DescriptionRandom Walks in Random Environments (RWREs) are models for the motion of particles in media containing inhomogeneities or impurities. As an example, think of a liquid composed of water and oil and a molecule that moves differently in each medium. In this setup one can turn to RWRE to characterize the long-term behavior of the trajectory \((X_n)_{n\in \mathbb{N}}\) of the molecule.In the case of symmetric environment, it is clear that \(\mathbb{E}[X_n] = 0\) for every \(n \in \mathbb{N}\). The main objective of the project is to deduce symmetry properties of the environment from the available statistics of the RWRE trajectories. Starting from the converse of the above statement (in dimension one), i.e., if \(\mathbb{E}[X_n] = 0\) for all \(n \in \mathbb{N}\) then the random environment is symmetric, one would like to weaken the assumptions while still preserving the symmetry conclusion. Depending on student's interest, one could explore higher dimensions, or the assumption \(\mathbb{E}[X_n]=0\) for all \(n\geq n_0\) in dimension one etc. The project will allow to get familiar with the classical theory of RWRE and to use symbolic computational methods to examine symmetry. Furthermore, one could explore multiple complementary approaches borrowing tools from algebra (finding zeros of polynomials associated with \(\mathbb{E}[X_n]=0\)), analysis (looking at \(\mathbb{E}[X_n]\) as analytic functions), and probability (using localization methods for RWRE). PrerequisitesProbability II, open mind, readiness to combine ideas from different areas of mathematics.Resources |
email: Conrado da Costa