Gradient estimates for some diffusion semigroups

 Consider the semigroup $P_t$ of an elliptic diffusion; we describe a simple stochastic method providing gradient estimates on $P_tf$. If $N$ is a manifold endowed with a connection, the method can also be applied to the associated nonlinear semigroup $Q_t$ acting on $N$-valued maps. With a localization technique, we deduce gradient estimates for real harmonic functions or $N$-valued harmonic maps. Moreover, the results are extended to a class of hypoelliptic diffusions. Received: 31 July 2000 / Revised version: 20 September 2001 / Published online: 15 March 2002