Universal excursion and bridge shapes in ABBM/CIR/Bessel processes

Several years ago, in the context of physics hysteresis magnetic materials, a simple stochastic model has been introduced: ABBM model. Later, advocated as paradigm for broad class diverse phenomena, baptised "crackling noise phenomena". The reproduces many statistical features such intermittent signals, statistics burst (or avalanche) durations and sizes, with their power law exponents that would characterise dynamics critical. Beyond "critical exponents", measure average shape avalanche also proposed. Here, exact calculation fluctuations is presented, showing its normalised independent from external drive. Moreover, multi-avalanche shape, sequence avalanches fixed total duration, computed. Surprisingly, two quantities (avalanche shapes) are identical. This result obtained using solution model, leveraging equivalence Cox-Ingersoll-Ross process (CIR), through an "time change". A presentation this other known results provided: notably correspondence ABBM/CIR generalised Bessel process, describing modulus multi dimensional Ornstein-Uhlenbeck process. As consequence, between excursion (avalanche) bridge (multi-avalanche) distributions, turns to apply all aforementioned processes. In words: considering distance origin diffusive particles, (normalised) (and fluctuations) trajectory until return time T same, whether it returned before or not.