# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 378-412. doi: 10.1016/0022-0396(81)90065-6. [2] A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator, J. Math. Kyoto Univ., 36 (1996), 481-498. [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), 277-291. doi: 10.1080/03605300500357998. [4] B. Goldys and B. Maslowski, Exponential ergordicity for stochastic reaction-diffusion equations, in Stochastic Partial Differential Equations and Applications-VII, Lecture Notes Pure Appl. Math., 245, Chapman Hall/CRC Press, 2006, 115-131. doi: 10.1201/9781420028720.ch12. [5] B. Goldys and B. Maslowski, Lower estimates of transition density and bounds on exponential ergodicity for stochastic PDEs, Ann. Probab., 34 (2006), 1451-1496. doi: 10.1214/009117905000000800. [6] N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations, Current problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71-147, 256. [7] W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ., 9 (2009), 747-770. doi: 10.1007/s00028-009-0032-8. [8] W. Liu, Ergodicity of transition semigroups for stochastic fast diffusion equations, Front. Math. China, 6 (2011), 449-472. doi: 10.1007/s11464-011-0112-2. [9] W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012. [10] W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Diffe. Equat., 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014. [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), 447-457. doi: 10.1214/ECP.v16-1643. [12] W. Liu and F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl., 342 (2008), 651-662. doi: 10.1016/j.jmaa.2007.12.047. [13] E. Pardoux, Sur des equations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci., 275 (1972), A101-A103. [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. [15] J. Ren, M. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusion equations, J. Diff. Equat., 238 (2007), 118-152. doi: 10.1016/j.jde.2007.03.027. [16] M. Röckner and F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal., 203 (2003), 237-261. doi: 10.1016/S0022-1236(03)00165-4. [17] M. Röckner and F.-Y. Wang, Non-monotone stochastic generalized porous media equations, J. Diff. Equat., 245 (2008), 3898-3935. doi: 10.1016/j.jde.2008.03.003. [18] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350. doi: 10.1214/009117906000001204. [19] F.-Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer Briefs in Mathematics, Springer, New York, 2013. [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), 1250020, 19pp. doi: 10.1142/S0219025712500208. [21] F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 121 (2011), 2692-2710. doi: 10.1016/j.spa.2011.07.001.

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##### 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), 378-412. doi: 10.1016/0022-0396(81)90065-6. [2] A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator, J. Math. Kyoto Univ., 36 (1996), 481-498. [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), 277-291. doi: 10.1080/03605300500357998. [4] B. Goldys and B. Maslowski, Exponential ergordicity for stochastic reaction-diffusion equations, in Stochastic Partial Differential Equations and Applications-VII, Lecture Notes Pure Appl. Math., 245, Chapman Hall/CRC Press, 2006, 115-131. doi: 10.1201/9781420028720.ch12. [5] B. Goldys and B. Maslowski, Lower estimates of transition density and bounds on exponential ergodicity for stochastic PDEs, Ann. Probab., 34 (2006), 1451-1496. doi: 10.1214/009117905000000800. [6] N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations, Current problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71-147, 256. [7] W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ., 9 (2009), 747-770. doi: 10.1007/s00028-009-0032-8. [8] W. Liu, Ergodicity of transition semigroups for stochastic fast diffusion equations, Front. Math. China, 6 (2011), 449-472. doi: 10.1007/s11464-011-0112-2. [9] W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012. [10] W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Diffe. Equat., 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014. [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), 447-457. doi: 10.1214/ECP.v16-1643. [12] W. Liu and F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl., 342 (2008), 651-662. doi: 10.1016/j.jmaa.2007.12.047. [13] E. Pardoux, Sur des equations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci., 275 (1972), A101-A103. [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. [15] J. Ren, M. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusion equations, J. Diff. Equat., 238 (2007), 118-152. doi: 10.1016/j.jde.2007.03.027. [16] M. Röckner and F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal., 203 (2003), 237-261. doi: 10.1016/S0022-1236(03)00165-4. [17] M. Röckner and F.-Y. Wang, Non-monotone stochastic generalized porous media equations, J. Diff. Equat., 245 (2008), 3898-3935. doi: 10.1016/j.jde.2008.03.003. [18] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350. doi: 10.1214/009117906000001204. [19] F.-Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer Briefs in Mathematics, Springer, New York, 2013. [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), 1250020, 19pp. doi: 10.1142/S0219025712500208. [21] F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 121 (2011), 2692-2710. doi: 10.1016/j.spa.2011.07.001.
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