Random stirring of the real line $R_1$ is defined. This notion is derived from a generalization of the nearest-neighbor simple exclusion model on the one-dimensional lattices discussed by Spitzer and by Harris. Under the random stirring, the motion of an infinite particle system is Markovian and has a Poisson process as an invariant probability measure. An ergodic theorem is established concerning the convergence of a system to a Poisson process.
"Random Stirring of the Real Line." Ann. Probab. 2 (4) 580 - 592, August, 1974. https://doi.org/10.1214/aop/1176996605