|M.Sc Student||Berezin Roman|
|Subject||Limiting Behaviour of Contact Random Walk|
|Department||Department of Industrial Engineering and Management||Supervisor||PROF. Leonid Mytnik|
|Full Thesis text|
Interacting particle systems have been a point of great interest to mathematicians for a long time, in particular the behaviour of those systems, suitably scaled, as the number of particles in the initial configuration approaches infinity. In this work we study the limiting behaviour of an interacting particle system, under the rules of a random walk in a d-dimensional Euclidian space, and a contact process, when the branching speed is on a smaller scale than the random walk. In our scaling, we start with N particles and let them evolve independently, such that the random walk speed is on the scale of Nm, for some m > 1, and the contact part takes place on the scale of N. We prove that when the random walk speed is large, m > mc, the measure valued process associated with this particle system converges weakly to a super-Brownian motion with drift. We also show that in this régime the collisions, that come from the contact part of the process do not contribute to the limit. To prove our results we will use the methods developed by Durrett and Perkins, in their study of the long range contact random process. Our results will also give us some understanding of the behaviour of the process studied by Konno. His process differs from ours in the random walk part, which, in his case, is replaced by an exclusion process.