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We show how to use the Malliavin calculus to obtain density estimates of thelaw of general centered random variables. In particular, under a non-degeneracycondition, we prove and use a new formula for the density of a random variablewhich is measurable and differentiable with respect to a given isonormalGaussian process. Among other results, we apply our techniques to bound thedensity of the maximum of a general Gaussian process from above and below;several new results ensue, including improvements on the so-calledBorell-Sudakov inequality. We then explain what can be done when one is onlyinterested in or capable of deriving concentration inequalities, i.e. tailbounds from above or below but not necessarily both simultaneously.

Density estimates and concentration inequalities with Malliavin calculus