We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.
Kyprianou, A. E., Murillo-Salas, A., & Pérez, J. L. (2012). An application of the backbone decomposition to supercritical super-Brownian motion with a barrier. Journal of Applied Probability, 49(3), 671-684. https://doi.org/10.1239/jap/1346955325