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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
DOI: https://doi.org/10.1090/S0002-9939-04-07625-7
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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  • 1. D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New Trends in Stochastic Analysis (Eds. K.D. Elworthy, S. Kusuoka, I. Shigekawa), 43-75, World Scientific, 1997. MR 99m:60110
  • 2. D. Bakry and M. Ledoux, Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123(1996), 253-270. MR 97c:58162
  • 3. M. F. Chen and F. Y. Wang, Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc. 349(1997), 1239-1267. MR 97h:35175
  • 4. G. Da Prato and B. Goldys, Elliptic operators on $\mathbb R^d$with unbounded coefficients, J. Diff. Equat. 172 (2001), 333-358. MR 2002d:35228
  • 5. F.-Z. Gong and F.-Y. Wang, Heat kernel estimates with application to compactness of manifolds, Quart. J. Math. 52(2001), 171-180. MR 2002c:58039
  • 6. E. P. Hsu, Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm. Math. Phys. 189(1997), 9-16. MR 98i:58249
  • 7. F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Relat. Fields 109(1997), 417-424. MR 98i:58253
  • 8. F.-Y. Wang, Functional inequalities, semigroup properties and spectrum estimates, Inf. Dimens. Anal. Quant. Probab. Relat. Topics 3(2000), 263-295. MR 2002b:47083

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Additional Information

Feng-Yu Wang
Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China
Email: wangfy@bnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-04-07625-7
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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