Projection of the infinitesimal generator of a diffusion

Abstract

Consider a finite or infinite dimensional space X with a diffusion having an invariant measure μ. A typical example is the Wiener space with the Ornstein-Ühlenbeck process. Another example is the heat process on the Wiener space. To a nondegenerated map g from X into a finite dimensional manifold V, we associate the following invariants: (1) a Riemannian metric on V; (2) a vector field Z on V. Then the image g ∗ μ of the measure μ through the map g will have a density k with respect to the Riemannian volume. For k, the relation Z = ▽log k will hold. The projected infinitesimal generator will be equal to the sum Δ + Z▽, where Δ is the riemannian Laplacian on the manifold V. This result applied to the heat equation on X yields a non-autonomous parabolic equation which gives the law g ∗ μ . The second order operator involved in this parabolic equation is strictly elliptic. The method reduces the study of a hypoelliptic operator to that of a non-autonomous elliptic one.