Entropy flow and De Bruijn's identity for a class of stochastic differential equations driven by fractional Brownian motion
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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∈(1/4,1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback-Leibler divergence by means of Itô's formula, and present two applications where we relax the assumption to H ∈ (0,1). 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 ∈ (0,1/2], the entropy power is concave in time while for H ∈ (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.
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