On quantitative noise stability and influences for discrete and continuous models

Keller and Kindler recently established a quantitative version of the famous Benjamini~--Kalai--Schramm Theorem on noise sensitivity of Boolean functions. The result was extended to the continuous Gaussian setting by Keller, Mossel and Sen by means of a Central Limit Theorem argument. In this work, we present an unified approach of these results, both in discrete and continuous settings. The proof relies on semigroup decompositions together with a suitable cut-off argument allowing for the efficient use of the classical hypercontractivity tool behind these results. It extends to further models of interest such as families of log-concave measures and Cayley and Schreier graphs. In particular we obtain a quantitative version of the B-K-S Theorem for the slices of the Boolean cube.