Ergodicity of transition semigroups for stochastic fast diffusion equations

Abstract

In this paper, we first show the uniqueness of invariant measures for the stochastic fast diffusion equation, which follows from an obtained new decay estimate. Then we establish the Harnack inequality for the stochastic fast diffusion equation with nonlinear perturbation in the drift and derive the heat kernel estimate and ultrabounded property for the associated transition semigroup. Moreover, the exponential ergodicity and the existence of a spectral gap are also investigated.

Keywords

Harnack inequality / invariant measure / ergodicity / fast diffusion equation / heat kernel / spectral gap

References

[1] Arnaudon M, Thalmaier A, Wang F Y. Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull Sci Math, 2006, 130: 223-233

[2] Barbu V, Da Prato G. Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation. Stoc Proc Appl, 2010, 120: 1247-1266

[3] Bendikov A, Maheux P. Nash type inequalities for fractional powers of nonnegative self-adjoint operators. Trans Amer Math Soc, 2007, 359: 3085-3097

[4] Bogachev V I, Da Prato G, Röckner M. Invariant measures of generalized stochastic porous medium equations. Dokl Math, 2004, 69: 321-325

[5] Bogachev V I, Röckner M, Zhang T S. Existence and uniqueness of invariant measures: an approach via sectorial forms. Appl Math Optim, 2000, 41: 87-109

[6] Croke C B. Some isoperimetric inequalities and eigenvalue estimates. Ann Sci Éc Norm Super, 1980, 13: 419-435

[7] 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 Equ, 2006, 31: 277-291

[8] Da Prato G, Röckner M, Wang F Y. Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J Funct Anal, 2009, 257: 992-1017

[9] Da Prato G, Zabczyk J. Ergodicity for Infinite-dimensional Systems. London Mathematical Society Lecture Note Series, 229. Cambridge: Cambridge University Press, 1996

[10] Daskalopoulos P, Kenig C E. Degenerate Diffusions: Initial Value Problems and Local Regularity Theory. EMS Tracts in Mathematics 1. Zürich: European Mathematical Society, 2007

[11] Doob J L. Asymptotics properties of Markov transition probabilities. Trans Amer Math Soc, 1948, 63: 393-421

[12] Driver B, Gordina M. Integrated Harnack inequalities on Lie groups. J Diff Geom, 2009, 83: 501-550

[13] Gess B, Liu W, Röckner M. Random attractors for a class of stochastic partial differential equations driven by general additive noise. J Differential Equations, 2011,

[14] Goldys B, Maslowski B. Exponential ergodicity for stochastic reaction-diffusion equations. In: Lecture Notes Pure Appl Math, Vol 245. Boca Raton: Chapman Hall/CRC Press, 2004, 115-131

[15] Goldys B, Maslowski B. Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s. Ann Probab, 2006, 34: 1451-1496

[16] Gong F Z, Wang F Y. Heat kernel estimates with application to compactness of manifolds. Q J Math, 2001, 52(2): 171-180

[17] Gyöngy I. On stochastic equations with respect to semimartingale III. Stochastics, 1982, 7: 231-254

[18] Hairer M. Coupling stochastic PDEs. In: XIVth International Congress on Mathematical Physics, 2005, 281-289

[19] Krylov N V, Rozovskii B L. Stochastic evolution equations. Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki, 1979, 14: 71-146

[20] Liu W. Harnack inequality and applications for stochastic evolution equations with monotone drifts. J Evol Equ, 2009, 9: 747-770

[21] Liu W. On the stochastic p-Laplace equation. J Math Anal Appl, 2009, 360: 737-751

[22] Liu W. Large deviations for stochastic evolution equations with small multiplicative noise. Appl Math Optim, 2010, 61: 27-56

[23] Liu W, Röckner M. SPDE in Hilbert space with locally monotone coefficients. J Funct Anal, 2010, 259: 2902-2922

[24] Liu W, Tölle J M. Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts. Preprint

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

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

[27] Pardoux E. Stochastic partial differential equations and filtering of diffusion processes. Stochastics, 1979, 3: 127-167

[28] Ren J, Röckner M, Wang F Y. Stochastic generalized porous media and fast diffusion equations. J Differential Equations, 2007, 238: 118-152

[29] Röckner M, Wang F Y. Non-monotone stochastic porous media equation. J Differential Equations, 2008, 245: 3898-3935

[30] Seidler J. Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math J, 1997, 47: 277-316

[31] Vázquez J L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Oxford Lecture Notes in Mathematics and Its Applications, Vol 33. Oxford: Oxford University Press, 2006

[32] Wang F Y. Functional inequalities, semigroup properties and spectrum estimates. Infin Dimens Anal Quant Probab Relat Top, 2000, 3: 263-295

[33] Wang F Y. Dimension-free Harnack inequality and its applications. Front Math China, 2006, 1: 53-72

[34] Wang F Y. Harnack inequality and applications for stochastic generalized porous media equations. Ann Probab, 2007, 35: 1333-1350

[35] Wang F Y. Harnack Inequalities on Manifolds with Boundary and Applications. J Math Pures Appl, 2010, 94: 304-321

[36] Zhang X. On stochastic evolution equations with non-Lipschitz coefficients. Stoch Dyn, 2009, 9: 549-595

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