Kazumasa Kuwada

Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods

We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow, and no further a priori curvature bound is required. As an application, we construct a coupling by reflection which yields a nice estimate of coupling time, and hence a gradient estimate for the associated semigroups.

On large deviations for random currents induced from stochastic line integrals

Abstract We study the large deviation and its rate function in the framework of current-valued processes. The class of processes we consider are determined by stochastic line integrals of 1-forms on a compact manifold. We obtain an explicit expression of the rate function, which enables us to observe the difference from the rate function which governs the moderate deviation. As a corollary, our result includes the large deviation for empirical laws in the space of currents.