Large Deviations for Brownian Motion in Evolving Riemannian Manifolds

#### R. Versendaal

2021, v.27, Issue 3, 381-412

ABSTRACT

We prove large deviations for $g(t)$-Brownian motion in a complete, evolving Riemannian manifold $M$ with respect to a collection $\{g(t)\}_{t\in[0,1]}$ 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\tire Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and time.

Keywords: large deviations, $g(t)$-Brownian motion, time-dependent geometry, evolving manifold, Schilder's theorem, frame bundle, horizontal lift, anti-development, Freidlin\tire Wentzell theory