  We revisit the variational characterization of conservative diffusion as
entropic gradient flow and provide for it a probabilistic interpretation based
on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that,
for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability
density flow maximizes the rate of relative entropy dissipation, as measured by
the distance traveled in the ambient space of probability measures with finite
second moments, in terms of the quadratic Wasserstein metric. We obtain novel,
stochastic-process versions of these features, valid along almost every
trajectory of the diffusive motion in the backward direction of time, using a
very direct perturbation analysis. By averaging our trajectorial results with
respect to the underlying measure on path space, we establish the maximal rate
of entropy dissipation along the Fokker-Planck flow and measure exactly the
deviation from this maximum that corresponds to any given perturbation. As a
bonus of our trajectorial approach we derive the HWI inequality relating
relative entropy (H), Wasserstein distance (W) and relative Fisher information
(I).
