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

Lucian Beznea (1, 2) , Madalina Deaconu (3, 4) , Oana Lupascu (5)

1 IMAR - "Simion Stoilow" Institute of Mathematics
2 UniBuc - University of Bucharest
3 IECL - Institut Élie Cartan de Lorraine
4 TOSCA - TO Simulate and CAlibrate stochastic models
5 ISMMA - Gheorghe Mihoc - Caius Iacob Institute of Mathematical Statistics and Applied Mathematics

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.

Mots clés: numerical methods; stochastic equation of fragmentation; Fragmentation equation and kernel; branching process and kernel; avalanche; space of fragmentation sizes; measure-valued process

Domaines: Probabilités [math.PR]

Lucian Beznea, Madalina Deaconu, Oana Lupascu. Numerical approach for stochastic differential equations of fragmentation; application to avalanches. Mathematics and Computers in Simulation, 2019, 160, pp.111-125. ⟨10.1016/j.matcom.2018.12.004⟩. ⟨hal-01667319⟩