This thesis is concerned with large deviations for processes in Riemannian manifolds. In particular, we study the extensions of large deviations for random walks and Brownian motion to the geometric setting. Furthermore, we also consider large deviations for random walks in Lie groups. The additional group structure allows for improvements over the general case. Finally, we also study large deviations for Brownian motion in evolving Riemannian manifolds. For this, we use the so-called 'rolling without slipping' construction of Riemannian Brownian motion, adepted to the time-inhomogeneous setting. @en

We generalize classical large deviation theorems to the setting of complete, smooth Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using viscosity solutions for Hamilton–Jacobi equations. As a corollary, we also obtain the analogue of Cramér's theorem. The approach also provides a new proof of Schilder's theorem. Additionally, we provide a proof of Schilder's theorem by using an embedding into Euclidean space, together with Freidlin–Wentzell theory.

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We provide a direct proof of Cramér’s theorem for geodesic random walks in a complete Riemannian manifold (M; g). We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in M. Furthermore, we reveal the geometric obstructions one runs into. To overcome these obstructions, we provide a Taylor expansion of the inverse Riemannian exponential map, together with appropriate bounds. Furthermore, we compare the differential of the Riemannian exponential map to parallel transport. Finally, we show how far geodesics, possibly starting in different points, may spread in a given amount of time. With all geometric results in place, we obtain the analogue of Cramér’s theorem for geodesic random walks by showing that the curvature terms arising in this geometric analysis can be controlled and are negligible on an exponential scale.

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We prove R-bisectoriality and boundedness of the (Formula presented.)-functional calculus in (Formula presented.) for all (Formula presented.) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.@en