Transition kernel

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 \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) for all s \in S and all i \in \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 du) \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 dt) \int_U \kappa^2((s,t), \mathrm du) \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 dt)\; 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} == References ==