log Hessian estimates and the Talagrand Conjecture

Motivated by Talagrand's conjecture about the regularization effect of the Ornstein-Uhlenbeck semi-group, we investigate lower bounds on the log Hessian of a family of diffusion semi-group (essentially perturbation of the Ornstein-Uhlenbeck semi-group) and prove that the conjecture holds. On the other side, we will also investigate similar questions for the (discrete) M/M/infinity queuing process on the integers. Joint work with N. Gozlan, X.-M Li, M. Madiman and P.-M. Samson.