A two-sided estimate for the Gaussian noise stability deficit

The Gaussian noise-stability of a set A in R^n is defined by S_rho(A) = P (X in A and Y in A) where X and Y are standard Gaussian vectors whose correlation is rho. Borell's inequality states that for all 0 < rho < 1, among all sets A with a given Gaussian measure, the quantity S_rho(A) is maximized when A is a half-space. We give a novel short proof of this fact, 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 S_rho(H) - S_rho(A) 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 establish the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit rho->1, we get an improved dimension free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also valid for a the more general version of stability where more than two correlated vectors are considered.