  We consider a variant of the classical notion of noise on the Boolean
hypercube which gives rise to a new approach to inequalities regarding noise
stability. We use this approach to give a new proof of the Majority is Stablest
theorem by Mossel, O'Donnell, and Oleszkiewicz, improving the dependence of the
bound on the maximal influence of the function from logarithmic to polynomial.
We also show that a variant of the conjecture by Courtade and Kumar regarding
the most informative Boolean function, where the classical noise is replaced by
our notion, holds true. Our approach is based on a stochastic construction that
we call the renormalized Brownian motion, which facilitates the use of
inequalities in Gaussian space in the analysis of Boolean functions.
