On Malliavin's proof of Hörmander's theorem

Abstract

The aim of this note is to provide a short and self-contained proof of Hörmander's theorem about the smoothness of transition probabilities for a diffusion under Hörmander's "brackets condition". While both the result and the technique of proof are well-known, the exposition given here is novel in two aspects. First, we introduce Malliavin calculus in an "intuitive" way, without using Wiener's chaos decomposition. While this may make it difficult to prove some of the standard results in Malliavin calculus (boundedness of the derivative operator in $L^p$ spaces for example), we are able to bypass these and to replace them by weaker results that are still sufficient for our purpose. Second, we introduce a notion of "almost implication" and "almost truth" (somewhat similar to what is done in fuzzy logic) which allows, once the foundations of Malliavin calculus are laid out, to give a very short and streamlined proof of Hörmader's theorem that focuses on the main ideas without clouding it by technical details.