- Random Processes in Random Medium, Branching Random Walks
Defense date and place:
- June 6, 1997, Academic Council of Faculty of Mechanics and Mathematics of Moscow State University
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
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.
The main results of the dissertation are contained in the following papers:
(see Papers for details)
- "Two Problems about Random Walks in a Random Field of Traps"
- "Random Walks in Random Labyrinths"
- "Branching Markov Chains: qualitative characteristics"