In my thesis, I study the extremal process of the branching Brownian motion. I am interested in a model in which particles are independent, move in space according to a Brownian motion and branch. To study the extremal process, I first introduce some basic definitions and facts on extreme value theory. Then, I define branching random walks and branching Brownian motion. I first study the maximum of both models and achieve important results. After that I focus on the extremal process of the branching Brownian motion, which is studied through a point process. The main theorem tells us that such a process converges to a Poisson cluster point process in the limit of large times. The thesis also contains a deep study of the paths of extremal particles: the result shows that such particles can move only in an admissible region that can be determined. Moreover, I prove an interesting theorem on the genealogy of the extremal particles: in the limit of large times, they descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0 or of order 1 from time t.

### Analysis of extremes in the branching Brownian motion

#### Abstract

In my thesis, I study the extremal process of the branching Brownian motion. I am interested in a model in which particles are independent, move in space according to a Brownian motion and branch. To study the extremal process, I first introduce some basic definitions and facts on extreme value theory. Then, I define branching random walks and branching Brownian motion. I first study the maximum of both models and achieve important results. After that I focus on the extremal process of the branching Brownian motion, which is studied through a point process. The main theorem tells us that such a process converges to a Poisson cluster point process in the limit of large times. The thesis also contains a deep study of the paths of extremal particles: the result shows that such particles can move only in an admissible region that can be determined. Moreover, I prove an interesting theorem on the genealogy of the extremal particles: in the limit of large times, they descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0 or of order 1 from time t.
##### Scheda Scheda DC
2019-04-19
96
branching Brownian motion
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.12608/27414`