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 ?
- 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
- Under , ,
- Under , ,
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 .