1. Probabilistic content of Hörmander’s condition
Theorem 1 Suppose that the Lie algebra generated by and brackets fills . Then the bilinear form on is non-degenerate
By Blumenthal’s 0/1-law, is not random. We prove by contradiction that (this will suffice to prove the theorem). Introduce
Let be orthogonal to (and thus to for ). Since is orthogonal to all , . But for all vectorfields , satisfies (first line is Stratonovich, the second is Ito)
thus for all ,
By uniqueness of the solution of an SDE, this implies that for all and . Replacing with shows that
Iterating the procedure shows orthogonality of with all iterated brackets, and thus .
2. Probabilistic proof of hypoellipticity
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 .
The proof we are about to give is due to a large extent to Bismut, although many details are skipped in Bismut’s original paper. We use more elementary tools. We shall rely on the following standard fact.
2.1. Girsanov’s theorem
Let a Brownian motion on Euclidean space. Add an absolutely continuous process, i.e. such that
is not a martingale any more, but this can be recovered by changing the probability measure.
Theorem 3 (Girsanov) is a Brownian measure with respect to the mesure whose density with respect to is
In other words, if is a functional on the space of Brownian motions, then
2.2. A criterion for a measure to have a smooth density
We want to prove that for . We use the following criterion.
Lemma 4 Let be a probability measure on some manifold, viewed as a distribution. Assume that for all and all test functions ,
Then has a smooth density.
2.3. Proof of Theorem 2
Fix . Identify with . We apply Girsanov’s theorem to where takes values in and . The modified flow is denoted by . Let be a function to be specified later. Up to introducing the density , nothing changes, and
does not depend on . Let us differentiate with respect to at .
Remember that SDE can be formally differentiated with respect to a parameter. Notation: . Get
This suggests choosing
With this choice,
By assumption, is invertible, so we take
where is to be specified later. This yields
for some rather complicated expression . Iteration gives
from which we get the estimate
The right hand side involves only polynomial expressions, except and its derivatives with respect to . These have to be computed and estimated too. Then the Lemma applies, it shows that the distribution of has a smooth density.
3. Subjects I could not cover
There was no time to treat
- the short time asymptotics of the heat kernel,
- bounds on the lifetime of Brownian motion (differentiating , leads to the Laplacian of the distance function, and to Ricci curvature).
- Bismut’s interpolation between the geodesic flow and an hypoelliptic diffusion.
There is more that probability theory can do for sub-Riemannian geometry and hypoelliptic PDE’s.