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Large deviation principle for stochastic heat equation with memory
Exponential convergence of non-linear monotone SPDEs
1. | School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
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. |
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), 378-412.
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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