A ratio limit theorem for erased branching Brownian motion

In this paper we study a branching Brownian motion inside a bounded, smooth domain with killing at the boundary. The paths of the process are transformed--roughly speaking--so that all the branches which reach the boundary are totally erased with unit speed for a given time [rho], 0[less-than-or-equals, slant][rho][less-than-or-equals, slant][infinity], starting from the tip of the branch. A limit theorem for the ratio of the number of particles in the erased process and the original one is proved. This may be viewed as a generalisation of a result for Galton-Watson processes due to Athreya and Ney (1972).