Abstract
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.
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Received: 31 July 2000 / Revised version: 20 September 2001 / Published online: 15 March 2002
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Picard, J. Gradient estimates for some diffusion semigroups. Probab Theory Relat Fields 122, 593–612 (2002). https://doi.org/10.1007/s004400100185
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DOI: https://doi.org/10.1007/s004400100185