Large Deviations for Drifted Random Walks

Abstract

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.