

for all
s\inS

and all
i\in\N

.

Operations

In this section, let

(S,lS)

(T,lT)

and

(U,lU)

be measurable spaces and denote the product σ-algebra of

and

with

lS ⊗ lT

Product of kernels

Definition

Let

\kappa¹

be a s-finite kernel from

and

\kappa²

be a s-finite kernel from

S x T

. Then the product

\kappa¹ ⊗ \kappa²

of the two kernels is defined as

\kappa¹ ⊗ \kappa²\colonS x (lT ⊗ lU)\to[0,infty]

\kappa¹ ⊗ \kappa^2(s,A)=\int_T\kappa^1(s,dt)\int_U\kappa^2((s,t),du)1_A(t,u)

for all

A\inlT ⊗ lU

Properties and comments

The product of two kernels is a kernel from

T x U

. It is again a s-finite kernel and is a

\sigma

-finite kernel if

\kappa¹

and

\kappa²

are

\sigma

-finite kernels. The product of kernels is also associative, meaning it satisfies

(\kappa¹ ⊗ \kappa²⁾ ⊗ \kappa³⁼\kappa¹ ⊗ (\kappa^{2 ⊗}\kappa³⁾

for any three suitable s-finite kernels

\kappa^1,\kappa^2,\kappa³

The product is also well-defined if

\kappa²

is a kernel from

. In this case, it is treated like a kernel from

S x T

that is independent of

. This is equivalent to setting

\kappa((s,t),A):=\kappa(t,A)

for all

A\inlU

and all

s\inS

Composition of kernels

Definition

Let

\kappa¹

be a s-finite kernel from

and

\kappa²

a s-finite kernel from

S x T

. Then the composition

\kappa¹ ⋅ \kappa²

of the two kernels is defined as

\kappa¹ ⋅ \kappa²\colonS x lU\to[0,infty]

(s,B)\mapsto\int_T\kappa^1(s,dt)\int_U\kappa^2((s,t),du)1_B(u)

for all

s\inS

and all

B\inlU

Properties and comments

The composition is a kernel from

that is again s-finite. The composition of kernels is associative, meaning it satisfies

(\kappa¹ ⋅ \kappa²⁾ ⋅ \kappa³⁼\kappa¹ ⋅ (\kappa² ⋅ \kappa³⁾

for any three suitable s-finite kernels

\kappa^1,\kappa^2,\kappa³

. Just like the product of kernels, the composition is also well-defined if

\kappa²

is a kernel from

An alternative notation is for the composition is

\kappa¹\kappa²

Kernels as operators

Let

lT^+,lS⁺

be the set of positive measurable functions on

(S,lS),(T,lT)

Every kernel

\kappa

from

can be associated with a linear operator

A_\kappa\colonlT⁺\tolS⁺

given by

(A_\kappaf)(s)=\int_T\kappa(s,dt) f(t).

The composition of these operators is compatible with the composition of kernels, meaning

A
	\kappa¹

A
	\kappa²

A
	\kappa¹ ⋅ \kappa²

References

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Notes and References

Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 29-30. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.
Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 30. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.
Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 180.
Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 281.
Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 33. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.
Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 279.