My PhD

In the thesis I investigate different behavior of processes constructed for modeling domain growth, deposition processes or current of particles, many phenomena occurring in every day's life. The main feature in these models is that they include many randomly behaving objects, interacting with each other. I consider a family of such processes here in a common framework, which contains some of the well-known models from the area of interacting particle systems, e.g. the simple exclusion and the zero range processes. As a slight generalization of the latter, I also introduce the bricklayers' process in the present thesis.

After describing many phenomena connected to these types of models, I give a precise definition of our systems, and show how to couple them. The evolution of a coupled pair of models is partially driven by joint randomness, hence their evolution is as close to each other as possible. By coupling I introduce the second class particle, an object playing an essential role in our methods. In the first part of the thesis, I refer to the connection between our processes and some first-order non linear partial differential equations, namely, the conservation laws. By a simple argument, I also indicate for our new bricklayers' model the connection of the second class particle to the so-called shock solutions of the corresponding partial differential equation. This is a well-known phenomenon from other works in this field. Based on this relation, I construct a class of distributions in a type of bricklayers' models, which exactly corresponds to a class of shock solutions in the model's partial differential equation. Since these distributions have extremely simple structure, as far as I know, no such simple distributions were found showing the properties of shocks in a microscopic level.

In the second part, I examine the fluctuations of our processes' growth. By using martingale techniques, I get to the point where the use of second class particles becomes very natural. However, the arguments require knowing some special properties of the second class particle, which are quite difficult to establish since the second class particle performs a random motion in the random environment created by the model. Hence long arguments are set in this part to establish the required properties of the second class particle. These arguments refine the coupling methods providing the ability of coupling second class particles between different models.

In the first two parts I dealt with systems of which the dynamics has, as far as I know, not yet been rigorously constructed. In some situations, these processes perform faster growth than the ones already constructed, hence in the third part I establish stochastic bounds which are sufficient to show that these models stay under control while they evolve. Of course, these bounds only apply in case of appropriate initial conditions. Although the construction of dynamics is in progress, and so it can not be part of the present thesis, our results in this direction probably mean the harder part of the problem.

The thesis

in PDF format


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