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 fg(t)gt is an element of[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 FM 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.