In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.

Let ${\displaystyle (X,{\mathcal {A}})}$ and ${\displaystyle (Y,{\mathcal {B}})}$ be measurable spaces. A Markov kernel with source ${\displaystyle (X,{\mathcal {A}})}$ and target ${\displaystyle (Y,{\mathcal {B}})}$, sometimes written as ${\displaystyle \kappa :(X,{\mathcal {A}})\to (Y,{\mathcal {B}})}$, is a function ${\displaystyle \kappa :{\mathcal {B}}\times X\to [0,1]}$ with the following properties:

In other words it associates to each point ${\displaystyle x\in X}$ a probability measure ${\displaystyle \kappa (dy|x):B\mapsto \kappa (B,x)}$ on ${\displaystyle (Y,{\mathcal {B}})}$ such that, for every measurable set ${\displaystyle B\in {\mathcal {B}}}$, the map ${\displaystyle x\mapsto \kappa (B,x)}$ is measurable with respect to the ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {A}}}$.

Take ${\displaystyle X=Y=\mathbb {Z} }$, and ${\displaystyle {\mathcal {A}}={\mathcal {B}}={\mathcal {P}}(\mathbb {Z} )}$ (the power set of ${\displaystyle \mathbb {Z} }$). Then a Markov kernel is fully determined by the probability it assigns to singletons ${\displaystyle \{m\},\,m\in Y=\mathbb {Z} }$ for each ${\displaystyle n\in X=\mathbb {Z} }$:

Now the random walk ${\displaystyle \kappa }$ that goes to the right with probability ${\displaystyle p}$ and to the left with probability ${\displaystyle 1-p}$ is defined by

where ${\displaystyle \delta }$ is the Kronecker delta. The transition probabilities ${\displaystyle P(m|n)=\kappa (\{m\}|n)}$ for the random walk are equivalent to the Markov kernel.

More generally take ${\displaystyle X}$ and ${\displaystyle Y}$ both countable and ${\displaystyle {\mathcal {A}}={\mathcal {P}}(X),\ {\mathcal {B}}={\mathcal {P}}(Y)}$. Again a Markov kernel is defined by the probability it assigns to singleton sets for each ${\displaystyle i\in X}$

We define a Markov process by defining a transition probability ${\displaystyle P(j|i)=K_{ji}}$ where the numbers ${\displaystyle K_{ji}}$ define a (countable) stochastic matrix ${\displaystyle (K_{ji})}$ i.e.