To conclude this course, we are going to provide an elementary proof of the Stroock–Varadhan support theorem which is based on rough paths theory. We first remind that the support of a random variable which defined on a metric space is the smallest closed such that . In particular if and only if for every open ball , .
Let be a -dimensional Brownian motion. We can see as a random variable that takes its values in , . The following theorem describes the support of this random variable.
Proposition: Let . The support of in is , that is the closure for the -variation distance of the set of smooth paths starting at 0.
Proof: The key argument is a clever application of the Cameron-Martin theorem. Let us recall that this theorem says that if
then the distribution of is equivalent to the distribution of .
Let us denote by the support of . It is clear that , because the paths of have bounded variation for .
Let now . We have for , . From the Cameron Martin theorem, we deduce then for , . This shows that . We can find a sequence of smooth that converges to in -variation. From the previous argument and converges to 0. Thus and using the same argument shows then that is included in . This proves that
The following theorem due to Stroock and Varadhan describes the support of solutions of stochastic differential equations. As in the previous proof, we denote
Theorem: Let and let be -Lipschitz vector fields on . Let . Let be the solution of the Stratonovitch differential equation:
Let . The support of in is the closure in the -variation topology of the set:
where is the solution of the ordinary differential equation
Proof: We denote by the piecewise linear process which is obtained from by interpolation along a subdivision which is such that and whose mesh goes to 0. We know that and that almost surely converges in -variation to . As a consequence almost surely takes its values in the closure of:
This shows that the support of is included in the closure of . The converse inclusion is a little more difficult and relies on the Lyons’ continuity theorem. It can be proved by using similar arguments as for (details are let to the reader) that the support of is the is the closure in the -variation topology of the set:
where denotes, as usual, the lift in the Carnot group of step 2. Take and . By the Lyons’ continuity theorem, there exists therefore such that implies . Therefore
In particular, we have . This proves that is in the support of . So, the proof now boils down to the statement that the support of is the closure in the -variation topology of the set: