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doi: 10.3934/cpaa.2022091
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Freidlin-Wentzell's large deviation principle for stochastic integral evolution equations

1. 

School of Statistics and Data Science, Nankai University, Tianjin 300071, China

2. 

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

3. 

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

* Corresponding author

Received  October 2021 Revised  March 2022 Early access May 2022

Fund Project: This work is supported by NSFC (No. 12171208 11831014, 12090011) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20-2324)

The main aim of this work is to investigate the large deviation principle for a class of stochastic integral evolution equations. As applications, our results can be applied to a large class of stochastic models with hereditary or memory effects such as stochastic integral porous medium equations, stochastic integral $ p $-Laplace equations and stochastic integral 2D Navier-Stokes equations.

Citation: Xiaomin Huang, Yanpei Jiang, Wei Liu. Freidlin-Wentzell's large deviation principle for stochastic integral evolution equations. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022091
References:
[1]

V. BarbuS. Bonaccorsi and L. Tubaro, Existence and asymptotic behavior for hereditary stochastic evolution equations, Appl. Math. Optim., 69 (2014), 273-314.  doi: 10.1007/s00245-013-9224-2.

[2]

W. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[3]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.  doi: 10.1080/13550280152537111.

[4]

Y. Chen and H. Gao, Well-posedness and large deviations for a class of SPDEs with Lévy noise, J. Differ. Equ., 263 (2017), 5216-5252.  doi: 10.1016/j.jde.2017.06.016.

[5]

P. L. Chow, Large deviation problem for some parabolic itô equations, Commun. Pure Appl. Math., 45 (1992), 97-120.  doi: 10.1002/cpa.3160450105.

[6]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[7]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03311-7.

[8]

Z. DongJ. L. WuR. Zhang and T. Zhang, Large deviation principles for first-order scalar conservation laws with stochastic forcing, Ann. Appl. Probab., 30 (2020), 324-367.  doi: 10.1214/19-AAP1503.

[9]

P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations, Wiley Series in Probability and Statistics: Probability and Statistics, New York, 1997. doi: 10.1002/9781118165904.

[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Translated from the Russian by Joseph Szücs. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.

[11]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differ. Equ., 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[12]

B. GessW. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differ. Equ., 269 (2020), 3414-3455.  doi: 10.1016/j.jde.2020.03.002.

[13]

W. HongS. Li and W. Liu, Freidlin-Wentzell type large deviation principle for multi-scale locally monotone SPDEs, SIAM J. Math. Anal., 53 (2021), 6517-6561.  doi: 10.1137/21M1404612.

[14]

X. Huang, W. Hong and W. Liu, Stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients, Front. Math. China., 17 (2022), In press.

[15]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Current Prob. Math., 14 (1979), 71-147. 

[16]

Y. LiY. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst., 35 (2015), 5221-5237.  doi: 10.3934/dcds.2015.35.5221.

[17]

D. Lipshutz, Exit time asymptotics for small noise stochastic delay differential equations, Discrete Contin. Dyn. Syst., 38 (2018), 3099-3138.  doi: 10.3934/dcds.2018135.

[18]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56.  doi: 10.1007/s00245-009-9072-2.

[19]

W. Liu, Well-posedness of stochastic partial differential equations with Lyapunov condition, J. Differ. Equ., 255 (2013), 572-592.  doi: 10.1016/j.jde.2013.04.021.

[20]

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.

[21]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[22]

W. LiuM. Röckner and J. L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM J. Math. Anal., 50 (2018), 2588-2607.  doi: 10.1137/17M1144593.

[23]

W. Liu, M. Röckner and J. L. da Silva, Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations, J. Funct. Anal., 281 (2021), No.109135, 34pp. doi: 10.1016/j.jfa.2021.109135.

[24]

W. LiuC. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., 63 (2020), 1181-1202.  doi: 10.1007/s11425-018-9440-3.

[25]

S. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 881-893.  doi: 10.3934/dcdsb.2006.6.881.

[26]

E. Pardoux, Équations Aux Dérivées partielles stochastiques de type monotone, (French) Collège de France, Paris, 1975.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, Monogr. Math, 87., Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations, J. Funct. Anal., 254 (2008), 3148-3172.  doi: 10.1016/j.jfa.2008.02.010.

[29]

M. Riedle and J. Zhai, Large deviations for stochastic heat equations with memory driven by Lévy-type noise, Discrete Contin. Dyn. Syst., 38 (2018), 1983-2005.  doi: 10.3934/dcds.2018080.

[30]

M. RöcknerB. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condensed Matter Physics., 11 (2018), 247-259. 

[31]

M. RöcknerF. Y. Wang and L. Wu, Large deviations for stochastic generalized porous media equations, Stochastic Process. Appl., 116 (2006), 1677-1689.  doi: 10.1016/j.spa.2006.05.007.

[32]

M. Röckner and T. Zhang, Stochastic 3D tamed Navier-Stokes equations: existence, uniqueness and small time large deviation principles, J. Differ. Equ., 252 (2012), 716-744.  doi: 10.1016/j.jde.2011.09.030.

[33]

M. RöcknerT. Zhang and X. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-285.  doi: 10.1007/s00245-009-9089-6.

[34]

B. L. Rozovskii and S. V. Lototsky, Stochastic evolution systems. Linear theory and applications to non-linear filtering, Second edition, Probability Theory and Stochastic Modelling 89, Springer, Cham, 2018. doi: 10.1007/978-3-319-94893-5.

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic. Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.

[36]

D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[37]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303.

[38]

F. Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204.

[39]

J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2842-2874.  doi: 10.3150/17-BEJ947.

[40]

J. Zhai and T. Zhang, Large deviations for stochastic models of two-dimensional second grade fluids, Appl. Math. Optim., 75 (2017), 471-498.  doi: 10.1007/s00245-016-9338-4.

[41]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stoch. Dyn., 9 (2009), 549-595.  doi: 10.1142/S0219493709002774.

show all references

References:
[1]

V. BarbuS. Bonaccorsi and L. Tubaro, Existence and asymptotic behavior for hereditary stochastic evolution equations, Appl. Math. Optim., 69 (2014), 273-314.  doi: 10.1007/s00245-013-9224-2.

[2]

W. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[3]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.  doi: 10.1080/13550280152537111.

[4]

Y. Chen and H. Gao, Well-posedness and large deviations for a class of SPDEs with Lévy noise, J. Differ. Equ., 263 (2017), 5216-5252.  doi: 10.1016/j.jde.2017.06.016.

[5]

P. L. Chow, Large deviation problem for some parabolic itô equations, Commun. Pure Appl. Math., 45 (1992), 97-120.  doi: 10.1002/cpa.3160450105.

[6]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[7]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03311-7.

[8]

Z. DongJ. L. WuR. Zhang and T. Zhang, Large deviation principles for first-order scalar conservation laws with stochastic forcing, Ann. Appl. Probab., 30 (2020), 324-367.  doi: 10.1214/19-AAP1503.

[9]

P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations, Wiley Series in Probability and Statistics: Probability and Statistics, New York, 1997. doi: 10.1002/9781118165904.

[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Translated from the Russian by Joseph Szücs. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.

[11]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differ. Equ., 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[12]

B. GessW. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differ. Equ., 269 (2020), 3414-3455.  doi: 10.1016/j.jde.2020.03.002.

[13]

W. HongS. Li and W. Liu, Freidlin-Wentzell type large deviation principle for multi-scale locally monotone SPDEs, SIAM J. Math. Anal., 53 (2021), 6517-6561.  doi: 10.1137/21M1404612.

[14]

X. Huang, W. Hong and W. Liu, Stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients, Front. Math. China., 17 (2022), In press.

[15]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Current Prob. Math., 14 (1979), 71-147. 

[16]

Y. LiY. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst., 35 (2015), 5221-5237.  doi: 10.3934/dcds.2015.35.5221.

[17]

D. Lipshutz, Exit time asymptotics for small noise stochastic delay differential equations, Discrete Contin. Dyn. Syst., 38 (2018), 3099-3138.  doi: 10.3934/dcds.2018135.

[18]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56.  doi: 10.1007/s00245-009-9072-2.

[19]

W. Liu, Well-posedness of stochastic partial differential equations with Lyapunov condition, J. Differ. Equ., 255 (2013), 572-592.  doi: 10.1016/j.jde.2013.04.021.

[20]

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.

[21]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[22]

W. LiuM. Röckner and J. L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM J. Math. Anal., 50 (2018), 2588-2607.  doi: 10.1137/17M1144593.

[23]

W. Liu, M. Röckner and J. L. da Silva, Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations, J. Funct. Anal., 281 (2021), No.109135, 34pp. doi: 10.1016/j.jfa.2021.109135.

[24]

W. LiuC. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., 63 (2020), 1181-1202.  doi: 10.1007/s11425-018-9440-3.

[25]

S. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 881-893.  doi: 10.3934/dcdsb.2006.6.881.

[26]

E. Pardoux, Équations Aux Dérivées partielles stochastiques de type monotone, (French) Collège de France, Paris, 1975.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, Monogr. Math, 87., Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations, J. Funct. Anal., 254 (2008), 3148-3172.  doi: 10.1016/j.jfa.2008.02.010.

[29]

M. Riedle and J. Zhai, Large deviations for stochastic heat equations with memory driven by Lévy-type noise, Discrete Contin. Dyn. Syst., 38 (2018), 1983-2005.  doi: 10.3934/dcds.2018080.

[30]

M. RöcknerB. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condensed Matter Physics., 11 (2018), 247-259. 

[31]

M. RöcknerF. Y. Wang and L. Wu, Large deviations for stochastic generalized porous media equations, Stochastic Process. Appl., 116 (2006), 1677-1689.  doi: 10.1016/j.spa.2006.05.007.

[32]

M. Röckner and T. Zhang, Stochastic 3D tamed Navier-Stokes equations: existence, uniqueness and small time large deviation principles, J. Differ. Equ., 252 (2012), 716-744.  doi: 10.1016/j.jde.2011.09.030.

[33]

M. RöcknerT. Zhang and X. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-285.  doi: 10.1007/s00245-009-9089-6.

[34]

B. L. Rozovskii and S. V. Lototsky, Stochastic evolution systems. Linear theory and applications to non-linear filtering, Second edition, Probability Theory and Stochastic Modelling 89, Springer, Cham, 2018. doi: 10.1007/978-3-319-94893-5.

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic. Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.

[36]

D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[37]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303.

[38]

F. Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204.

[39]

J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2842-2874.  doi: 10.3150/17-BEJ947.

[40]

J. Zhai and T. Zhang, Large deviations for stochastic models of two-dimensional second grade fluids, Appl. Math. Optim., 75 (2017), 471-498.  doi: 10.1007/s00245-016-9338-4.

[41]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stoch. Dyn., 9 (2009), 549-595.  doi: 10.1142/S0219493709002774.

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