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Continuous dependence of the weak limit of iterates of some random-valued vector functions.
- Source :
-
Aequationes Mathematicae . Aug2023, Vol. 97 Issue 4, p753-776. 24p. - Publication Year :
- 2023
-
Abstract
- Given a probability space (Ω , A , P) , a complete separable Banach space X with the σ -algebra B (X) of all its Borel subsets, an operator Λ : Ω → L (X , X) and ξ : Ω → X we consider the B (X) ⊗ A -measurable function f : X × Ω → X given by f (x , ω) = Λ (ω) x + ξ (ω) and investigate the continuous dependence of a weak limit π f of the sequence of iterates (f n (x , ·)) n ∈ N of f, defined by f 0 (x , ω) = x , f n + 1 (x , ω) = f (f n (x , ω) , ω n + 1) for x ∈ X and ω = (ω 1 , ω 2 , ⋯) . Moreover for X taken as a Hilbert space we characterize π f via the functional equation φ f (u) = ∫ Ω φ f (Λ (ω) u) φ ξ (u) P (d ω) with the aid of its characteristic function φ f . We also indicate the continuous dependence of a solution of that equation. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00019054
- Volume :
- 97
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Aequationes Mathematicae
- Publication Type :
- Academic Journal
- Accession number :
- 164224983
- Full Text :
- https://doi.org/10.1007/s00010-023-00959-w