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doi: 10.3934/cpaa.2021175
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## Large deviation principle for stochastic Burgers type equation with reflection

 1 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 2 CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

* Corresponding author

Received  February 2021 Revised  September 2021 Early access October 2021

Fund Project: The first author is partially supported by the National Natural Science Foundation of China (Nos. 11871382 and 12071361) and the Fundamental Research Funds for the Central Universities 2042020kf0031. The second author is partially supported by the National Natural Science Foundation of China (Nos. 11971456, 11671372 and 11721101) and the School Start-up Fund (USTC) KY0010000036

In this paper, we establish a large deviation principle for stochastic Burgers type equation with reflection perturbed by the small multiplicative noise. The main difficulties come from the highly non-linear coefficient and the singularity caused by the reflection. Here, we adopt a new sufficient condition for the weak convergence criteria, which is proposed by Matoussi, Sabbagh and Zhang [14].

Citation: Ran Wang, Jianliang Zhai, Shiling Zhang. Large deviation principle for stochastic Burgers type equation with reflection. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021175
##### References:
 [1] A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.   Google Scholar [2] A. Budhiraja and P. Dupuis, Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods, Springer, New York, 2019. doi: 10.1007/978-1-4939-9579-0.  Google Scholar [3] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems continuous time processes, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.  Google Scholar [4] A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), 725-747.  doi: 10.1214/10-AIHP382.  Google Scholar [5] V. Cardon-Weber, Large deviations for a Burgers'-type SPDE, Stochastic Process. Appl., 84 (1999), 53-70.  doi: 10.1016/S0304-4149(99)00047-2.  Google Scholar [6] R. Dalang, C. Mueller and L. Zambotti, Hitting properties of parabolic S.P.D.E.'s with reflection, Ann. Probab., 34 (2006), 1423-1450.  doi: 10.1214/009117905000000792.  Google Scholar [7] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.  Google Scholar [8] Z. Dong, J. Wu, R. 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.  Google Scholar [9] Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar [10] T. Funaki and S. Olla, Fluctuations for $\nabla \varphi$ interface model on a wall, Stochastic Process. Appl., 94 (2001), 1-27.  doi: 10.1016/S0304-4149(00)00104-6.  Google Scholar [11] W. Liu, C. 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.  Google Scholar [12] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Berlin, Springer, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar [13] K. Magdalena, Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 160–181. doi: 10.1214/11-AIHP444.  Google Scholar [14] A. Matoussi, W. Sabbagh and T. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim., 83 (2021), 849-879.  doi: 10.1007/s00245-019-09570-5.  Google Scholar [15] D. Nualart and E. Pardoux, White noise driven by quasilinear SPDEs with reflection, Probab. Theory Related Fields, 93 (1992), 77-89.  doi: 10.1007/BF01195389.  Google Scholar [16] J. Ren and J. Wu, On uniform large deviations principle for multi-valued SDEs via the viscosity solution approach, Chin. Ann. Math. Ser. B, 40 (2019), 285-308.  doi: 10.1007/s11401-019-0133-9.  Google Scholar [17] J. Ren, J. Wu and H. Zhang, General large deviations and functional iterated logarithm law for multivalued stochastic differential equations, J. Theoret. Probab., 28 (2015), 550-586.  doi: 10.1007/s10959-013-0531-y.  Google Scholar [18] J. Ren, S. Xu and X. Zhang, Large deviation for multivalued stochastic differential equations, J. Theoret. Probab., 23 (2010), 1142-1156.  doi: 10.1007/s10959-009-0274-y.  Google Scholar [19] R. Wang, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations, J. Differ. Equ., 258 (2015), 3363-3390.  doi: 10.1016/j.jde.2015.01.008.  Google Scholar [20] J. Wu, Uniform large deviations for multivalued stochastic differential equations with Poisson jumps, Kyoto J. Math., 51 (2011), 535-559.  doi: 10.1215/21562261-1299891.  Google Scholar [21] J. Xiong and J. L. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2429-2460.  doi: 10.3150/17-BEJ947.  Google Scholar [22] T. Xu and T. Zhang, White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles, Stochastic Process. Appl., 119 (2009), 3453-3470.  doi: 10.1016/j.spa.2009.06.005.  Google Scholar [23] S. Yang and T. Zhang, Strong solutions to reflecting stochastic differential equations with singular drift, preprint, arXiv: 2002.12150. Google Scholar [24] T. Zhang, Large deviations for invariant measures of SPDEs with two reflecting walls, Stochastic Process. Appl., 122 (2012), 3425-3444.  doi: 10.1016/j.spa.2012.06.003.  Google Scholar [25] T. Zhang, Lattice approximations of reflected stochastic partial differential equations driven by space-time white noise, Ann. Appl. Probab., 26 (2016), 3602-3629.  doi: 10.1214/16-AAP1186.  Google Scholar [26] T. Zhang, Stochastic Burgers type equations with reflection: existence, uniqueness, J. Differ. Equ., 267 (2019), 4537-4571.  doi: 10.1016/j.jde.2019.05.008.  Google Scholar [27] W. Zheng, J. Zhai and T. Zhang, Moderate deviations for stochastic models of two-dimensional second-grade fluids driven by Lévy noise, Commun. Math. Stat., 6 (2018), 583–612. doi: 10.1007/s40304-018-0165-6.  Google Scholar

show all references

##### References:
 [1] A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.   Google Scholar [2] A. Budhiraja and P. Dupuis, Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods, Springer, New York, 2019. doi: 10.1007/978-1-4939-9579-0.  Google Scholar [3] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems continuous time processes, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.  Google Scholar [4] A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), 725-747.  doi: 10.1214/10-AIHP382.  Google Scholar [5] V. Cardon-Weber, Large deviations for a Burgers'-type SPDE, Stochastic Process. Appl., 84 (1999), 53-70.  doi: 10.1016/S0304-4149(99)00047-2.  Google Scholar [6] R. Dalang, C. Mueller and L. Zambotti, Hitting properties of parabolic S.P.D.E.'s with reflection, Ann. Probab., 34 (2006), 1423-1450.  doi: 10.1214/009117905000000792.  Google Scholar [7] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.  Google Scholar [8] Z. Dong, J. Wu, R. 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.  Google Scholar [9] Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar [10] T. Funaki and S. Olla, Fluctuations for $\nabla \varphi$ interface model on a wall, Stochastic Process. Appl., 94 (2001), 1-27.  doi: 10.1016/S0304-4149(00)00104-6.  Google Scholar [11] W. Liu, C. 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.  Google Scholar [12] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Berlin, Springer, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar [13] K. Magdalena, Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 160–181. doi: 10.1214/11-AIHP444.  Google Scholar [14] A. Matoussi, W. Sabbagh and T. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim., 83 (2021), 849-879.  doi: 10.1007/s00245-019-09570-5.  Google Scholar [15] D. Nualart and E. Pardoux, White noise driven by quasilinear SPDEs with reflection, Probab. Theory Related Fields, 93 (1992), 77-89.  doi: 10.1007/BF01195389.  Google Scholar [16] J. Ren and J. Wu, On uniform large deviations principle for multi-valued SDEs via the viscosity solution approach, Chin. Ann. Math. Ser. B, 40 (2019), 285-308.  doi: 10.1007/s11401-019-0133-9.  Google Scholar [17] J. Ren, J. Wu and H. Zhang, General large deviations and functional iterated logarithm law for multivalued stochastic differential equations, J. Theoret. Probab., 28 (2015), 550-586.  doi: 10.1007/s10959-013-0531-y.  Google Scholar [18] J. Ren, S. Xu and X. Zhang, Large deviation for multivalued stochastic differential equations, J. Theoret. Probab., 23 (2010), 1142-1156.  doi: 10.1007/s10959-009-0274-y.  Google Scholar [19] R. Wang, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations, J. Differ. Equ., 258 (2015), 3363-3390.  doi: 10.1016/j.jde.2015.01.008.  Google Scholar [20] J. Wu, Uniform large deviations for multivalued stochastic differential equations with Poisson jumps, Kyoto J. Math., 51 (2011), 535-559.  doi: 10.1215/21562261-1299891.  Google Scholar [21] J. Xiong and J. L. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2429-2460.  doi: 10.3150/17-BEJ947.  Google Scholar [22] T. Xu and T. Zhang, White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles, Stochastic Process. Appl., 119 (2009), 3453-3470.  doi: 10.1016/j.spa.2009.06.005.  Google Scholar [23] S. Yang and T. Zhang, Strong solutions to reflecting stochastic differential equations with singular drift, preprint, arXiv: 2002.12150. Google Scholar [24] T. Zhang, Large deviations for invariant measures of SPDEs with two reflecting walls, Stochastic Process. Appl., 122 (2012), 3425-3444.  doi: 10.1016/j.spa.2012.06.003.  Google Scholar [25] T. Zhang, Lattice approximations of reflected stochastic partial differential equations driven by space-time white noise, Ann. Appl. Probab., 26 (2016), 3602-3629.  doi: 10.1214/16-AAP1186.  Google Scholar [26] T. Zhang, Stochastic Burgers type equations with reflection: existence, uniqueness, J. Differ. Equ., 267 (2019), 4537-4571.  doi: 10.1016/j.jde.2019.05.008.  Google Scholar [27] W. Zheng, J. Zhai and T. Zhang, Moderate deviations for stochastic models of two-dimensional second-grade fluids driven by Lévy noise, Commun. Math. Stat., 6 (2018), 583–612. doi: 10.1007/s40304-018-0165-6.  Google Scholar
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