  We analyze F\"urth's 1933 classical uncertainty relations in the modern
language of stochastic differential equations. Our interest is motivated by
applications to non-equilibrium classical statistical mechanics. We show that
F\"urth's uncertainty relations are a property enjoyed by martingales under the
measure of a diffusion process. This result implies a lower bound on
fluctuations in current velocities of entropic quantifiers of transitions in
stochastic thermodynamics. In cases of particular interest, we recover an
inequality well known in optimal mass transport relating the mean kinetic
energy of the current velocity and the squared quadratic Wasserstein distance
between the probability distributions of the entropy. In performing our
analysis, we also avail us of an unpublished argument due to Krzysztof
Gaw\c{e}dzki to derive a lower bound to the entropy production by transition
described by Langevin-Kramers process in terms of the squared quadratic
Wasserstein distance between the initial and final states of the transition.
Finally, we illustrate how F\"urth's relations admit a straightforward
extension to piecewise deterministic processes. We thus show that the results
in the paper concern properties enjoyed by general Markov processes.
