A Feller Transition Kernel with Measure Supports Given by a Set-Valued Mapping

Assume that $X$ is a topological space and $Y$ is a separable metric space. Let these spaces be equipped with Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, respectively. Suppose that $P(x,B)$ is a stochastic transition kernel; i.e., the mapping $x \mapsto P(x,B)$ is measurable for all $B \in \mathcal{B}_Y$ and the mapping $B\mapsto P(x, B)$ is a probability measure for any $x \in X$. Denote by $\mathrm{supp}(P(x,\cdot))$ the topological support of the measure $B\mapsto P(x, B)$. If the transition kernel $P(x,B)$ satisfies the Feller property, i.e., the mapping $x \mapsto P(x,\cdot)$ is continuous in the weak topology on the space of probability measures, then the set-valued mapping $x\mapsto\mathrm{supp}(P(x,\cdot))$ is lower semicontinuous. Conversely, consider a set-valued mapping $x\mapsto S(x)$, where $x\in X$ and $S(x)$ is a nonempty closed subset of a Polish space $Y$. If $x \mapsto S(x)$ is lower semicontinuous, then, under some general assumptions on the space $X$, there exists a Feller transition kernel such that $\mathrm{supp}(P(x,\cdot))=S(x)$ for all $x\in X$.