Harnack inequality and applications for stochastic evolution equations with monotone drifts

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.

Aida S., Zhang T.: On the small time asymptotics of diffusion processes on path groups. Pot. Anal. 16, 67–78 (2002)

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)

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)

Bobkov S.G., Gentil I., Ledoux M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001)

Chen, M.-F, From Markov Chains to Non-equilibrum Particle Systems 2 ed. World Scientific, 2004.

Doob J.L.: Asymptotics properties of Markoff transition probabilities. Trans. Amer. Math. Soc. 63, 393–421 (1948)

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)

Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions Encyclopedia of Mathematics and its Applications, Cambridge University Press. 1992.

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)

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.

Goldys B., Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s. Ann. Probab. 34, 1451–1496 (2006)

Gyöngy I., Millet A.: On discretization schemes for etochastic evolution equations. Pot. Anal. 23, 99–134 (2005)

Gong F.-Z., Wang F.-Y.: Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52, 171–180 (2001)

Gong F.-Z., Wang F.-Y.: On Gromov’s theorem and L 2-Hodge decomposition. Int. Math. Math. Sci. 1, 25–44 (2004)

Harrier, M., Coupling stochastic PDEs XIVth International Congress on Mathematical Physics (2005), 281–289.

Kawabi H.: The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application. Potential Analysis 22, 61–84 (2005)

Kim J.U.: On the stochastic porous medium equation. J. Diff. Equat. 220, 163–194 (2006)

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

Liu, W. Large deviations for stochastic evolution equations with small multiplicative noise Appl. Math. Optim.(2009), doi: 10.1007/s00245-009-9072-2 .

Liu W., Wang F.-Y.: Harnack inequality and Strong Feller property for stochastic fast diffusion equations. J. Math. Anal. Appl. 342, 651–662 (2008)

Maslowski B., Seidler J.: Invariant measure for nonlinear SPDE’s: Uniqueness and Stability. Archivum Math. 34, 153–172 (1999)

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

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.

Mueller C.: Coupling and invariant measures for the heat equation with noise. Ann. Probab. 21, 2189–2199 (1993)

Pardoux, E. Equations aux dérivées partielles stochastiques non linéaires monotones Thesis, Université Paris XI, 1975.

Prévôt, C. and Röckner, M., A Concise Course on Stochastic Partial Differential Equations Lecture Notes in Mathematics 1905, Springer, 2007.

Ren J., Röckner M., Wang F.-Y.: Stochastic generalized porous media and fast diffusion equations. J. Diff. Equat. 238, 118–152 (2007)

Röckner M., Wang F.-Y.: Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15, 893–921 (2003)

Röckner M., Wang F.-Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203, 237–261 (2003)

Seidler J.: Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47, 277–316 (1997)

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.

Wang F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probability Theory Relat. Fields 109, 417–424 (1997)

Wang F.-Y.: Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants. Ann. Probab. 27, 653–663 (1999)

Wang F.-Y.: Functional inequalities, semigroup properties and spectrum estimates. Infin. Dimens. Anal. Quant. Probab. Relat. Topics 3, 263–295 (2000)

Wang F.-Y.: Logarithmic Sobolev inequalities: conditions and counterexamples. J. Operator Theory 46, 183–197 (2001)

Wang F.-Y.: Harnack Inequality and Applications for Stochastic Generalized Porous Media equations. Ann. Probab. 35, 1333–1350 (2007)

Wu L.: Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172, 301–376 (2000)

Zhang, X., On stochastic evolution equations with non-Lipschitz coefficients BiBoS-Preprint 08-03-279.