Abstract

           The project develops stochastic analysis and nonlinear equations aspects related to fragmentation processes, with applications to avalanches’ modeling. The approach is based on modern infinite dimensional stochastic analysis techniques, stochastic differential equations, jump processes, semimartingale theory, but also on some applied mathematical tools from numerical analysis. A main difficulty is to relate the physical deterministic models on the fragmentation and branching stochastic processes. We first invesitigate fragmentation processes depending on  several fragmentation kernels and on the position of the fragments in a given space.  The motivation comes from real-life rupture models  where  at each space position not only the mass fragmentation  occurs but also the velocity is “fragmented”. A new stochastic differential equation of  multi-fragmentation, a specific space of fragmentation sizes and a nonlinear integral equation will occur. Second, we study the dual of  the time continuous branching and fragmentation processes, in the sens of the weak duality for Markov processes, giving a precise mathematical meaning to a statement of J. Bertoin. We investigate the semimartingale and the quasimartingale structures of  the superprocesses, necessary in order to consider stochastic integrals against such processes. Third, we treat analytical, probabilistic, and geometrical problems related to first order operators, which play the role of the operator induced by a gradient in infinit dimensions. We construct semiflows induced by Markovian Lp-semigroups and we present a method of using the Girsanov’s transformation to improve some classical inequalities. Fourth, we develop mesh free methods, as an alternative to finite element discretization techniques, for the onset of a shallow dense avalanche, viewed as a fragmentation process modeled by a nonlinear PDE. The rupture or fragmentation problem is reduced to the minimization of of a mesh free functional.