  Let $L$ be an infinitely degenerate second-order linear operator defined on a
bounded smooth Euclidean domain. Under weaker conditions than those of
H\"ormander, we show that the Dirichlet problem associated with $L$ has a
unique smooth classical solution. The proof uses the Malliavin calculus. At
present, there appears to be no proof of this result using classical analytic
techniques.

Analysis of PDEs