Minimal displacement of random variables under Lipschitz random maps

Ismag Beg

DOI: http://dx.doi.org/10.12775/TMNA.2002.020

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.

Keywords


Minimal displacement; random contraction; nonexpansive random map; measurable space; metric space

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