New limit theorems for regular diffusion processes with finite speed measure

We derive limit theorems for diffusion processes that have a finite speed measure. First we prove a number of asymptotic properties of the density $rho_t = dmu_t /dmu$ of the empirical measure $mu_t$ with respect to the normalized speed measure $mu$. These results are then used to derive finite dimensional and uniform central limit theorems for integrals of the form $sqrt{tint (rho_t-1),dnu$, where $nu$ is an arbitrary finite, signed measure on the state space of the diffusion. We also consider a number of interesting special cases, such as uniform central limit theorems for Lebesgue integrals and functional weak convergence of the empirical distribution function.