Stochastic proof of the sharp symmetrized Talagrand inequality (Q6644105)

The aim of the paper is to give a short stochastic proof of the sharp symmetrized Talagrand inequality. Stochastic proofs of functional inequalities, in particular using Brownian motion and Girsanov's theorem, go back to Borell's stochastic proof of the Prekópa-Leindler inequality. To analyze the relations between various inequalities and their proofs, the authors take a retrospective look on the connection between the mentioned inequality and the functional Blaschke-Santaló inequality, and also provide a list of papers with various proofs of both. Compared to stochastic proofs of other Gaussian functional inequalities, the new idea in the present paper is a certain coupling induced by time-reversed martingale representations. The importance of the coupling based on time-reversal is highlighted. The presented proof is indeed concise and elegant.