January  2022, 21(1): 213-238. doi: 10.3934/cpaa.2021175

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 Published  January 2022 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 and Applied Analysis, 2022, 21 (1) : 213-238. 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. 

[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.

[3]

A. BudhirajaP. 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.

[4]

A. BudhirajaP. 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.

[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.

[6]

R. DalangC. 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.

[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.

[8]

Z. DongJ. 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]

Z. DongJ. XiongJ. 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.

[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.

[11]

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.

[12]

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

[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.

[14]

A. MatoussiW. 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.

[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.

[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.

[17]

J. RenJ. 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.

[18]

J. RenS. 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.

[19]

R. WangJ. 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.

[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.

[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.

[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.

[23]

S. Yang and T. Zhang, Strong solutions to reflecting stochastic differential equations with singular drift, preprint, arXiv: 2002.12150.

[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.

[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.

[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.

[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.

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. 

[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.

[3]

A. BudhirajaP. 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.

[4]

A. BudhirajaP. 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.

[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.

[6]

R. DalangC. 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.

[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.

[8]

Z. DongJ. 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]

Z. DongJ. XiongJ. 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.

[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.

[11]

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.

[12]

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

[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.

[14]

A. MatoussiW. 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.

[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.

[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.

[17]

J. RenJ. 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.

[18]

J. RenS. 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.

[19]

R. WangJ. 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.

[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.

[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.

[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.

[23]

S. Yang and T. Zhang, Strong solutions to reflecting stochastic differential equations with singular drift, preprint, arXiv: 2002.12150.

[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.

[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.

[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.

[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.

[1]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[2]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[3]

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

[4]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[5]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048

[6]

Naoki Fujino, Mitsuru Yamazaki. Burgers' type equation with vanishing higher order. Communications on Pure and Applied Analysis, 2007, 6 (2) : 505-520. doi: 10.3934/cpaa.2007.6.505

[7]

Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835

[8]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic and Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034

[9]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

[10]

Rémi Lassalle, Jean Claude Zambrini. A weak approach to the stochastic deformation of classical mechanics. Journal of Geometric Mechanics, 2016, 8 (2) : 221-233. doi: 10.3934/jgm.2016005

[11]

Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091

[12]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[13]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193

[14]

A. Guillin, R. Liptser. Examples of moderate deviation principle for diffusion processes. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 803-828. doi: 10.3934/dcdsb.2006.6.803

[15]

Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1939-1964. doi: 10.3934/dcds.2012.32.1939

[16]

Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247

[17]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[18]

Diogo Poças, Bartosz Protas. Transient growth in stochastic Burgers flows. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2371-2391. doi: 10.3934/dcdsb.2018052

[19]

Zhaosheng Feng, Qingguo Meng. Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 285-291. doi: 10.3934/dcdsb.2007.7.285

[20]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1029-1054. doi: 10.3934/dcdsb.2021079

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (183)
  • HTML views (143)
  • Cited by (0)

Other articles
by authors

[Back to Top]