As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.
"Finitary isomorphisms of Poisson point processes." Ann. Probab. 47 (5) 3055 - 3081, September 2019. https://doi.org/10.1214/18-AOP1332