Beckner's inequality is a series of inequalities indexed by a parameter between 1 and 2 which interpolate between the Poincare inequality and the logarithmic Sobolev inequality, originally proved for the standard Gaussian measure. I will discuss this inequality in various settings from the very simple two point distribution to the path space over a compact Riemannian manifold and show the rich content of this inequality in relation to probability theory and, in particular,
stochastic analysis.

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