Let $X$ and $Y$ be Poisson point processes on the real numbers with rates $l_1$ and $l_2$ respectively. We show that if $l_1 > l_2$, then there exists a deterministic map $f$ such that $f(X)$ and $Y$ have the same distribution, the joint distribution of $(X, f(X))$ is translation-invariant, and which is monotone in the sense that for all intervals $I$, $f(X)(I) \leq X(I)$, almost surely.
"Poisson Thinning by Monotone Factors." Electron. Commun. Probab. 10 60 - 69, 2005. https://doi.org/10.1214/ECP.v10-1134