  We analyze dynamical properties of the Shannon information entropy of a
continuous probability distribution, which is driven by a standard diffusion
process. This entropy choice is confronted with another option, employing the
conditional Kullback-Leibler entropy. Both entropies discriminate among various
probability distributions, either statically or in the time domain. An
asymptotic approach towards equilibrium is typically monotonic in terms of the
Kullback entropy. The Shannon entropy time rate needs not to be positive and is
a sensitive indicator of the power transfer processes (removal/supply) due to
an active environment. In the case of Smoluchowski diffusions, the Kullback
entropy time rate coincides with the Shannon entropy "production" rate.
