Abstract
We give a novel short proof of Borell's Gaussian noise stability inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set A and its corresponding half-space H (namely the distance between the two centroids), we show that the deficit between the noise stability of A and H can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors.
As a consequence, we also manage to get the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit where the correlation goes to 1, we get an improved dimension free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also vali d for a the more general version of stability where more than two correlated vectors are considered.