A Stein deficit for the logarithmic Sobolev inequality
classification
🧮 math.PR
math.FA
keywords
deficitgaussianinequalityinformationlogarithmicsobolevsteinterms
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We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory.
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