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 , S ) {\displaystyle (S,{\mathcal {S}})} , ( T , T ) {\displaystyle (T,{\mathcal {T}})} be two measurable spaces. A function
κ : S × T → [ 0 , + ∞ ] {\displaystyle \kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]}
is called a (transition) kernel from S {\displaystyle S} to T {\displaystyle T} if the following two conditions hold:
For any fixed B ∈ T {\displaystyle B\in {\mathcal {T}}} , the mapping
s ↦ κ ( s , B ) {\displaystyle s\mapsto \kappa (s,B)}
is S / B ( [ 0 , + ∞ ] ) {\displaystyle {\mathcal {S}}/{\mathcal {B}}([0,+\infty ])} -measurable; For every fixed s ∈ S {\displaystyle s\in S} , the mapping
B ↦ κ ( s , B ) {\displaystyle B\mapsto \kappa (s,B)}
is a measure on ( T , T ) {\displaystyle (T,{\mathcal {T}})} .
Classification of transition kernels Transition kernels are usually classified by the measures they define. Those measures are defined as
κ s : T → [ 0 , + ∞ ] {\displaystyle \kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]}
with
κ s ( B ) = κ ( s , B ) {\displaystyle \kappa _{s}(B)=\kappa (s,B)}
for all B ∈ T {\displaystyle B\in {\mathcal {T}}} and all s ∈ S {\displaystyle s\in S} . With this notation, the kernel κ {\displaystyle \kappa } is called
a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all κ s {\displaystyle \kappa _{s}} are sub-probability measures a Markov kernel, stochastic kernel or probability kernel if all κ s {\displaystyle \kappa _{s}} are probability measures a finite kernel if all κ s {\displaystyle \kappa _{s}} are finite measures a σ {\displaystyle \sigma } -finite kernel if all κ s {\displaystyle \kappa _{s}} are σ {\displaystyle \sigma } -finite measures a s {\displaystyle s} -finite kernel if κ {\displaystyle \kappa } can be written as a countable sum of finite kernels (so that in particular, all κ s {\displaystyle \kappa _{s}} are s {\displaystyle s} -finite measures). a uniformly σ {\displaystyle \sigma } -finite kernel if there are at most countably many measurable sets B 1 , B 2 , … {\displaystyle B_{1},B_{2},\dots } in T {\displaystyle T} with κ s ( B i ) < ∞ {\displaystyle \kappa _{s}(B_{i})<\infty } for all s ∈ S {\displaystyle s\in S} and all i ∈ N {\displaystyle i\in \mathbb {N} } .
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