• Previous Article
    Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence
  • CPAA Home
  • This Issue
  • Next Article
    Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term
doi: 10.3934/cpaa.2021175
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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

[1]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & 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 & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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 & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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 & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[17]

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

[18]

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

[19]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[20]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (88)
  • HTML views (67)
  • Cited by (0)

Other articles
by authors

[Back to Top]