Weak limit of iterates of some random-valued functions and its application

Abstract

Given a probability space \( (\Omega , {\mathcal {A}}, P) \), a complete and separable metric space X with the \( \sigma \)-algebra \( {\mathcal {B}} \) of all its Borel subsets, a \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable and contractive in mean \( f: X \times \Omega \rightarrow X \), and a Lipschitz F mapping X into a separable Banach space Y we characterize the solvability of the equation

$$\begin{aligned} \varphi (x)=\int _{\Omega }\varphi \left( f(x,\omega )\right) P(d\omega )+F(x) \end{aligned}$$

in the class of Lipschitz functions \(\varphi : X \rightarrow Y\) with the aid of the weak limit \(\pi ^f\) of the sequence of iterates \(\left( f^n(x,\cdot )\right) _{n \in {\mathbb {N}}}\) of f, defined on \( X \times \Omega ^{{\mathbb {N}}}\) by \(f^0(x, \omega ) = x\) and \( f^n(x, \omega ) = f\left( f^{n-1}(x, \omega ), \omega _n\right) \) for \(n \in {\mathbb {N}}\), and propose a characterization of \(\pi ^f\) for some special rv-functions in Hilbert spaces.

Change history

  • 27 January 2020

    In the original publication, Example 2.2 was incorrectly published.

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Acknowledgements

I thank Professor Gregory Derfel for calling my attention to the paper by O. K. Zakusilo.

This research was supported by the University of Silesia Mathematics Department (Iterative Functional Equations and Real Analysis program).

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Correspondence to Karol Baron.

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Dedicated to Professor János Aczél on his 95th birthday.

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Baron, K. Weak limit of iterates of some random-valued functions and its application. Aequat. Math. 94, 415–425 (2020). https://doi.org/10.1007/s00010-019-00650-z

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Keywords

  • Random-valued functions
  • Iterates
  • Weak limit
  • Iterative equations
  • Lipschitzian solutions
  • Bochner integral
  • Gaussian measures

Mathematics Subject Classification

  • Primary 39B12
  • 26A18
  • Secondary 60B12
  • 58D20