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A character of the gradient estimate for diffusion semigroups

Author: Feng-Yu Wang
Journal: Proc. Amer. Math. Soc. 133 (2005), 827-834
MSC (2000): Primary 47D07, 60H10
Published electronically: September 29, 2004
MathSciNet review: 2113933
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Abstract: Let $P_t$ be the semigroup of the diffusion process generated by $L:= \sum_{i,j}a_{ij}\partial_i\partial_j +\sum_ib_i\partial_i$ on $\mathbb{R} ^d$. It is proved that there exists $c\in \mathbb{R} $ and an $\mathbb{R} ^d$-valued function $b=(b_i)$ such that $\vert\nabla P_tf\vert\le \text{\rm {e}} ^{ct}P_t\vert\nabla f\vert$ holds for all $t>0$ and all $f\in C_b^1(\mathbb{R} ^d)$ if and only if $a=(a_{ij})$ satisfies the formula $\partial_k a_{ij}+\partial_ja_{ki} +\partial_i a_{kj}=0$ for all $i,j,k.$

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Additional Information

Feng-Yu Wang
Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China

Received by editor(s): February 6, 2002
Received by editor(s) in revised form: November 15, 2003
Published electronically: September 29, 2004
Additional Notes: This work was supported in part by NNSFC(10025105, 10121101), TRAPOYT in China and the 973-Project.
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2004 American Mathematical Society

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