Mehler formula provides a representation of the Ornstein-Uhlenbeck semigroup, associated with a general Poisson point measure, in terms of thinning operators. In this talk, we shall explore two applications of such a result that are relevant in stochastic geometry: (i) the proof of explicit Berry-Essen bounds expressed in terms of second-order Poincaré inequalities, and (ii) the derivation of Phi-Sobolev inequalities and of associated concentration estimates. Applications to random geometric graphs will be evoked throughout the talk.

Based on joint works with Günter Last and Matthias Schulte (Karlsruhe) and Sascha Bachmann (Osnabrück).

slides