Applicability of Gaussian proof techniques to the Boolean hypercube case

Ascertain whether the Gaussian-space proof based on Föllmer’s process, coupling, and stochastic calculus can be adapted to prove Talagrand’s convolution conjecture for the heat semigroup on the Boolean hypercube.

Background

The Gaussian analogue of the conjecture has been fully resolved using stochastic calculus techniques (Föllmer’s process, semi-log-convexity, and refined coupling methods). While the Boolean case is related to the Gaussian via central limit considerations, it is unclear whether these Gaussian techniques transfer to the discrete setting.

The authors explicitly state that the usefulness of the Gaussian proof for the Boolean cube remains unknown.