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 is an element of (0, 1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback-Leibler divergence by means of Ito'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 is an element of (0, 1/2], the entropy power is concave in time while for H is an element of (1/2, 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.