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

TL;DR

This paper reveals a deep equivalence between the mean-field avalanche theory in elastic interface dynamics and super-Brownian motion in probability theory, enabling cross-field insights and transfer of results.

Contribution

It establishes a precise correspondence between the Brownian force model and super-Brownian motion, connecting two previously separate theoretical frameworks.

Findings

Mapping of the instanton equation to Dawson-Watanabe duality

Comparison of results across physics and probability theory

Scaling limit of branching Brownian motion related to mean-field avalanches

Abstract

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