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

Abstract

Assume that X is a topological space and Y is a separable metric space. Let these spaces be equipped with Borel σ-algebras BX and BY, respectively. Suppose that P(x, B) is a stochastic transition kernel; i.e., the mapping x ↦ P(x,B) is measurable for all B ∈ BY and the mapping B ↦ P(x, B) is a probability measure for any x ∈ X. Denote by supp(P(x, ·) the topological support of the measure B ↦ P(x,B). If the transition kernel P(x,B) satisfies the Feller property, i.e., the mapping x ↦ P(x, ·) is continuous in the weak topology on the space of probability measures, then the set-valued mapping x ↦ supp(P(x, ·) is lower semicontinuous. Conversely, consider a set-valued mapping x ↦ S(x), where x ∈ X and S(x) is a nonempty closed subset of a Polish space Y. If x ↦ 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, ·) = S(x) for all x ∈ X.