November  2015, 35(11): 5221-5237. doi: 10.3934/dcds.2015.35.5221

Large deviation principle for stochastic heat equation with memory

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, China, China

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

Received  September 2013 Revised  March 2014 Published  May 2015

In this work, using the weak convergence argument, we prove a Freidlin-Wentzell's large deviation principle for a class of stochastic heat equations with memory and Dirichlet boundary conditions, where the nonlinear term is allowed to be of polynomial growth.
Citation: Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (2003).   Google Scholar

[2]

M. Boué and P. Dupuis, A variational representation for certain functionals of Brownian motion,, Ann. of Prob., 26 (1998), 1641.  doi: 10.1214/aop/1022855876.  Google Scholar

[3]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion,, Probab. Math. Statist., 20 (2000), 39.   Google Scholar

[4]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems,, Ann. of Prob., 36 (2008), 1390.  doi: 10.1214/07-AOP362.  Google Scholar

[5]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms,, Nonlin. Diff. Eqs. Appl., 10 (2003), 399.  doi: 10.1007/s00030-003-1004-2.  Google Scholar

[6]

T. Caraballo, I. D. Chueshov and J. Real, Pullback attractors for stochastic heat equation in materials with memory,, Discrete and Continuous Dynamical Systems - Series B, 9 (2005), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[7]

T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory,, Discrete Contin. Dyn. Syst., 18 (2007), 253.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

[8]

P. Clement and G. Da Prato, White noise perturbation of the heat equations in materials with memory,, Dyn. Syst. Appl., 6 (1997), 441.   Google Scholar

[9]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations,, Wiley, (1997).  doi: 10.1002/9781118165904.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, On small random perturbations of dynamical system,, Russian Math. Surveys, 25 (1970), 1.   Google Scholar

[11]

C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333.  doi: 10.1007/s000300050049.  Google Scholar

[12]

B. Goldys, M. Röckner and X. Zhang, Martingale solutions and Markov selections for stochastic evolution equations,, Stoch. Proc. and Appl., 119 (2009), 1725.  doi: 10.1016/j.spa.2008.08.009.  Google Scholar

[13]

O. Kallenberg, Foundations of Modern Probability,, Second edition, (2002).  doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[14]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise,, Appl. Math. Optim., 61 (2010), 27.  doi: 10.1007/s00245-009-9072-2.  Google Scholar

[15]

J. Ren and X. Zhang, Schilder theorem for the Brownian motion on the diffeomorphism group of the circle,, J. Func. Anal., 224 (2005), 107.  doi: 10.1016/j.jfa.2004.08.006.  Google Scholar

[16]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations,, J. Func. Anal., 254 (2008), 3148.  doi: 10.1016/j.jfa.2008.02.010.  Google Scholar

[17]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions,, Condensed Matter Physics, 11 (2008), 247.  doi: 10.5488/CMP.11.2.247.  Google Scholar

[18]

D. W. Stroock, An Introduction to the Theory of Large Deviations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4613-8514-1.  Google Scholar

[19]

X. Zhang, A variational representation for random functionals on abstract Wiener spaces,, J. Math. Kyoto Univ., 49 (2009), 475.   Google Scholar

[20]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients,, Stochastic and Dynamics, 9 (2009), 549.  doi: 10.1142/S0219493709002774.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (2003).   Google Scholar

[2]

M. Boué and P. Dupuis, A variational representation for certain functionals of Brownian motion,, Ann. of Prob., 26 (1998), 1641.  doi: 10.1214/aop/1022855876.  Google Scholar

[3]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion,, Probab. Math. Statist., 20 (2000), 39.   Google Scholar

[4]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems,, Ann. of Prob., 36 (2008), 1390.  doi: 10.1214/07-AOP362.  Google Scholar

[5]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms,, Nonlin. Diff. Eqs. Appl., 10 (2003), 399.  doi: 10.1007/s00030-003-1004-2.  Google Scholar

[6]

T. Caraballo, I. D. Chueshov and J. Real, Pullback attractors for stochastic heat equation in materials with memory,, Discrete and Continuous Dynamical Systems - Series B, 9 (2005), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[7]

T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory,, Discrete Contin. Dyn. Syst., 18 (2007), 253.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

[8]

P. Clement and G. Da Prato, White noise perturbation of the heat equations in materials with memory,, Dyn. Syst. Appl., 6 (1997), 441.   Google Scholar

[9]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations,, Wiley, (1997).  doi: 10.1002/9781118165904.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, On small random perturbations of dynamical system,, Russian Math. Surveys, 25 (1970), 1.   Google Scholar

[11]

C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333.  doi: 10.1007/s000300050049.  Google Scholar

[12]

B. Goldys, M. Röckner and X. Zhang, Martingale solutions and Markov selections for stochastic evolution equations,, Stoch. Proc. and Appl., 119 (2009), 1725.  doi: 10.1016/j.spa.2008.08.009.  Google Scholar

[13]

O. Kallenberg, Foundations of Modern Probability,, Second edition, (2002).  doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[14]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise,, Appl. Math. Optim., 61 (2010), 27.  doi: 10.1007/s00245-009-9072-2.  Google Scholar

[15]

J. Ren and X. Zhang, Schilder theorem for the Brownian motion on the diffeomorphism group of the circle,, J. Func. Anal., 224 (2005), 107.  doi: 10.1016/j.jfa.2004.08.006.  Google Scholar

[16]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations,, J. Func. Anal., 254 (2008), 3148.  doi: 10.1016/j.jfa.2008.02.010.  Google Scholar

[17]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions,, Condensed Matter Physics, 11 (2008), 247.  doi: 10.5488/CMP.11.2.247.  Google Scholar

[18]

D. W. Stroock, An Introduction to the Theory of Large Deviations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4613-8514-1.  Google Scholar

[19]

X. Zhang, A variational representation for random functionals on abstract Wiener spaces,, J. Math. Kyoto Univ., 49 (2009), 475.   Google Scholar

[20]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients,, Stochastic and Dynamics, 9 (2009), 549.  doi: 10.1142/S0219493709002774.  Google Scholar

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