November  2015, 35(11): 5239-5253. doi: 10.3934/dcds.2015.35.5239

Exponential convergence of non-linear monotone SPDEs

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  October 2013 Revised  October 2014 Published  May 2015

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\lambda>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{\mu(f^2)\le 1}||P_tf-\mu(f)||_\infty \le C e^{-\lambda t},\ \ t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.
Citation: Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
References:
[1]

D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains,, J. Diff. Equ., 39 (1981), 378.  doi: 10.1016/0022-0396(81)90065-6.  Google Scholar

[2]

A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator,, J. Math. Kyoto Univ., 36 (1996), 481.   Google Scholar

[3]

G. Da Prato, M. Röckner and B. L. Rozovskiĭ and F.-Y. Wang, Strong solutions of Generalized porous media equations: Existence, uniqueness and ergodicity,, Comm. Part. Diff. Equ., 31 (2006), 277.  doi: 10.1080/03605300500357998.  Google Scholar

[4]

B. Goldys and B. Maslowski, Exponential ergordicity for stochastic reaction-diffusion equations,, in Stochastic Partial Differential Equations and Applications-VII, (2006), 115.  doi: 10.1201/9781420028720.ch12.  Google Scholar

[5]

B. Goldys and B. Maslowski, Lower estimates of transition density and bounds on exponential ergodicity for stochastic PDEs,, Ann. Probab., 34 (2006), 1451.  doi: 10.1214/009117905000000800.  Google Scholar

[6]

N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations,, Current problems in mathematics, (1979), 71.   Google Scholar

[7]

W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts,, J. Evol. Equ., 9 (2009), 747.  doi: 10.1007/s00028-009-0032-8.  Google Scholar

[8]

W. Liu, Ergodicity of transition semigroups for stochastic fast diffusion equations,, Front. Math. China, 6 (2011), 449.  doi: 10.1007/s11464-011-0112-2.  Google Scholar

[9]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients,, J. Funct. Anal., 259 (2010), 2902.  doi: 10.1016/j.jfa.2010.05.012.  Google Scholar

[10]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions,, J. Diffe. Equat., 254 (2013), 725.  doi: 10.1016/j.jde.2012.09.014.  Google Scholar

[11]

W. Liu and J. M. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts,, Elect. Comm. Probab., 16 (2011), 447.  doi: 10.1214/ECP.v16-1643.  Google Scholar

[12]

W. Liu and F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations,, J. Math. Anal. Appl., 342 (2008), 651.  doi: 10.1016/j.jmaa.2007.12.047.  Google Scholar

[13]

E. Pardoux, Sur des equations aux dérivées partielles stochastiques monotones,, C. R. Acad. Sci., 275 (1972).   Google Scholar

[14]

E. Pardoux, Equations aux dérivées partielles stochastiques non lineaires monotones: Etude de solutions fortes de type Ito,, Thése Doct. Sci. Math. Univ. Paris Sud., (1975).   Google Scholar

[15]

J. Ren, M. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusion equations,, J. Diff. Equat., 238 (2007), 118.  doi: 10.1016/j.jde.2007.03.027.  Google Scholar

[16]

M. Röckner and F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups,, J. Funct. Anal., 203 (2003), 237.  doi: 10.1016/S0022-1236(03)00165-4.  Google Scholar

[17]

M. Röckner and F.-Y. Wang, Non-monotone stochastic generalized porous media equations,, J. Diff. Equat., 245 (2008), 3898.  doi: 10.1016/j.jde.2008.03.003.  Google Scholar

[18]

F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations,, Ann. Probab., 35 (2007), 1333.  doi: 10.1214/009117906000001204.  Google Scholar

[19]

F.-Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations,, Springer Briefs in Mathematics, (2013).   Google Scholar

[20]

F.-Y. Wang and L. Xu, Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012).  doi: 10.1142/S0219025712500208.  Google Scholar

[21]

F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications,, Stoch. Proc. Appl., 121 (2011), 2692.  doi: 10.1016/j.spa.2011.07.001.  Google Scholar

show all references

References:
[1]

D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains,, J. Diff. Equ., 39 (1981), 378.  doi: 10.1016/0022-0396(81)90065-6.  Google Scholar

[2]

A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator,, J. Math. Kyoto Univ., 36 (1996), 481.   Google Scholar

[3]

G. Da Prato, M. Röckner and B. L. Rozovskiĭ and F.-Y. Wang, Strong solutions of Generalized porous media equations: Existence, uniqueness and ergodicity,, Comm. Part. Diff. Equ., 31 (2006), 277.  doi: 10.1080/03605300500357998.  Google Scholar

[4]

B. Goldys and B. Maslowski, Exponential ergordicity for stochastic reaction-diffusion equations,, in Stochastic Partial Differential Equations and Applications-VII, (2006), 115.  doi: 10.1201/9781420028720.ch12.  Google Scholar

[5]

B. Goldys and B. Maslowski, Lower estimates of transition density and bounds on exponential ergodicity for stochastic PDEs,, Ann. Probab., 34 (2006), 1451.  doi: 10.1214/009117905000000800.  Google Scholar

[6]

N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations,, Current problems in mathematics, (1979), 71.   Google Scholar

[7]

W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts,, J. Evol. Equ., 9 (2009), 747.  doi: 10.1007/s00028-009-0032-8.  Google Scholar

[8]

W. Liu, Ergodicity of transition semigroups for stochastic fast diffusion equations,, Front. Math. China, 6 (2011), 449.  doi: 10.1007/s11464-011-0112-2.  Google Scholar

[9]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients,, J. Funct. Anal., 259 (2010), 2902.  doi: 10.1016/j.jfa.2010.05.012.  Google Scholar

[10]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions,, J. Diffe. Equat., 254 (2013), 725.  doi: 10.1016/j.jde.2012.09.014.  Google Scholar

[11]

W. Liu and J. M. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts,, Elect. Comm. Probab., 16 (2011), 447.  doi: 10.1214/ECP.v16-1643.  Google Scholar

[12]

W. Liu and F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations,, J. Math. Anal. Appl., 342 (2008), 651.  doi: 10.1016/j.jmaa.2007.12.047.  Google Scholar

[13]

E. Pardoux, Sur des equations aux dérivées partielles stochastiques monotones,, C. R. Acad. Sci., 275 (1972).   Google Scholar

[14]

E. Pardoux, Equations aux dérivées partielles stochastiques non lineaires monotones: Etude de solutions fortes de type Ito,, Thése Doct. Sci. Math. Univ. Paris Sud., (1975).   Google Scholar

[15]

J. Ren, M. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusion equations,, J. Diff. Equat., 238 (2007), 118.  doi: 10.1016/j.jde.2007.03.027.  Google Scholar

[16]

M. Röckner and F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups,, J. Funct. Anal., 203 (2003), 237.  doi: 10.1016/S0022-1236(03)00165-4.  Google Scholar

[17]

M. Röckner and F.-Y. Wang, Non-monotone stochastic generalized porous media equations,, J. Diff. Equat., 245 (2008), 3898.  doi: 10.1016/j.jde.2008.03.003.  Google Scholar

[18]

F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations,, Ann. Probab., 35 (2007), 1333.  doi: 10.1214/009117906000001204.  Google Scholar

[19]

F.-Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations,, Springer Briefs in Mathematics, (2013).   Google Scholar

[20]

F.-Y. Wang and L. Xu, Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012).  doi: 10.1142/S0219025712500208.  Google Scholar

[21]

F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications,, Stoch. Proc. Appl., 121 (2011), 2692.  doi: 10.1016/j.spa.2011.07.001.  Google Scholar

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