Gamma-Limit of the Onsager--Machlup Functional on the Space of Curves

The Onsager--Machlup (OM) and Freidlin--Wentzell (FW) functionals are both widely used in seeking the most probable transition path between two states for a diffusion process. We study the relation between these two functionals on the space of curves. We prove that the $\Gamma$-limit of the OM functional on the space of curves is the geometric form of the FW functional in a proper time scale $T=T(\epsilon)$ as $\epsilon\to 0$. For other time scales, the limit of the OM functional is infinite in general. We then introduce the concept of renormalization for the OM functional and prove that the $\Gamma$-limit of the renormalized OM functional is the geometric FW functional in any time scale on the space of curves.