Classical large deviations theorems on complete Riemannian manifolds

TL;DR

This paper extends classical large deviations theorems to complete Riemannian manifolds, introducing new proofs and generalizations for key theorems like Mogulskii's, Cramér's, and Schilder's, using viscosity solutions and embedding techniques.

Contribution

It generalizes large deviations theorems to Riemannian manifolds and offers new proofs via viscosity solutions and Euclidean embeddings.

Findings

Generalization of Mogulskii's theorem to Riemannian manifolds

Extension of Cramér's theorem in the manifold setting

New proof of Schilder's theorem using embedding and Freidlin-Wentzell theory

Abstract

We generalize classical large deviations theorems to the setting of complete Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using visocity solutions for Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of Cram\'er'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.