Markov kernel

A Markov kernel from a measurable space $(E,\mathcal{E})$ to a measurable space $(F,\mathcal{F})$ is a function $K:E\times \mathcal{F}\to [0,1]$ satisfying two conditions: for each $x\in E$, $A\mapsto K(x,A)$ is a Probability measure; for each $A\in \mathcal{F}$, $x\mapsto K(x,A)$ is a measurable function. The second condition makes integrals against a distribution of current states well defined.

If the current state has probability distribution $\mu$, the next-state distribution isOn a finite or countable state space with all subsets measurable, a transition matrix gives $K(i,A)={\sum }_{j\in A}{P}_{ij}$. A deterministic measurable map $T:E\to F$ gives $K(x,A)=1_A(T(x))$, so the kernel formalism includes deterministic dynamics as well as random transitions.

Successive kernels compose by $(KL)(x,A)={\int }_{F}L(y,A)K(x,dy)$. The tower property of conditional expectation explains this rule: first condition on the intermediate state, then average over it. For a homogeneous Markov chain, the $n$-step transition kernel is $K^n$; its composition law is the Chapman-Kolmogorov equation.

A probability density function for the conditional future state of a Markov process given its current state. Integrating $p_t(x,y)$ over a measurable region gives its transition probability. A transition kernel need not admit a density; discrete or partly atomic transition laws require a different reference measure or a separate atomic part.