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

## TL;DR

This paper extends De Bruijn's identity to stochastic differential equations driven by fractional Brownian motion, revealing how entropy behaves over time depending on the Hurst parameter, with applications to Gaussian distributions.

## Contribution

It derives a generalized De Bruijn's identity for fractional Brownian motion-driven SDEs and explores entropy power behavior, linking it to Gaussian and Stein's identities.

## Key findings

- Entropy power is concave for H ≤ 1/2 and convex for H > 1/2 with Gaussian initial distributions.
- Generalized De Bruijn's identity is established using Itô's formula for fractional Brownian motion.
- The work highlights the significant role of the Hurst parameter in entropy dynamics.

## Abstract

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 $H \in (0,1/2]$, the entropy power is concave in time while for $H \in (1/2,1)$ 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.