| In this thesis, we generalize earlier results on branching annihilating random walk in one dimension. In this model, particles perform random walks, they branch into new particles, and they annihilate each other upon collision. The result we prove in this thesis is that for branching rate sufficiently low, the system is guaranteed to go to extinction. That is, if the process starts with an even number of particles, all particles will eventually disappear through annihilations, and if it starts with infinitely many particles, the particles become arbitrarily sparse.; Our motivation for studying branching annihilating random walks is its usefulness for analyzing traffic flow. We focus on a traffic flow model, which we call the Discrete Traffic Model. The model, which is a special case of a model of Gray and Griffeath, is a probabilistic cellular automaton. In this thesis, we conjecture that for this model, with suitable parameter values, the distribution of traffic approaches a limiting distribution which is a mixture of "traffic jam" and "freeflow" traffic. In other words, as time goes on, the traffic becomes separated into increasingly large regions of high traffic density (traffic jam) which alternate with increasingly large regions of low density (freeflow). This conjecture is not completely proved, and we leave a short list of unproved statements which would establish the conjecture. |
