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

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?

• Maybe the mode is simpler or more tractable than the mean, and that is enough. May 6, 2022 at 13:33