Reflected Stochastic Burgers Equation with Jumps

doi:10.1007/s42967-023-00305-6

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1282-1307.doi: 10.1007/s42967-023-00305-6

• ORIGINAL PAPERS • 上一篇下一篇

Reflected Stochastic Burgers Equation with Jumps

Hongchao Qian¹, Jun Peng¹, Ruizhi Li², Yewei Gui²

1. School of Mathematics and Statistics, Central South University, Changsha, 410075, Hunan, China;
2. State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, 621000, Sichuan, China

收稿日期:2023-03-14 修回日期:2023-08-04 接受日期:2023-08-09 出版日期:2023-10-18 发布日期:2023-10-18
通讯作者: Jun Peng,E-mail:pengjun2015@csu.edu.cn E-mail:pengjun2015@csu.edu.cn
作者简介:Hongchao Qian, E-mail:hcqian@csu.edu.cn;Ruizhi Li, E-mail:liruizhi2008@163.com;Yewei Gui, E-mail:guiyewei777@126.com

Reflected Stochastic Burgers Equation with Jumps

Hongchao Qian¹, Jun Peng¹, Ruizhi Li², Yewei Gui²

1. School of Mathematics and Statistics, Central South University, Changsha, 410075, Hunan, China;
2. State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, 621000, Sichuan, China

Received:2023-03-14 Revised:2023-08-04 Accepted:2023-08-09 Online:2023-10-18 Published:2023-10-18

摘要/Abstract

摘要： This paper is concerned with the reflected stochastic Burgers equation driven both by the Brownian motion and by the Poisson random measure. The existence and uniqueness of solutions are established. The penalization method plays an important role.

关键词: Stochastic Burgers equation, Reflection, Penalization, Jumps

Abstract: This paper is concerned with the reflected stochastic Burgers equation driven both by the Brownian motion and by the Poisson random measure. The existence and uniqueness of solutions are established. The penalization method plays an important role.

Key words: Stochastic Burgers equation, Reflection, Penalization, Jumps

中图分类号:

Hongchao Qian, Jun Peng, Ruizhi Li, Yewei Gui. Reflected Stochastic Burgers Equation with Jumps[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1282-1307.

参考文献

[1] Albeverio, S., Wu, J.L., Zhang, T.S.: Parabolic SPDEs driven by Poisson white noise. Stoch. Process. Appl. 74, 21-36 (1998)
[2] Assaad, O., Tudor, C.: Pathwise analysis and parameter estimation for the stochastic Burgers equation. Bulletin des Sciences Mathématiques 170, 102995 (2021)
[3] Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation. Commun. Math. Phys. 165(2), 211-232 (1994)
[4] Burgers, J.: The Nonlinear Diffusion Equation. Springer, Dordrecht (1974)
[5] Da Prato, G., Debussche, A., Teman, R.: Stochastic Burgers equation. Nonlinear Differ. Equ. Appl. 1, 389-402 (1994)
[6] Dalang, R., Mueller, C., Zambotti, L.: Hitting properties of parabolic SPDE’s with reflection. Ann. Probab. 34(4), 1423-1450 (2006)
[7] Debbi, L., Dozzi, M.: On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stoch. Process. Appl. 115, 1764-1781 (2005)
[8] Donati-Martin, C., Pardoux, E.: White noise driven SPDEs with reflection. Probab. Theory Relat. Fields 95(1), 1-24 (1993)
[9] Dong, Z.: On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes. J. Theoret. Probab. 21, 322-335 (2008)
[10] Dong, Z., Xu, T.: One-dimensional stochastic Burgers equation driven by Lévy processes. J. Funct. Anal. 234, 631-678 (2007)
[11] Duan, J., Peng, J.: White noise driven SPDEs with oblique reflection: existence and uniqueness. J. Math. Anal. Appl. 480(1), 123356 (2019)
[12] Essaky, E., Ouknine, Y.: Homogenization of multivalued partial differential equations via reflected backward stochastic differential equations. Stoch. Anal. Appl. 22(1), 307-336 (2004)
[13] Fournier, N.: Malliavin calculus for parabolic SPDEs with jumps. Stoch. Process. Appl. 87(1), 115-147 (2000)
[14] Fu, G., Horst, U., Qiu, J.: Maximum principle for quasi-linear reflected backward SPDEs. J. Math. Anal. Appl. 456, 307-336 (2017)
[15] Gyöngy, I.: Existence and uniqueness results for similinear stochastic partial differential equations. Stoch. Process. Appl. 73, 271-299 (1988)
[16] Gyöngy, I., Nualart, D.: On the stochastic Burgers equation in the real line. Ann. Probab. 27, 782-802 (1999)
[17] Hopf, E.: The partial differential equation. Comm. Pure Appl. Math. 3, 201-230 (1950)
[18] Menaldi, J.: Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32(5), 733-744 (1983)
[19] Menaldi, J., Robin, M.: Reflected diffusion processes with jumps. Ann. Probab. 13(2), 319-341 (1985)
[20] Mytnik, L.: Stochastic partial differential equations driven by stable noise. Probab. Theory Related Fields 123(2), 157-201 (2002)
[21] Nualart, D., Pardoux, E.: White noise driven quasilinear SPDEs with reflection. Probab. Theory Relat. Fields 93(1), 77-89 (1992)
[22] Truman, A., Wu, J.: Stochastic Burgers equation with Lévy space-time white noise. In: Davies, I.M., Truman, A., Hassan, O., Morgan, K., Weatherill, N.P. (eds) Probabilistic Methods in Fluids, Proceedings of the Swansea 2002 Workshop, pp. 298-323. World Sci. Publishing, River Edge, NJ (2003)
[23] Wu, J., Xie, B.: On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in R^d. Bulletin des Sciences Mathématiques 136(5), 484-506 (2012)
[24] Zambotti, L.: A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal. 180(1), 195-209 (2001)
[25] Zhang, T.: Systems of stochastic partial differential equations with reflection: existence and uniqueness. Stoch. Process. Appl. 33(2), 137-151 (2010)
[26] Zhang, T.: White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal. 121(6), 1356-1372 (2011)
[27] Zhang, T.: Stochastic Burgers type equations with reflection: existence, uniqueness. J. Diff. Equat. 267(8), 4537-4571 (2019)

Reflected Stochastic Burgers Equation with Jumps

Reflected Stochastic Burgers Equation with Jumps

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

编辑推荐

Metrics

本文评价