Minimal displacement of random variables under Lipschitz random maps

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.