Onsager-Machlup functional for uniformly elliptic time-inhomogeneous diffusion

In this paper we will make the computation of the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional will have formally the same picture as in the homogeneous case, the only difference come from the infinitesimal variation of the volume. For example in the Ricci flow case, we find some functional which is not so far to the ${\mathcal{L}}_0$ distance used by Lott to study this flow \cite{Lott:08}. We finish by a application to small ball probability for weighted sup norm, for inhomogeneous diffusion.