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A Feller Transition Kernel with Measure Supports Given by a Set-Valued Mapping
S. N. Smirnov Lomonosov Moscow State University
Abstract:
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
$x\in X$.
Keywords:
Feller property, transition kernel, topological support of a measure, lower semicontinuous set-valued mapping, continuous branch (selection).
UDC:
519.216,
519.866.2
MSC: 60J35,
91B25 Received: 13.07.2018
Revised: 16.11.2018
Accepted: 19.11.2018
DOI:
10.21538/0134-4889-2019-25-1-219-228
Fulltext:
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