Temps de Branchement du Mouvement Brownien Branchant Inhomogène
This thesis studies the behavior of particles that are maximal at time t in branching random walk and branching Brownian motion on R, for large values of t. Precisely, we look at the behavior of the maximum in a branching random walk in a time-inhomogeneous environment, where the law of the increments varies with respect to time. We compare with known or simplified models such as the model where random walks are taken to be i.i.d. and the branching random walk in a time-homogeneous environment model. We then take a look at the correlations between maximal particles in a branching brownian motion. Specifically, we look at the branching time between those maximal particles. Finally, we apply results and methods from the first chapters to study those same correlations in branching Brownian motion in a inhomogeneous environment. The thesis’ main result establishes existence of branching time at the center of the interval [0, t] for the branching Brownian motion in a inhomogeneous environment, which is not the case for standard branching brownian motion.We also present results of simulations that agree with theoretical results and help establishing new hypotheses for future research.
Branching random walk Branching Brownian motion Branching time