De Bruijn identities in different Markovian channels

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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. For more information see https://ejde.math.txstate.edu/Volumes/2023/12/abstr.html