Large deviation for pinned covering diffusion

We shall establish large deviations for the pinned motions of a periodic di ffusion process on n dimensional Euclidean space and the Brownian motion on 2 dimensional Lobachevsky space. It will be shown that the rate functions for the large deviations are corresponding to the infinitesimal generators of the diffusion processes obtained from the above processes through a kind of harmonic transform by positive principal eigenfunctions. (C) 2001 Editi ons scientifiques et medicales Elsevier SAS.