This thesis proves a Cramér-type large deviation principle for drifted geodesic random walks on a complete Riemannian manifold. A geodesic random walk is the manifold analogue of a sum of independent increments: at each step a random tangent vector is drawn and the walk movesalong the geodesic that this vector defines. We add a deterministic drift, given by a smooth vector field V , so that each step follows the geodesic determined by the random increment together with the value of V at the current point. For the empirical average of such a walk we establish a large deviation principle at speed n, with an explicit good rate function obtained as the Legendre transform of the increment law, corrected by the drift. The proof rests on a single construction, the drift map, which sends a noise path to the path obtained by carrying it on top of the flow of V . We show that this map is continuous, and the large deviation principle then follows from the Riemannian version of Mogulskii’s theorem by two applications of the contraction principle. Setting the drift to zero recovers the undrifted theorem of Kraaij, Redig and Versendaal. A final chapter removes the bounded-increment hypothesis. It is replaced by Cram´er’s condition on the increment law together with a bounded-geometry assumption on the manifold, and the same rate function is obtained. ... Read more

This thesis is concerned with the broad question:
How is the emergence of deterministic macroscopic laws from random microscopic dynamics influenced by the geometry of the underlying space?

The rigorous derivation of macroscopic equations from microscopic models is a central theme of mathematical physics. In the probabilistic setting, interacting particle systems provide tractable microscopic models made of many particles evolving according to stochastic rules. Under suitable scaling limits, the empirical conserved quantities of these systems are expected to converge to deterministic solutions of partial differential equations. Such results are called hydrodynamic limits.... ... Read more

In this bachelor thesis, we will look a variant of the random walk, namely the elephant random walk. The elephant random walk, named after the elephant due to its excellent memory, is a random walk where future steps are based on previous steps taken in the walk based on the memory parameter p. To that end, we will be looking at the differences between the normal random walk and elephant random walk: we will look at the differences in behavior and properties of both types of random walks. In particular, we will focus on the gambler’s ruin problem, where we discuss the chances of player A winning, ν(α), and the expected number of steps until either player wins, e(α). To do so, we will make use of martingales and known theorems about martingales to prove the theorems about the elephant random walk. For the gambler’s ruin problem for the elephant random walk, we use simulations to look at the behavior of the gambler’s ruin for the elephant random walk.

We found that in general, the elephant random walk behaves differently from the normal random walk: while the expected value of the normal random walk increases or decreases linearly or stays at zero, the elephant random walk either diverges to infinity less than linearly or converges to zero based on the memory parameter p. In addition to this, one finds that the elephant random walk has three different diffusion regimes based on p: compared to the normal random walk, which only has a diffusive regime, the elephant random walk has a diffusive, marginally superdiffusive and superdiffusive regime. Looking at the properties, we see that some of the results of the normal random walk also hold for the elephant random walk, in particular the law of large numbers. However, it turns out that the central limit theorem only holds for certain values of p, and that the central limit theorem stops holding for larger p due to the dependence of steps on each other being too high. In addition, we found that the memory parameter p and initial parameter q for the gambler’s ruin for the elephant random walk affects how ν(α) and e(α) behaves, with the influence of q increasing as p increases on both ν(α) and e(α). The total capital N has little influence on the behavior of the gambler’s ruin for the elephant random walk, with N only affecting the maximal expected number of steps. ... Read more