## Intermittency for Branching random walks with heavy tails.

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## Abstract

iic 2016Asmaa GetanALL RIGHTS RESERVEDiiiABSTRACT Branching Markov process models populations in which each individual in gener-ation n produces some random number of individuals in the next generation, n + 1, according to a certain probability distribution.Branching processes play important role in the study of the evolution of various population plants, where members of the population may die or produce ospring independently of the rest. They can be used to model reproduction of bacteria where each bacteria generates several ospring with some probability in a single time unit. And they can be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor.In our dissertation, we consider a long time behavior for a model of Branchingrandom walk problem of a population of particles on the d- dimensional lattice Z^d.In this model, the number of particles increases exponentially by duplicating, with a constant rate of birth (each particle can split into two particles), and the particles spread everywhere by jumping to not necessary a neighbor place (it could be a faraway distance) under probability of jumps that is described to be a Heavy tailed probability. Branching or jumping of each particle occurs independently of the other particles.Under these two conditions (constant rate of birth and heavy tailed probability of jumps), the front of propagation (where local growth occurs) has been found to be moving exponentially fast.A well developed non-uniformity concept called intermittency, is used to investigatethe uniformity of the distribution of the particles, on, inside and outside of the front.A random field is called intermittent if it is distributed very non-uniformly, where huge values can appear with a very small probability. For instance, the magnetic field of the sun is highly intermittent, as almost all of its energy is concentrated in black spots which covers only small parts of the surface of the sun.In our work, we found that particles on the front exhibit intermittent behavior. We proved that, the same is true for some region inside the front. Despite that the front of propagation of particles moves exponentially fast, the front of intermittency moves with a small power rate, |x| > t^γ, inside the first front. In the area between those two fronts, the particles are concentrated in very sparse spots with clustered density. This means that, the zone of non-intermittency extends with that rate too.This rate has been found exactly.