We consider a real-valued branching Brownian motion where particles are killed at rate $\mu$ and split at rate $\lambda \leq \mu$ into two independent offspring particles. The process dies out almost surely, so it reaches some lowest level. We prove a decomposition of the branching Brownian path at its minimum. The post-minimum path is just branching Brownian motion conditioned never to go beneath its initial point. The pre-minimum piece is independent of the post-minimum piece, and has the same law as the post-minimum piece reweighted by a functional of the endpoints of the tree. Applications to branching polymers are discussed.
"Decomposing the Branching Brownian Path." Ann. Appl. Probab. 2 (4) 973 - 986, November, 1992. https://doi.org/10.1214/aoap/1177005584