Transporting random measures on the line and embedding excursions into Brownian motion

We consider two jointly stationary and ergodic random measures ξ and η on the real line R with equal intensities. An allocation is an equivariant random mapping from R to R. We give sufficient and partially necessary conditions for the existence of allocations transporting ξ to η. An important ingredient of our approach is a transport kernel balancing ξ and η, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on −∞, 0], an excursion distributed according to a conditional Itô measure and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion measure.