Ph.D Student | Berezin Roman |
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Subject | Limiting Behavior of Some Interacting Particle Systems |

Department | Department of Industrial Engineering and Management |

Supervisor | Professor Leonid Mytnik |

Full Thesis text |

In this work we are going to study the behaviour of two processes. First, a contact process combined with a simple random walk process is studied. Then we will consider a more complicated model where an exclusion process (stirring) takes the place of the random walk. We are going to look at the first process as a measure valued process on a d-dimensional Euclidian space and rescale it in such a way that the limiting process will be diffusive. To that end we will use standard methods of martingale problems. The construction will be similar to the work done by Durrett and Perkins for the long range contact process. Durrett and Perkins showed that in dimensions two or higher a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with a drift. As a consequence of this result they improved the result of Bramson, Durrett and Swindle, replacing their order of magnitude estimates of how close the critical value of survival of the process is to 1, by sharp asymptotic. DeMasi, Ferrari and Lebowitz studied interacting particle systems on a lattice under the combined influence of spin flip and simple exchange dynamics. They proved that when the stirrings occur on a fast time scale of a certain order the macroscopic density, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations around the deterministic macroscopic evolution were found explicitly. The authors showed that they grow, with time, to become infinite when the deterministic solution is unstable. Durrett and Neuhauser considered a more general setting of processes defined on a general state space. Using the connection between a convergent sequence of such particle systems to a solution of a reaction-diffusion equation, Durrett and Neuhauser proved results about the existence of phase transitions for many systems when the stirring rate is large. Studying the contact random walk is the first step in understanding the sharp asymptotic of the second process of interest -- a process previously studied by Konno - contact process with rapid stirring, in a setting similar to that of Bramson, Durrett and Swindle. The main point of interest is to find the sharp asymptotic of the smallest birth rate such that this process survives indefinably with positive probability.