Entropy flow and De Bruijn's identity for a class of stochastic differential equations driven by fractional Brownian motion

Motivated by the classical De Bruijn’s identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (0,1)$. We derive generalized De Bruijn’s identity for Shannon entropy and Kullback-Leibler divergence by means of It^o’s formula, and present two applications. In the first application we demonstrate its equivalence with Stein’s identity for Gaussian distributions, while in the second application, we show that for $H\in (0,1\mathrm{/}2]$, the entropy power is concave in time while for $H\in (1\mathrm{/}2,1)$ it is convex in time when the initial distribution is Gaussian. Compared with the classical case of $H = 1/2$, the time parameter plays an interesting and significant role in the analysis of these quantities.