Noise Stability - Beyond the Boolean cube
summary
Replaces the heat semigroup by a renormalized Brownian motion that martingales the harmonic extension of Boolean functions and embeds the hypercube into Wiener space. Proves an almost-sure quadratic variation domination between that martingale and a Gaussian/majority martingale via a coupling, which yields domination for any convex test (not just L2), including binary entropy. Delivers a polynomial quantitative bound improving the prior sublogarithmic error in Majority is Stablest and gives positive progress on Courtade–Kumar’s most-informative-bit under a different noise model.
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