TITLE: Big Picture: What is the connection of Malliavin calculus with differential geometry?
QUESTION [18 upvotes]: I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of mathematics? What has this to do with Hörmander's theorem (if so)?

REPLY [7 votes]: Some of Malliavin's ideas are well-explained in his own book: Stochastic Analysis. 
I think the main geometric idea behind his proof of Hormander's theorem is the idea of submersion. More precisely, if we consider a stochastic differential equation in Stratonovitch form
$dX_t=V_0(X_t) dt +\sum_{i=1}^n V_i(X_t) \circ dW^i_t
$
where $W$ is a Wiener process and the $V_i$'s vector fields on $R^n$ (or a manifold), we can see the solution $X_t$ as function of the path $\{ W_s , s \le t\}$. We thus have a map $\Pi$ from the Wiener space $\mathbb{W}$, which is the path space of $W$ and $\mathbb{R}^n$. 
A key insight of Malliavin is now to see the Wiener space $\mathbb{W}$ as an infinite-dimensional manifold with tangent space the Cameron-Martin space (space of functions with derivatives in $L^2$) and to realize that Hormander's conditions make the  map $\Pi$ a submersion. As a consequence the Wiener measure is mapped into a smooth measure, which means that if Hormander's conditions are satisfied then the distribution of $X_t$ has a smooth density. It implies the hypoellipticity of the generator of the Markov process $(X_t)_{t \ge 0}$ which is $L=V_0+\frac{1}{2} \sum_{i=1}^n V_i^2$.
Of course, this has to be made rigorous and some care has to be given because we have to use Sobolev type derivatives instead of Frechet derivatives.
This idea that there is a "natural" manifold structure on path spaces is quite fruitful and besides Malliavin's proof of Hormander's theorem explains and leads to several interesting properties of the operator $V_0+\frac{1}{2} \sum_{i=1}^n V_i^2$.