Large Deviations for Brownian Motion in Evolving Riemannian Manifolds

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.