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Dissertation title:

• Random Processes in Random Medium, Branching Random Walks

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Scientific advisor:

• Professor Mikhail Menshikov, Moscow State University, Russia

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Official referees:

• Professor Valentin Kolchin, Steklov Mathematical Institute of the Russian Academy of Sciences
• Dr. Boris Gurevich, Moscow State University, Russia

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Defense date and place:

• June 6, 1997, Academic Council of Faculty of Mechanics and Mathematics of Moscow State University

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Abstract:

The dissertation is devoted to an investigation of different problems about random walk in stochastic media and consists of three chapters.
    In Chapter 1 we study simple random walks on three and more dimensional lattice in a random field of traps, the density of which tends to zero at infinity. We distinguish between the quenched problem (when the traps are fixed) and the annealed problem (when the traps are updated each unit of time). Our goal is not only to find a criterion for zero vs. positive probability of survival, but also to show three different methods that can be applied in this area. These methods are based on Lyapunov functions, capacity and mean hitting time respectively.
    Chapter 2 contains results related to stochastic graphs which we call `random labyrinths'. A random labyrinth is a disordered environment of scatters on a lattice, through which a beam of light travels being reflected off the scatters. We study the set of illuminated vertices under the assumption that there is a positive density of points, called `normal points', at which the light behaves as a simple random walk. The ensuing `random walk in a labyrinth is found to be recurrent in two dimensions and transient in three and higher dimensions, subject to the assumption that the density of `non-trivial' scatterers is sufficiently small. The results are based on percolation theory and on the method of electrical networks.
    In Chapter 3 we study random walks with branching. We introduce the proper notion of recurrence and transience for these processes and provide criteria for them. For the Lamperti problem and many-dimensional random walks with branching we find the critical (for transience vs. recurrence) decaying speed of density of average number of off-springs at a point with respect to its distance to the origin.

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The main results of the dissertation are contained in the following papers:

• "Two Problems about Random Walks in a Random Field of Traps"
• "Random Walks in Random Labyrinths"
• "Branching Markov Chains: qualitative characteristics"

(see Papers for details)