This is a soft question.

I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.

Given a measure $\mu$ on metric space $(X,d)$, if there is a function $F$ on $X$ for which

$$\lim_{\varepsilon\to 0}\frac{\mu(B_\varepsilon(z_2))}{\mu(B_\varepsilon(z_1))}=\exp\left(F(z_2)-F(z_1)\right)$$

(and the limit always exists) then $F$ is called the Onsager-Machlup function for $\mu$. The minimizers of $F$ are often called the modes of $\mu$.

When considering a probability measure of some space of paths, if the minimizer exists then it is referred to as the "mode" of the stochastic process whose law is that measure.

I have a bit of a soft question -

When does the mode of a stochastic process capture its behavior better than its mean?