@article{20591,
  abstract     = {In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. On the other hand, we show that assuming only fast decay of the tails of the initial datum does not suffice to guarantee uniform log-Hessian upper bounds.},
  author       = {Brigati, Giovanni and Pedrotti, Francesco},
  issn         = {1083-589X},
  journal      = {Electronic Communications in Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Heat flow, log-concavity, and Lipschitz transport maps}},
  doi          = {10.1214/25-ECP717},
  volume       = {30},
  year         = {2025},
}