**1. Stochastic flows of diffeomorphisms **

We continue our study of SDE . Up to now, the starting point was fixed. Now we exploit the dependance on .

** 1.1. Random continuous paths of diffeomorphisms **

Let us introduce the random set of starting points whose trajectory is still alive at time . Then

- is open (in fact, the lifetime is lower semi-continuous in ).
- is a diffeomorphism onto an open subset of .
- is continuous: .

Furthermore, under mild growth conditions on vectorfields and their derivatives (for instance, if is compact), for all .

Consider the tangent flow on . It solves the formally differentiated SDE

** 1.2. Crucial observation **

Let us transport a vectorfield under our stochastic flow. We get a random vectorfield . This means that, for a test function ,

Maillavin’s covariance matrix is defined as follows. For ,

This is a random smooth section of over . We shall see later that the condition we need to make this nondegenerate is Hörmander’s condition.

On may view

as a linear map from to . Its adjoint is a linear map from to . Then may be viewed as an endomorphism of .

**Lemma 1** * The SDE satisfied by is *

*
** *

In particular, if commutes with vectorfields , .

**2. Stochastic flows and hypoellipticity **

We assume that all constant coefficient combinations of the are complete. The flow defines two canonical measures,

- The distribution of , ,
- Green’s measure .

Let us study the following Dirichlet boundary problem

The solution takes the following form (Feyman-Kac formula).

** 2.1. Hörmander’s condition **

**Question**: When do and have a density ?

Let

- denote the Lie algebra generated by the vectorfield ,
- the Lie algebra generated by only,
- by and brackets ,
- by and on .

Hörmander’s theorem states

- hypoellipticity of under ,
- hypoellipticity of under .

It follows that

where the densities are smooth.

** 2.2. A probabilistic proof of hypoellipticity ? **

In 1970, in his Kyoto lectures, Paul Maillavin proposed a toolbox to prove this, called Maillavin Calculus. This calculus deals with infinite dimensional path spaces.

Instead, I will describe a more direct root. The existence of smooth densities and in turn imply hypoellipticity, so it suffices to prove this.

**Theorem 2** * Assume that and there derivatives satisfy suitable growth conditions. Assume that the bilinear form is non-degenerate and *

*
** for all . Then with a smooth density . *

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/