Talagrand's Convolution Conjecture on Gaussian Space

Abstract

Smoothing properties of the noise operator on the discrete cube and on Gaussian space have played a pivotal role in many fields. In particular, these smoothing effects have seena broad range of applications in theoretical computer science. We exhibit new regularization properties of the noise operator on Gaussian space. More specifically, we show that the mass on level sets of a probability density decays uniformly under the Ornstein-Uhlenbeck semi group. This confirms positively the Gaussian case of Talagrand's convolution conjecture (1989)on the discrete cube. A major theme is our use of an It o process (the Follmer drift)which can be seen as an entropy-optimal coupling between the Gaussian measure and another given measure on Gaussian space. To analyze this process, we employ stochastic calculus and Girsanov's change of measure formula. The ideas and tools employed here provide a new perspective on hyper contractivity in Gaussian space and the discrete cube. In particular, our work gives a new way of studying small sets in product spaces (e.g., Sets of size 2o(n) in the discrete cube) using a form of regularized online gradient descent.