Universal Excursion and Bridge shapes in ABBM/CIR/Bessel processes

Abstract

The exact calculation of the average (as well as the fluctuations) of the avalanche shape for the ABBM model is presented, showing that its normalised shape does not depend on the external drive. Moreover, the average (and the fluctuations) of the multi-avalanche shape, that is the average shape of a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are exactly the same. This result is obtained using the exact solution of the ABBM model, which is obtained leveraging the equivalence with the Cox-Ingersoll-Ross model, rigorously obtained with a so called "time change". A simple presentation of this and other known relevant exact results is provided: notably the correspondence of the ABBM/CIR model with the Rayleigh model and, more importantly, with the generalised Bessel process, which describes the dynamics of the modulus of the multi dimensional Ornstein-Uhlenbeck process (exactly as the Bessel process does for the Brownian process). As a consequence, our main finding, that is the correspondence between the excursion (avalanche) and bridge (multiavalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: if we consider the distance from the origin of such diffusive particles in any dimension, the (normalised) average shape of its trajectory (and the fluctuations around that) until a return in a time T is the same, whether it has returned before T or not.

Generating stochastic trajectories with global dynamical constraints

“…Simulating Brownian bridges can then be easily done by discretizing the effective Langevin equation over small time increments. Effective Langevin equations have been obtained for several constrained processes such as excursions, Brownian meanders, Ornstein-Uhlenbeck processes [26][27][28][29][30][31] and more recently for interacting particles, such as non-intersecting Brownian bridges [32]. It was also recently shown that this concept can be applied to discrete-time random walks with arbitrary jump distributions, including fat-tailed ones [33], as well as to non-Markovian processes, such as the run-and-tumble motion [34].…”

Generating constrained run-and-tumble trajectories

“…The effective Langevin equation (3) can be discretised over time to numerically generate Brownian bridge trajectories with the appropriate statistical weight. The concept of effective Langevin equation is quite robust and can be easily extended to other types of constrained Brownian motions such as excursions, meanders and non-intersecting Brownian motions [24][25][26][27][28]. In addition, the concept was recently extended to the case of discrete-time random walks with arbitrary jump distributions, including fat-tailed distributions, and was also shown to be quite a versatile method [29].…”

Generating stochastic trajectories with global dynamical constraints

“…Simulating Brownian bridges can then be easily done by discretizing the effective Langevin equation over small time increments. Effective Langevin equations have been obtained for several constrained processes such as excursions, meanders [25][26][27][28] and more recently for interacting particles, such as non-intersecting Brownian bridges [29]. It was also recently shown that this concept can be applied to discrete-time random walks with arbitrary jump distributions, including fat-tailed ones [30], as well as to non-Markovian processes, such as the run-and-tumble motion [31].…”