zbMATH — the first resource for mathematics

Dimension-independent Harnack inequalities for subordinated semigroups. (English) Zbl 1219.43006
Authors’ abstract: Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power \(\alpha \) satisfies a dimension-free Harnack inequality provided \(\alpha \in(\frac{1}{2}, 1)\), and it satisfies the log-Harnack inequality for all \(\alpha \in (0,1)\). Some infinite-dimensional examples are also presented.

43A80 Analysis on other specific Lie groups
58J65 Diffusion processes and stochastic analysis on manifolds
60J60 Diffusion processes
Full Text: DOI arXiv
[1] Aida, S.: Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158(1), 152–185 (1998) MR MR1641566 (2000d:60125) · Zbl 0914.47041 · doi:10.1006/jfan.1998.3286
[2] Aida, S., Kawabi, H.: Short time asymptotics of a certain infinite dimensional diffusion process. In: Stochastic Analysis and Related Topics, VII (Kusadasi, 1998), Progr. Probab., vol. 48, pp. 77–124. Birkhäuser, Boston, MA, USA (2001) MR MR1915450 (2003m:60219) · Zbl 0976.60077
[3] Aida, S., Zhang, T.: On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16(1), 67–78 (2002) MR MR1880348 (2003e:58052) · Zbl 0993.60026 · doi:10.1023/A:1024868720071
[4] Arnaudon, M., Thalmaier, A., Wang, F.-Y.: Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130(3), 223–233 (2006) MR MR2215664 (2007i:58032) · Zbl 1089.58024 · doi:10.1016/j.bulsci.2005.10.001
[5] Arnaudon, M., Thalmaier, A., Wang, F.-Y.: Gradient estimates and harnack inequalities on non-compact riemannian manifolds. Stoch. Process. Their Appl. 119(10), 3653–3670 (2009) · Zbl 1178.58013 · doi:10.1016/j.spa.2009.07.001
[6] Bobkov, S.G., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80(7), 669–696 (2001) MR MR1846020 (2003b:47073) · Zbl 1038.35020 · doi:10.1016/S0021-7824(01)01208-9
[7] Da Prato, G., Röckner, M., Rozovskii, B.L., Wang, F.-Y.: Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31(1–3), 277–291 (2006) MR MR2209754 (2007b:60153) · Zbl 1158.60356 · doi:10.1080/03605300500357998
[8] Da Prato, G., Röckner, M.: Singular dissipative stochastic equations in Hilbert spaces. Probab. Theor. Relat. Fields 124(2), 261–303 (2002) MR MR1936019 (2003k:60151) · Zbl 1036.47029 · doi:10.1007/s004400200214
[9] Da Prato, G., Röckner, M., Wang, F.-Y.: Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257(4), 992–1017 (2009) MR MR2535460 · Zbl 1193.47047 · doi:10.1016/j.jfa.2009.01.007
[10] Driver, B.K., Gordina, M.: Heat kernel analysis on infinite-dimensional Heisenberg groups. J. Funct. Anal. 255(9), 2395–2461 (2008) MR MR2473262 · Zbl 1163.43005 · doi:10.1016/j.jfa.2008.06.021
[11] Driver, B.K., Gordina, M.: Integrated Harnack inequalities on Lie groups. J. Differ. Geom. 3, 501–550 (2009) · Zbl 1205.53044
[12] Jacob, N.: Pseudo differential operators and Markov processes. In: Fourier analysis and semigroups, vol. 1. Imperial College Press, London (2001) MR MR1873235 (2003a:47104) · Zbl 0987.60003
[13] Liu, W.: Fine Properties of Stochastic Evolution Equations and Their Applications. Ph.D. thesis, Doctor-Thesis, Bielefeld University (2009) · Zbl 1213.60010
[14] Liu, W., Wang, F.-Y.: Harnack inequality and strong Feller property for stochastic fast-diffusion equations. J. Math. Anal. Appl. 342(1), 651–662 (2008) MR MR2440828 (2009k:60137) · Zbl 1151.60032 · doi:10.1016/j.jmaa.2007.12.047
[15] Ouyang, S.-X.: Harnack Inequalities and Applications for Stochastic Equations. Ph.D. thesis, Ph.D. thesis, Bielefeld University (2009) · Zbl 1213.60012
[16] Ouyang, S.-X., Röckner, M., Wang, F.-Y.: Harnack inequalities and applications for Ornstein-Uhlenbeck semigroups with jumps (2010, submitted)
[17] Röckner, M., Wang, F.-Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203(1), 237–261 (2003) MR MR1996872 (2005d:47077) · Zbl 1059.47051 · doi:10.1016/S0022-1236(03)00165-4
[18] Wang, F.-Y.: Heat kernel inequalities for curvature and second fundamental form. Doc. Math. (to appear)
[19] Wang, F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109(3), 417–424 (1997) MR MR1481127 (98i:58253) · Zbl 0887.35012 · doi:10.1007/s004400050137
[20] Wang, F.-Y.: Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35(4), 1333–1350 (2007) MR MR2330974 (2008e:60192) · Zbl 1129.60060 · doi:10.1214/009117906000001204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.