Density Formula and Concentration Inequalities with Malliavin Calculus

We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and differentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not re- fer to the divergence operator δ (dual of the Malliavin derivative D). The formula is based on an auxilliary random variable G :=〈DZ,−DL−1Z〉H, where L is the generator of the so- called Ornstein-Uhlenbeck semigroup. The use of G was first discovered by Nourdin and Peccati (Probab. Theory Relat. Fields 145, 2009) in the context of rates of convergence in law. Here, thanks to G, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for Z in our framework.