Let $X$ and $Y$ be Polish spaces and $K$ a Markov kernel from $X$ to $Y$. That is, $K$ is a mapping $X \times \mathcal{B}\_Y \rightarrow [0,1]$ (where $\mathcal{B}\_Y$ is the $\sigma$-algebra of Borel sets on $Y$) s.t.


* For every $A \in \mathcal{B}\_Y$, the mapping $K^A: X \rightarrow [0,1]$ defined by $K^A(x):=K(x,A)$ is Borel measurable.
* For every $x \in X$, the mapping $K\_x: \mathcal{B}\_Y \rightarrow [0,1]$ defined by $K\_x(A):=K(x,A)$ is a probability measure on $Y$.


It seems natural to consider the additional condition on $K$ that $K\_x$ depends continuously on $x$ in the sense of the given topology on $X$ and the *weak* topology on the space $\mathcal{P}(Y)$ of probability measures on $Y$. For example, if $K$ is deterministic (i.e. if for any $x \in X$, $K\_x$ is a Dirac measure) this condition means that $K$ defines a continuous mapping from $X$ to $Y$.



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> What is the standard name for this condition?
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What do you call a Markov kernel continuous w.r.t. the weak topology?
As stated in the comment, this seems to be some kind of [Feller continuity](https://en.wikipedia.org/wiki/Feller-continuous_process).


Having said that, I should emphasize that there is some confusion about the name "Feller" with regard to the properties of a Markov process, Markov transition function or a Markov kernel, and link to [Martin Hairer's comment about that](https://mathoverflow.net/a/280390/108637).


