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 L 0 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 L 0 . 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.