When is the mode of a stochastic process a better statistic than the mean?

Question

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?

Answer 1

A "mode" of the Onsager-Machlup action functional identifies a locally most probable transition pathway between metastable states. If there is a single minimizer then mode and mean will be equally informative, but there may well be multiple local minima of the action functional, and then the mean does not tell you which are the relevant transition pathways.