On the rate of change of Varentropy for Markov diffusion processes

Relying on the reverse-time space-time harmonic property of the ratio of two solutions of the Fokker-Plank equation, we establish an explicit formula for derivative of the \emph{varentropy} for a Markov diffusion process. The formula involves a nonlinear function of the {\em local free energy} $\ln(p_t(x)/\bar{p}(x))$.We then verify that our formula yields the correct result in the simple case of a scalar Gaussian diffusion. In the latter case, varentropy is exponentially decaying to zero.