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2012

Generalized Curvature Condition for Subelliptic Diffusion Processes

Wang, Feng-Yu

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By using a general version of curvature condition, derivative inequalities are established for a large class of subelliptic diffusion semigroups.

As applications, the Harnack/cost-entropy/cost-variance inequalities for the diffusion semigroups, and the Poincar\'e/log-Sobolev inequalities for the associated Dirichlet forms in the symmetric case, are derived.
Our results largely generalize and partly improve the corresponding ones obtained recently in [BB].

Generalized Curvature Condition for Subelliptic Diffusion Processes · Around

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Similar

D. Bakry, M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), 253–270
1996
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F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Relat. Fields 109(1997), 417–424
1997
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Z. Qian, A comparison theorem for an elliptic operator, Potential Anal. 8 (1998), 137¨C142
1998
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B. K. Driver, T. Melcker, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221(2005), 340–365
2005
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F. Baudoin, M. Bonnefont, Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality, to appear in J. Funct. Anal
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F. Baudoin, M. Bonnefont, N. Garofalo, A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality, arXiv:1007.1600
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F. Baudoin, N. Garofalo, Curvature-dimesnion inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, arXiv:1101.3590v3
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F. Baudoin, M. Gordina, T. Melcher, Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups, arXiv:1108.1527
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A. Guillin, F.-Y. Wang, Degerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, accessible on arXiv:1103.2817v2
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F.-Y. Wang, Derivative formula and gradient estimates for Gruschin type semigroups, arXiv:1109.6738
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F.-Y. Wang, Derivative formula and Harnack inequality for jump processes, arXiv:1104.5531
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Then

H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236(2006), 369–394
2006
Later among the works it cites.
M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stoch. Proc. Appl. 119(2009), 3653–3670
2009
Later among the works it cites.
F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. 94(2010), 304–321
2010
Later among the works it cites.
F.-Y. Wang, C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl. 121(2011), 2692–2710
2011
Later among the works it cites.

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