On Malliavinʼs proof of Hörmanderʼs theorem

By Martin Hairer. 2011

131 citations recorded by the source corpus.

The aim of this note is to provide a short and self-contained proof of Hormanderʼs theorem about the smoothness of transition probabilities for a diffusion under Hormanderʼs “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 Wienerʼs chaos decomposition. While this may make it difficult to prove some of the standard results in Malliavin calculus (boundedness of the derivative operator in Lp 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 Hormaderʼs theorem that focuses on the main ideas without clouding it by technical details.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cdb13ee1e481536697b0e3b8586bdda5b8c31c97

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In Mathematical Physics

Citations in the corpus

Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “The probabilistic proof of Hörmander’s theorem is intricate, and hard to summarise in a few lines, see [18] for a relatively short exposition.”