  We study the relationship between information- and estimation-theoretic
quantities in time-evolving systems. We focus on the Fokker-Planck channel
defined by a general stochastic differential equation, and show that the time
derivatives of entropy, KL divergence, and mutual information are characterized
by estimation-theoretic quantities involving an appropriate generalization of
the Fisher information. Our results vastly extend De Bruijn's identity and the
classical I-MMSE relation.
