  We consider degenerate diffusion equations of the form $\partial_tp_t =
\Delta f(p_t)$ on a bounded domain and subject to no-flux boundary conditions,
for a class of nonlinearities $f$ that includes the porous medium equation. We
derive for them a trajectorial analogue of the entropy dissipation identity,
which describes the rate of entropy dissipation along every path of the
diffusion. Our approach is based on applying stochastic calculus to the
underlying probabilistic representations, which in our context are stochastic
differential equations with normal reflection on the boundary. This
trajectorial approach also leads to a new derivation of the Wasserstein
gradient flow property for nonlinear diffusions, as well as to a simple proof
of the HWI inequality in the present context.
