×

zbMATH — the first resource for mathematics

Harnack inequality and strong Feller property for stochastic fast-diffusion equations. (English) Zbl 1151.60032
As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equation, Ann. Probab. 35, No. 4, 1333–1350 (2007; Zbl 1129.60060)], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, in this very nice paper the authors obtain analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation has weaker dissipativity than the porous medium equation, such results are harder to prove. As a compensation for the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some convincing concrete examples are presented to illustrate the main results.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35Q35 PDEs in connection with fluid mechanics
47D07 Markov semigroups and applications to diffusion processes
60J35 Transition functions, generators and resolvents
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aida, S., Uniformly positivity improving property, Sobolev inequalities and spectral gap, J. funct. anal., 158, 152-185, (1998) · Zbl 0914.47041
[2] Aida, S.; Kawabi, H., Short time asymptotics of certain infinite dimensional diffusion process, (), 77-124 · Zbl 0976.60077
[3] Aida, S.; Zhang, T., On the small time asymptotics of diffusion processes on path groups, Potential anal., 16, 67-78, (2002) · Zbl 0993.60026
[4] Arnaudon, M.; Thalmaier, A.; Wang, F.-Y., Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. sci. math., 130, 223-233, (2006) · Zbl 1089.58024
[5] Aronson, D.G., The porous medium equation, Lecture notes in math., vol. 1224, (1986), Springer Berlin, pp. 1-46 · Zbl 0626.76097
[6] Bendikov, A.; Maheux, P., Nash type inequalities for fractional powers of nonnegative self-adjoint operators, Trans. amer. math. soc., 359, 3085-3097, (2007) · Zbl 1122.47014
[7] Bobkov, S.G.; Gentil, I.; Ledoux, M., Hypercontractivity of hamilton – jacobi equations, J. math. pures appl., 80, 7, 669-696, (2001) · Zbl 1038.35020
[8] Da Prato, G.; Röckner, M.; Rozovskii, B.L.; Wang, F.-Y., Strong solutions to stochastic generalized porous media equations: existence, uniqueness and ergodicity, Comm. partial differential equations, 31, 277-291, (2006) · Zbl 1158.60356
[9] Davies, E.B., Heat kernels and spectral theory, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006
[10] Gong, F.-Z.; Wang, F.-Y., Heat kernel estimates with application to compactness of manifolds, Q. J. math., 52, 2, 171-180, (2001) · Zbl 1132.58302
[11] Hambly, B.M.; Kumagai, T., Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London math. soc. (3), 78, 431-458, (1999) · Zbl 1027.60087
[12] Kim, J.U., On the stochastic porous medium equation, J. differential equations, 220, 163-194, (2006) · Zbl 1099.35187
[13] Krylov, N.V.; Rozovskii, B.L., Stochastic evolution equations, (), 71-146, translated from: · Zbl 0396.60058
[14] Ren, J.; Röckner, M.; Wang, F.-Y., Stochastic generalized porous media and fast diffusion equations, J. differential equations, 238, 118-152, (2007) · Zbl 1129.60059
[15] Röckner, M.; Wang, F.-Y., Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds, Forum math., 15, 893-921, (2003) · Zbl 1062.47044
[16] Röckner, M.; Wang, F.-Y., Harnack and functional inequalities for generalized mehler semigroups, J. funct. anal., 203, 237-261, (2003) · Zbl 1059.47051
[17] Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. theory related fields, 109, 417-424, (1997) · Zbl 0887.35012
[18] Wang, F.-Y., Functional inequalities, semigroup properties and spectrum estimates, Infin. dimens. anal. quantum probab. relat. top., 3, 263-295, (2000) · Zbl 1037.47505
[19] Wang, F.-Y., Harnack inequality and applications for stochastic generalized porous media equations, Ann. probab., 35, 1333-1350, (2007) · Zbl 1129.60060
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.