Super-Brownian motion with reflecting historical paths. II: Convergence of approximations (Q2571008)

The authors complete the project started by \textit{K. Burdzy} and \textit{J.-F. Le Gall} [Probab. Theory Relat. Fields 121, No. 4, 447--491 (2001; Zbl 1001.60086)]. In that paper, tightness was proved for a sequence of branching particle systems which the authors expected should converge to a limit representing super-Brownian motion with reflecting historical paths, but uniqueness of a limit law was left open. Some properties of the historical paths under any limiting distribution were also proved. In the present paper it is shown that the sequence of approximations indeed converges, and not only in distribution, but also in probability in an appropriate space. The proof uses Le Gall's Brownian snake.