Markov kernel

In probability theory, a Markov kernel is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.

Formal definition

Let $(X,\mathcal A)$, $(Y,\mathcal B)$ be measurable spaces. A Markov kernel with source $(X,\mathcal A)$ and target $(Y,\mathcal B)$ is a map K that associates to each point $x \in X$ a probability measure K(x) on $(Y,\mathcal B)$ such that, for every measurable set $B\in\mathcal B$, the map $x\mapsto K(x)(B)$ is measurable with respect to the σ-algebra $\mathcal A$.

Let $\mathcal P(X,\mathcal A)$ denote the set of all probability measures on the measurable space $(X,\mathcal A)$. If K is a Markov kernel with source $(X,\mathcal A)$ and target $(Y,\mathcal B)$ then we can naturally associate to K a map $\widehat K:\mathcal P(X,\mathcal A)\to\mathcal P(Y,\mathcal B)$ defined as follows: given P in $\mathcal P(X,\mathcal A)$, we set $\widehat K(P)(B)=\int_XK(x)(B)\,\mathrm dP(x)$, for all B in $\mathcal B$.

References

• Bauer, Heinz (1996), Probability Theory, de Gruyter, ISBN 3-11-013935-9 

§36. Kernels and semigroups of kernels

• Reiss, R D (1993), A Course on Point Processes, Springer-Verlag, ISBN 0387979247