Onsager-Machlup functional for uniformly elliptic time-inhomogeneous diffusion

TL;DR

This paper computes the Onsager-Machlup functional for inhomogeneous uniformly elliptic diffusions, extending known results from homogeneous cases and exploring applications to small ball probabilities.

Contribution

It derives the Onsager-Machlup functional for inhomogeneous diffusions, highlighting differences from homogeneous cases and connecting to Ricci flow and small ball probability analysis.

Findings

Functional similar to homogeneous case but with volume variation

Identifies connection to $\\ ext{L}_0$ distance in Ricci flow

Application to small ball probability for weighted sup norm

Abstract

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.