  In this paper we consider diffusion semigroups generated by second order
differential operators of degenerate type. The operators that we consider do
not, in general, satisfy the Hormander condition and are not hypoelliptic. In
particular, instead of working under the Hormander paradigm, we consider the
so-called UFG condition, introduced by Kusuoka and Strook in the eighties. The
UFG condition is weaker than the uniform Hormander condition, the smoothing
effect taking place only in certain directions (rather than in every direction,
as it is the case when the Hormander condition is assumed). Under the UFG
condition, Kusuoka and Strook deduced sharp small time asymptotic bounds for
the derivatives of the semigroup in the directions where smoothing occurs. In
this paper, we study the large time asymptotics for the gradients of the
diffusion semigroup in the same set of directions and under the same UFG
condition. In particular, we identify conditions under which the derivatives of
the diffusion semigroup in the smoothing directions decay exponentially in
time. This paper constitutes therefore a stepping stone in the analysis of the
long time behaviour of diffusions which do not satisfy the Hormander
condition.
