Electron. J. Differential Equations, Vol. 2023 (2023), No. 12, pp. 1-11.

De Bruijn identities in different Markovian channels

Hassan Emamirad, Arnaud Rougirel

Abstract:
De Bruijn's identity in information theory states that if u is the solution of the heat equation, then the time derivative of the Shannon entropy for this solution is equal to the amount of Fisher information at u. In this article, we show how this identity changes if we replace the heat channel by the Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck channels. Through these passages we investigate the different properties of these solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck semigroup given by the Mehler formula expression.

Submitted October 27, 2022. Published February 6, 2023.
Math Subject Classifications: 94A17, 94A40.
Key Words: Fokker Planck; relative Fisher information; Kullback-Leibler divergence.
DOI: https://doi.org/10.58997/ejde.2023.12

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Hassan Emamirad
Laboratoire de Mathématiques
Université de Poitiers
teleport 2, BP 179
86960 Chassneuil du Poitou, Cedex, France
email: emamirad@math.univ-poitiers.fr
Arnaud Rougirel
Laboratoire de Mathématiques
Université de Poitiers
teleport 2, BP 179
86960 Chassneuil du Poitou, Cedex, France
email: rougirel@math.univ-poitiers.fr

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