Random walks are interesting objects in the study of several physical phenemonon; and are extensively used to models polymers, financial markets, to estimate the size of the web. It is well known that the transience and recurrence of random walks on integer lattices is deeply connected to its Green's function. Furthermore, the unique solution to discrete Laplace/Dirichlet boundary problems is given by the hitting/exit time of random walks. This provides a probablistic approach using martingales (harmonic functions) to solve important problems from PDE.
It is well known that for a random walk with finite second moment, the rate of decay of the Green's function at x (denoted by G(0, x)) is of polynomial order of the distance of x, that is, G(0,x) ∼ C |x| 2-d in dimensions d ≧ 3, where |x| denotes the Euclidean distance of x (see Lawler 1991 and Spitzer, 2001). For the proposed project, we will be primarily interested in heavy-tailed random walks with tail exponent 0 < α < 2. We will try to investigate under what conditions does the Green's function for heavy-tailed random walks have similar polynomial decay, that is, G(0, x) ∼ |x| α-d L(x), for some slowly varying function L. This will enable us to investigate which are the massive sets for the random walk under consideration. We will also consider the Green's function for the absorbing boundary random walk, which is the equivalent of the Gambler ruin problem in this case, see Kesten 1961a, 1961b, and Bendikov and Cygan, 2015.
The prospects of this project includes both a thorough literature survey as well as, some interesting simuations for random walks. A basic knowledge of probability theory is required. It is recommended that you revise the contents of the modules Probability I and II. Any advanced knowledge of martingales and Fourrier transforms that we may require can be picked up as a part of the project.
The interested candidates may look at these references, and the references therein. For further details, please write to me at debleena.thacker@durham.ac.uk.