Topological Methods in Nonlinear Analysis

Minimal displacement of random variables under Lipschitz random maps

Ismag Beg

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Abstract

Let $(\Omega ,\Sigma )$ be a measurable space and $X$ be a separable metric space. It is shown that for measurable maps $\zeta ,\eta \colon \Omega \rightarrow X$, if a random map $T\colon\Omega \times X\rightarrow X$ satisfies $d(T(\omega ,\zeta (\omega )),T(\omega ,\eta (\omega )))\leq \alpha d(\zeta (\omega ),\eta (\omega ))+\gamma $ then $\inf\{d(\xi (\omega ),T(\omega ,\xi (\omega )))\}\leq \gamma/ (1-\alpha)$, where $\gamma > 0$, $\alpha \in (0,1)$ and $\inf$ is taken over all measurable maps $\xi \colon \Omega \rightarrow X$. Several consequences of this result are also obtained.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 19, Number 2 (2002), 391-397.

Dates
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1470138770

Mathematical Reviews number (MathSciNet)
MR1921055

Zentralblatt MATH identifier
1030.47047

Citation

Beg, Ismag. Minimal displacement of random variables under Lipschitz random maps. Topol. Methods Nonlinear Anal. 19 (2002), no. 2, 391--397. https://projecteuclid.org/euclid.tmna/1470138770


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