S.N. Smirnov. A Feller transition kernel with measure supports given by a set-valued mapping ... P. 219-228

Vol. 25, no. 1, 2019

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 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 $x\mapsto$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 supp$(P(x,\cdot))=S(x)$ for all $x\in X$.

Keywords: Feller property, transition kernel, topological support of a measure, lower semicontinuous set-valued mapping, continuous branch (selection)

Received July 13, 2018

Revised November 16, 2018

Accepted November 19, 2018

Funding Agency: This study was carried out at the Faculty of Computational Mathematics and Cybernetics of Moscow State University within the project “Optimization Methods in Control Problems for Complex Systems under Available Information” (state registration no. AAAA-A16-116021110324-8).

Sergei Nikolaevich Smirnov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: s.n.smirnov@gmail.com

Cite this article as: S.N. Smirnov. A Feller transition kernel with measure supports given by a set-valued mapping, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 219-228. 

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