Establishing regularity of transition probabilities is a standard focus for solutions to stochastic differential equations (SDEs). For diffusions in finite-dimensional spaces, the Hormander ``bracket generating'' condition for an SDE is a standard geometric assumption that ensures smoothness of the solution. The Hormander condition also often induces a natural geometry on the space which is tied to the analysis of the diffusion.
The situation in infinite dimensions is more complicated and less understood. We'll consider a class of infinite dimensional spaces where we propose a different but equivalent analytic formulation of the Hormander condition. Under this assumption, we discuss the related geometry and establish some regularity properties of the associated diffusion.