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 ∈ (0, 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. 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.

中文翻译：

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分数布朗运动驱动的一类随机微分方程的熵流和 DE BRUIJN 恒等式

受加性高斯噪声通道的经典 De Bruijn 恒等式的启发，在本文中，我们考虑了一个广义设置，其中通道通过由带有 Hurst 参数的分数布朗运动驱动的随机微分方程建模H∈ (0, 1)。我们通过Itô公式推导出了香农熵和Kullback-Leibler散度的广义De Bruijn恒等式，并给出了两种应用。在第一个应用程序中，我们证明了它与高斯分布的 Stein 恒等式，而在第二个应用程序中，我们证明了H∈ (0, 1/2]，熵幂在时间上是凹的，而对于H∈ (1/2, 1) 当初始分布为高斯分布时，它在时间上是凸的。与经典案例相比H= 1/2，时间参数在这些量的分析中起着有趣且重要的作用。