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Harnack inequality and applications for stochastic evolution equations with monotone drifts. (English) Zbl 1239.60058
Summary: As a Generalization to [F.-Y. Wang, Ann. Probab. 35, No. 4, 1333–1350 (2007; Zbl 1129.60060)] where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic \(p\)-Laplace equation in Hilbert space.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35R50 PDEs of infinite order
60F10 Large deviations
60H40 White noise theory
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References:
[1] Aida, S. and Kawabi, H., Short time asymptotics of certain infinite dimensional diffusion process ”Stochastic Analysis and Related Topics”, VII (Kusadasi,1998); in Progr. Probab. Vol. 48(2001), 77–124. · Zbl 0976.60077
[2] Aida S., Zhang T.: On the small time asymptotics of diffusion processes on path groups. Pot. Anal. 16, 67–78 (2002) · Zbl 0993.60026 · doi:10.1023/A:1024868720071
[3] 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 · doi:10.1016/j.bulsci.2005.10.001
[4] Aronson D.G., Peletier L.A.: Large time behaviour of solutions of the porous medium equation in bounded domains. J. Diff. Equ. 39, 378–412 (1981) · Zbl 0475.35059 · doi:10.1016/0022-0396(81)90065-6
[5] Bobkov S.G., Gentil I., Ledoux M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001) · Zbl 1038.35020
[6] Chen, M.-F, From Markov Chains to Non-equilibrum Particle Systems 2 ed. World Scientific, 2004.
[7] Doob J.L.: Asymptotics properties of Markoff transition probabilities. Trans. Amer. Math. Soc. 63, 393–421 (1948) · Zbl 0041.45406
[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. Part. Diff. Equat. 31, 277–291 (2006) · Zbl 1158.60356 · doi:10.1080/03605300500357998
[9] Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions Encyclopedia of Mathematics and its Applications, Cambridge University Press. 1992. · Zbl 0761.60052
[10] Döblin W.: Exposé sur la théorie des chaînes simples constantes de Markoff à un nombre finid’états. Rev. Math. Union Interbalkanique 2, 77–105 (1938)
[11] Goldys B. and Maslowski B., Exponential ergodicity for stochastic reaction-diffusion equations, Stochastic Partial Differential Equations and Applications VII. Lecture Notes Pure Appl. Math. 245(2004), 115-131. Chapman Hall/CRC Press. · Zbl 1091.35118
[12] Goldys B., Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s. Ann. Probab. 34, 1451–1496 (2006) · Zbl 1121.60066 · doi:10.1214/009117905000000800
[13] Gyöngy I., Millet A.: On discretization schemes for etochastic evolution equations. Pot. Anal. 23, 99–134 (2005) · Zbl 1067.60049 · doi:10.1007/s11118-004-5393-6
[14] Gong F.-Z., Wang F.-Y.: Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52, 171–180 (2001) · Zbl 1132.58302 · doi:10.1093/qjmath/52.2.171
[15] Gong F.-Z., Wang F.-Y.: On Gromov’s theorem and L 2-Hodge decomposition. Int. Math. Math. Sci. 1, 25–44 (2004) · Zbl 1082.58028 · doi:10.1155/S0161171204210365
[16] Harrier, M., Coupling stochastic PDEs XIVth International Congress on Mathematical Physics (2005), 281–289.
[17] Kawabi H.: The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application. Potential Analysis 22, 61–84 (2005) · Zbl 1118.60053
[18] Kim J.U.: On the stochastic porous medium equation. J. Diff. Equat. 220, 163–194 (2006) · Zbl 1099.35187 · doi:10.1016/j.jde.2005.02.006
[19] Krylov N.V., Rozovskii B.L.: Stochastic evolution equations. Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki 14, 71–146 (1979) Plenum Publishing Corp
[20] Liu, W. Large deviations for stochastic evolution equations with small multiplicative noise Appl. Math. Optim.(2009), doi: 10.1007/s00245-009-9072-2 .
[21] Liu W., Wang F.-Y.: Harnack inequality and Strong Feller property for stochastic fast diffusion equations. J. Math. Anal. Appl. 342, 651–662 (2008) · Zbl 1151.60032 · doi:10.1016/j.jmaa.2007.12.047
[22] Maslowski B., Seidler J.: Invariant measure for nonlinear SPDE’s: Uniqueness and Stability. Archivum Math. 34, 153–172 (1999) · Zbl 0914.60028
[23] Maslowski, B. and Seidler, J., Strong Feller infinite-Dimensional Diffusions Proceedings, Trento 2000, 373–389, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 2002 · Zbl 1007.60063
[24] Mattingly, J.C., On recent progress for the stochastic Navier Stokes equations Journées ”Équations aux Dérivées Partielles”, Exp. No. XI, 52 pp., Univ. Nantes, Nantes, 2003.
[25] Mueller C.: Coupling and invariant measures for the heat equation with noise. Ann. Probab. 21, 2189–2199 (1993) · Zbl 0795.60056 · doi:10.1214/aop/1176989016
[26] Pardoux, E. Equations aux dérivées partielles stochastiques non linéaires monotones Thesis, Université Paris XI, 1975.
[27] Prévôt, C. and Röckner, M., A Concise Course on Stochastic Partial Differential Equations Lecture Notes in Mathematics 1905, Springer, 2007. · Zbl 1123.60001
[28] Ren J., Röckner M., Wang F.-Y.: Stochastic generalized porous media and fast diffusion equations. J. Diff. Equat. 238, 118–152 (2007) · Zbl 1129.60059 · doi:10.1016/j.jde.2007.03.027
[29] Röckner M., Wang F.-Y.: Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15, 893–921 (2003) · Zbl 1062.47044
[30] Röckner M., Wang F.-Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203, 237–261 (2003) · Zbl 1059.47051 · doi:10.1016/S0022-1236(03)00165-4
[31] Seidler J.: Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47, 277–316 (1997) · Zbl 0935.60041 · doi:10.1023/A:1022821729545
[32] Walsh, J.B., An introduction to stochastic partial differential equations, Ecole d’Ete de Probabilite de Saint-Flour XIV (1984), P.L. Hennequin editor, Lecture Notes in Mathematics 1180 , 265–439.
[33] Wang F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probability Theory Relat. Fields 109, 417–424 (1997) · Zbl 0887.35012 · doi:10.1007/s004400050137
[34] Wang F.-Y.: Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants. Ann. Probab. 27, 653–663 (1999) · Zbl 0948.58023 · doi:10.1214/aop/1022677381
[35] Wang F.-Y.: Functional inequalities, semigroup properties and spectrum estimates. Infin. Dimens. Anal. Quant. Probab. Relat. Topics 3, 263–295 (2000) · Zbl 1037.47505
[36] Wang F.-Y.: Logarithmic Sobolev inequalities: conditions and counterexamples. J. Operator Theory 46, 183–197 (2001) · Zbl 0993.58019
[37] Wang F.-Y.: Harnack Inequality and Applications for Stochastic Generalized Porous Media equations. Ann. Probab. 35, 1333–1350 (2007) · Zbl 1129.60060 · doi:10.1214/009117906000001204
[38] Wu L.: Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172, 301–376 (2000) · Zbl 0957.60032 · doi:10.1006/jfan.1999.3544
[39] Zhang, X., On stochastic evolution equations with non-Lipschitz coefficients BiBoS-Preprint 08-03-279.
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