A Feller transition kernel with measure supports given by a set-valued mapping (Q2185797)

Let \((X, \mathcal{B}_X)\) and \((Y, \mathcal{B}_Y)\) be measurable spaces. A function \(Q\) defined on \( X\times \mathcal{B}_Y \) is called a stochastic transition kernel if a mapping \(x\mapsto Q(x, B)\) is \(\mathcal{B}_X\)-measurable for all \(B\in \mathcal{B}_Y\) and a function \(B\mapsto Q(x, B)\) is a probability measure for all \(x\in X\). Denote by \(\text{supp}(Q)\) a topological support of a probability measure \(Q\) and by \(\mathcal{P}(Y)\) the space of probability measures on \(Y\). We say that the transition kernel \(Q\) satisfies the Feller property if the mapping \(x\mapsto Q(x, \cdot)\) is continuous in the weak topology on the space \(\mathcal{P}(Y)\). The set \( \text{supp}(Q)\) is always closed. Moreover, for any closed subset \(F\) of \(Y\) there exists a measure \(Q\in \mathcal{P}(Y)\) such that its support is equal to \(F\). If \(Y\) is a separable metric space, then \(Q\mapsto \text{supp}(Q)\) is lower semicontinuous in the weak topology on \(\mathcal{P}(Y)\). If \(X\) is a topological space and \(Y\) is a metric space with Borel \(\sigma\)-algebras \(\mathcal{B}_X\) and \(\mathcal{B}_Y\) (respectively) and a mapping \(x\mapsto Q(x, \cdot)\in \mathcal{P}(Y)\), \(x\in X\), is continuous then \(Q=Q(x, B)\), \(x\in X\), \(B\in \mathcal{B}_Y\), is a transition kernel. Assuming that \(X\) is a topological space and \(Y\) is a separable metric space with Borel \(\sigma\)-algebras \(\mathcal{B}_X\) and \(\mathcal{B}_Y\) (respectively) and a transition kernel \(Q=Q(x, B)\) (\(x\in X\), \(B\in \mathcal{B}_Y\)) has the Feller property, then the set-valued mapping \(x\mapsto \text{supp}(Q(x, \cdot))\) is lower semicontinuous. Conversely: if \(X\) is a normal countably paracompact space, \(Y\) is a Polish space (with Borel \(\sigma\)-algebras \(\mathcal{B}_X\) and \(\mathcal{B}_Y\), respectively) and the set-valued mapping \(S\colon X\to 2^X\) with nonempty closed values is lower semicontinuous, then there exists a Feller transition kernel \(Q=Q(x, B)\) (\(x\in X\), \(B\in \mathcal{B}_Y\)) such that \(\text{supp}(Q(x, \cdot))\subseteq S(x)\). If in the above theorem we assume that \(X\) is perfectly normal, then \(\text{supp}(Q(x, \cdot))= S(x)\) for \(x\in X\).