A Variational Proof of Robust Gaussian Noise Stability

Using the calculus of variations, we prove that a Euclidean set of fixed Gaussian measure that nearly maximizes Gaussian noise stability is close to a half space. The main result proves a modification of a conjecture of Eldan from 2013: a robust Borell inequality that removes a logarithmic dependence on the distance of the set to a half space. For sets of Gaussian measure $1/2$, we prove Eldan's 2013 conjecture. The noise stability of a Euclidean set $A$ with correlation $\rho$ is the probability that $(X,Y)\in A\times A$, where $X,Y$ are standard Gaussian random vectors with correlation $\rho\in(-1,1)$. Barchiesi, Brancolini and Julin proved that a Euclidean set of fixed Gaussian measure that nearly minimizes Gaussian surface area is close to a half space, using a variational "penalty function" method. Our proof adapts their method to the more general setting of noise stability. We also show that half spaces are the only sets that are stable (in the sense of second variation) for noise stability, generalizing a result of McGonagle and Ross for Gaussian surface area.