The Annals of Probability

Harnack inequality and applications for stochastic generalized porous media equations

Feng-Yu Wang

Full-text: Open access

Abstract

By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the Lp-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.

Article information

Source
Ann. Probab. Volume 35, Number 4 (2007), 1333-1350.

Dates
First available: 8 June 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1181334247

Digital Object Identifier
doi:10.1214/009117906000001204

Mathematical Reviews number (MathSciNet)
MR2330974

Zentralblatt MATH identifier
1129.60060

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 76S05: Flows in porous media; filtration; seepage

Keywords
Harnack inequality stochastic generalized porous medium equation ultracontractivity

Citation

Wang, Feng-Yu. Harnack inequality and applications for stochastic generalized porous media equations. The Annals of Probability 35 (2007), no. 4, 1333--1350. doi:10.1214/009117906000001204. http://projecteuclid.org/euclid.aop/1181334247.


Export citation

References

  • Aida, S. and Kawabi, H. (2001). Short time asymptotics of certain infinite dimensional diffusion process. Stochastic Analysis and Related Topics 48 77--124.
  • Aida, S. and Zhang, T. (2002). On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16 67--78.
  • Arnaudon, M., Thalmaier, A. and Wang, F.-Y. (2005). Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130 223--233.
  • Aronson, D. G. (1986). The porous medium equation. Nonlinear Diffusion Problems (Montecatini Terme, 1985). Lecture Notes in Math. 1224 1--46. Springer, Berlin.
  • Bakry, D. and Ledoux, M. (1996). Lévy--Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 259--281.
  • Da Prato, G., Röckner, M., Rozovskii, B. L. and Wang, F.-Y. (2004). Strong solutions to stochastic generalized porous media equations: Existence, uniqueness and ergodicity. Comm. Partial Differential Equations 31 277--291.
  • Bobkov, S. G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton--Jacobi equations. J. Math. Pures Appl. 80 669--696.
  • Gong, F.-Z. and Wang, F.-Y. (2001). Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52 171--180.
  • Gong, F.-Z. and Wang, F.-Y. (2002). Functional inequalities for uniformly integrable semigroups and application to essential spectrums. Forum Math. 14 293--313.
  • Hambly, B. M. and Kumagai, T. (1999). Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. (3) 78 431--458.
  • Kawabi, H. (2005). The parabolic Harnack inequality for the time dependent Ginzburg--Landau type SPDE and its application. Potential Anal. 22 61--84.
  • Kim, J. U. (2006). On the stochastic porous medium equation. J. Differential Equations 220 163--194.
  • Krylov, N. V. and Rozovskii, B. L. (1979). Stochastic evolution equations. Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki 14 71--146. Plenum Publishing Corp.
  • Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, New York.
  • Ren, J., Röckner, M. and Wang, F.-Y. (2007). Stochastic generalized porous media and fast diffusion equations. J. Differential Equations. To appear.
  • Röckner, M. and Wang, F.-Y. (2003). Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15 893--921.
  • Röckner, M. and Wang, F.-Y. (2003). Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203 237--261.
  • Röckner, M., Wang, F.-Y. and Wu, L. (2007). Large deviations for stochastic generalized porous media equations. Stoch. Proc. Appl. 116 1677--1689.
  • Wang, F.-Y. (1997). Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 417--424.
  • Wang, F.-Y. (1999). Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants. Ann. Probab. 27 653--663.
  • Wang, F.-Y. (2000). Functional inequalities, semigroup properties and spectrum estimates. Infin. Dimens. Anal. Quantum Probab. Relat. Topics 3 263--295.
  • Wang, F.-Y. (2001). Logarithmic Sobolev inequalities: Conditions and counterexamples. J. Operator Theory 46 183--197.
  • Wu, L. (2000). Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 301--376.