Numerical approach for stochastic differential equations of fragmentation; application to avalanches

Abstract

This paper builds and develops an unifying method for the construction of a continuous time fragmentation-branching processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by discontinuous ones. This construction leads to a stochastic model for the fragmentation phase of an avalanche. We introduce also an approximation scheme for the process which solves the corresponding stochastic differential equations of fragmentation. A new achievement of the paper is to compute the distributions of the branching processes approximating the forthcoming branching-fragmentation process. This numerical approach of the associated branching-fragmentation process, is, to our knowledge, one of the first in this direction. We present also numerical results that confirm the validity of the fractal property which was emphasized by our model for an avalanche.

Scaling Property for Fragmentation Processes Related to Avalanches

“…We fix a set A ⊂ [0, 1], a point x ∈ [0, 1], a final time T ∈ R * + and n ∈ N * . In the first step, we sample values of X T starting from x as a solution of the corresponding stochastic differential equation of fragmentation with the discontinuous kernel F r , by using the algorithm developed and implemented in [3]. For the reader's convenience we recall it below, we can remark the fractal property of the resulting fragments after the splitting, property which holds according to assertion (iii) of Theorem 2.…”

“…In the papers [1], [2], and [3] we studied binary fragmentation processes (and associated non-local branching processes, cf. [4]) of an infinite particles system, including a numerical approach for the time evolution of the fragmentation phase of an avalanche.…”

Time‐average stochastic control based on a singular local Lévy model for environmental project planning under habit formation

“…Recent experimental and modeling studies suggested that the storage decrease is a function of time as with certain power coefficient [40]. In addition, erosion is a macroscopic emergence of stochastic avalanching events of sediment lumps [41]. We show that the previous jump‐driven stochastic process model [34] cannot capture the temporal scaling.…”