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We point out that the mean-field theory of avalanches in the dynamics ofelastic interfaces, the so-called Brownian force model (BFM) developed recentlyin non-equilibrium statistical physics, is equivalent to the so-calledsuper-Brownian motion (SBM) developed in probability theory, a continuum limitof branching processes related to {\it space-embedded} Galton-Watson trees. Inparticular the exact solvability property recently (re-)discovered from thefield theory in mean-field avalanches (the "instanton equation") maps onto theso-called Dawson-Watanabe 1968 duality property. In the light of thiscorrespondence we compare the results obtained independently in the two fields,and transport some of them from one field to the other. In particular, wediscuss a scaling limit of the branching Brownian motion which maps onto thecontinuum field theory of mean-field avalanches

Equivalence of mean-field avalanches and branching diffusions: From the Brownian force model to the super-Brownian motion