Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations
Abstract
The paper consists of two parts. At first, assuming that (Ω,𝒜,P) is a probability space and (X,ρ) is a complete and separable metric space with the σ-algebra 𝐵 of all its Borel subsets we consider the set 𝑅c of all 𝐵⊗𝒜 measurable and contractive in mean functions f:X×Ω→X with finite integral ∫Ωρ(f(x,ω),x)P(dω) for x∈X, the weak limit πf of the sequence of iterates of f∈𝑅c, and investigate continuity-like property of the function f↦πf, f∈𝑅c, and Lipschitz solutions ϕ that take values in a separable Banach space of the equation
ϕ(x) = ∫Ωϕ(f(x,ω))P(dω) + F(x).
Next, assuming that X is a real separable Hilbert space, Λ:X→X is linear and continuous with ‖Λ‖<1, and μ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions ϕ:X→ℂ of the equation
ϕ(x) = μ^(x)ϕ(Λx)
which characterizes the limit distribution πf for some special f∈𝑅c.
Keywords
random-valued functions; iterates; convergence in law; continuous dependence on the given function; Fourier transform; iterative functional equations; continuous and Lipschitz solutions
References
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Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
https://orcid.org/0000-0002-0807-2776
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