A Stein deficit for the logarithmic Sobolev inequality
arXiv:1602.08235 · doi:10.1007/s11425-016-0134-7
Abstract
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.
22 pages