Long-time behavior of random walks in random environment and asymptotics of a ternary coalescent process

Abstract

The first part of this thesis deals with random walks in a random environment on the d-dimensional lattice Zd, d 3. Such stochastic systems can be used to model the random motion of a particle in an inhomogeneous medium. The effects of irregularities caused by impurities or defects of the medium are from a mathematical point of view best described by randomizing the medium. In fact, randomness enters at two different levels: It governs the choice of the environment and also the movement of the particle. In our case, the random environment is modeled by independent and identically distributed random transition kernels (ωx(e))|e|=1, e ∈ Zd , x ∈ Zd, which are small isotropic perturbations of the homogeneous simple random walk kernel p(x, x+ ) 1/(2d). Given an environment ω, one considers the random walk with transition kernel pω(x, x + ) = ωx( ). First, we investigate exit distributions of such walks from large balls. We show that when the radius of the ball tends to infinity, the exit measure is approximately given by that of a simple random walk. More precisely, we transfer estimates on the total variation distance between these two measures and smoothed versions thereof from smaller to larger radii. Further, we compare the exit distributions on certain boundary portions. Finally, under an additional assumption on the measure governing the environment, we use the information on the spatial behavior to control mean sojourn times in large balls.
In the second part of the thesis, from page 91 onwards, we consider coagulation and fragmentation processes. Fragmentation processes describe a memoryless evolution of particles characterized by their masses, which split independently into (smaller) new particles. Conversely, coagulation or coalescent processes model the coagulation of particles, where the rate at which a family merges depends only on the members involved. One again assumes that the particles are determined by their masses and that the system develops in a Markovian way.
We study a ternary coalescent process where three particles of masses r, s, t > 0 coagulate at rate r + s + t + 3. Different representations in terms of quantities related to a one-dimensional simple random walk and random binary forests are used to establish various properties of this process. First, we show that time reversal results in a fragmentation process. Then we investigate asymptotic behavior. Starting from a fixed number of N particles of unit mass we let N tend to infinity and obtain under an appropriate rescaling a well-known binary coalescent process, the so-called standard additive coalescent. Finally, we look at particle densities and solve the associated Srnoluchowski coagulation equations.