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Stability of the Shannon-Stam inequality via the Föllmer process [article]

Ronen Eldan, Dan Mikulincer
2019 arXiv   pre-print
We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance.  ...  In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X,Y ∈R^d, the deficit in the Shannon-Stam inequality is  ...  We would also like to thank Tom Courtade for his thoughtful comments concerning a preliminary draft and for suggesting that we generalize the proof of Theorem 3 for arbitrary covariance structures.  ... 
arXiv:1903.07140v1 fatcat:lfg4yghwjnestpxvypcdg6zrva

Stability of the Shannon–Stam inequality via the Föllmer process

Ronen Eldan, Dan Mikulincer
2020 Probability theory and related fields  
We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance.  ...  In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X , Y ∈ R d , the deficit in the Shannon-Stam inequality  ...  We would also like to thank Tom Courtade for his thoughtful comments concerning a preliminary draft and for suggesting that we generalize the proof of Theorem 2 for arbitrary covariance structures.  ... 
doi:10.1007/s00440-020-00967-w fatcat:dpzp5fmbdbgttao7wkiwxbjr4y

Stability of the logarithmic Sobolev inequality via the Föllmer Process [article]

Ronen Eldan, Joseph Lehec, Yair Shenfeld
2020 arXiv   pre-print
We study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature  ...  For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized  ...  Comparison theorems In this section we prove Theorems 1 and 3 via the Föllmer process. Proof of Theorem 1.  ... 
arXiv:1903.04522v2 fatcat:psijpqucwrcubashpw3mm6sw2q

Equality cases in the Anantharam-Jog-Nair inequality [article]

Efe Aras, Thomas A. Courtade, Albert Zhang
2022 arXiv   pre-print
Anantharam, Jog and Nair recently unified the Shannon-Stam inequality and the entropic form of the Brascamp-Lieb inequalities under a common inequality.  ...  They left open the problems of extremizability and characterization of extremizers. Both questions are resolved in the present paper.  ...  Acknowledgement T.C. thanks Dan Mikulincer for his explanations of the properties of the Föllmer drift and the martingale embedding used in [11] .  ... 
arXiv:2206.11809v1 fatcat:grxf5iymnza7dflqiufgxw53wi

Trajectorial Otto calculus [article]

Ioannis Karatzas, Walter Schachermayer, Bertram Tschiderer
2020 arXiv   pre-print
We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time  ...  As a bonus of our perturbation analysis we derive the so-called HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).  ...  account of the results by Barron, Blachman, Brown, Linnik, Rényi, Shannon, Stam and many others in this vein, as well as to the paper [MV00] by Markowich and Villani, and the book [Vil03] by Villani  ... 
arXiv:1811.08686v4 fatcat:bhaed2v3arfsdin3paplsv6t4m