The rate function of hypoelliptic diffusions

Let be a hypoelliptic diffusion operator on a compact manifold M. Given an a priori smooth reference measure λ on M, we can then rewrite L as the sum of a λ-symmetric part L⁰ and a first-order drift part Y. The paper investigates the effect of the drift Y on the Donsker-Varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion. When Y is in the linear span of the first and second-order Lie brackets of the X_i's, we derive an affine bound relating the rate functions associated with L and L⁰. As soon as one point exists where Y is not in the linear span of the first and second-order Lie brackets of the X_i's, we show that such an affine bound is impossible. © 1994 John Wiley & Sons, Inc.