We consider a certain class of Riemannian submersions π:N→M and study lifted geodesic random walks from the base manifold M to the total manifold N. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle, i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian ΔH on N and the Laplace–Beltrami operator ΔM on M. In the setting where N is the orthonormal frame bundle O(M), this identity is central in the Malliavin–Eells–Elworthy construction of Riemannian Brownian motion.

We prove large deviations for g(t)-Brownian motion in a complete, evolving Riemannian manifold M with respect to a collection {g(t)}_t∈[0,1] of Riemannian metrics, smoothly depending on t. We show how the large deviations are obtained from the large deviations of the (time-dependent) horizontal lift of g(t)-Brownian motion to the frame bundle F M over M. The latter is proved by embedding the frame bundle into some Euclidean space and applying Freidlin – Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and time.

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.

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.