Illinois Journal of Mathematics

Exponential mixing for SPDEs driven by highly degenerate Lévy noises

Xiaobin Sun, Yingchao Xie, and Lihu Xu

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Abstract

By modifying a coupling method developed by the third author with much more delicate analysis, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include a finite dimensional α-stable process with α(0,2).

Article information

Source
Illinois J. Math., Volume 63, Number 1 (2019), 75-102.

Dates
Received: 26 April 2018
Revised: 1 February 2019
First available in Project Euclid: 29 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1559116823

Digital Object Identifier
doi:10.1215/00192082-7600360

Mathematical Reviews number (MathSciNet)
MR3959868

Zentralblatt MATH identifier
07064387

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J75: Jump processes 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Citation

Sun, Xiaobin; Xie, Yingchao; Xu, Lihu. Exponential mixing for SPDEs driven by highly degenerate Lévy noises. Illinois J. Math. 63 (2019), no. 1, 75--102. doi:10.1215/00192082-7600360. https://projecteuclid.org/euclid.ijm/1559116823


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References

  • [1] S. Albeverio, J. L. Wu, and T. S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl. 74 (1998), no. 1, 21–36.
  • [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Math. 121, Cambridge University Press, Cambridge, 1996.
  • [3] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, 1996.
  • [4] Z. Dong and Y. Xie, Ergodicity of stochastic 2D Naiver-Stokes equation with Lévy noise, J. Differential Equations 251 (2011), no. 1, 196–222.
  • [5] D. Zhao, L. Xu, and X. Zhang, Exponential ergodicity of stochastic Burgers equations driven by $\alpha $-stable processes, J. Stat. Phys. 154 (2014), no. 4, 929–949.
  • [6] T. Funaki and B. Xie, A stochastic heat equation with the distributions of Lévy processes as its invariant measures, Stochastic Process. Appl. 119 (2009), no. 2, 307–326.
  • [7] M. Hairer, Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields 124 (2002), no. 3, 345–380.
  • [8] M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (2006), no. 3, 993–1032.
  • [9] M. Hairer and J. C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs, Electron. J. Probab. 16 (2011), no. 23, 658–738.
  • [10] A. M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stochastic Process. Appl. 119 (2009), no. 2, 602–632.
  • [11] S. Kuksin and A. Shirikyan, A coupling approach to randomly forced nonlinear PDEs, I, Comm. Math. Phys. 221 (2001), no. 2, 351–366.
  • [12] C. Marinelli and M. Röckner, Well-posedness and ergodicity for stochastic reaction-diffusion equations with multiplicative Poisson noise, Electron. J. Probab. 15 (2010) 1528–1555.
  • [13] H. Masuda, Ergodicity and exponential $\beta $-mixing bounds for multidimensional diffusions with jumps, Stochastic Process. Appl. 117 (2007), no. 1, 35–56.
  • [14] C. Odasso, Exponential mixing for the 3D stochastic Navier-Stokes equations, Comm. Math. Phys. 270 (2007), no. 1, 109–139.
  • [15] C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case, Probab. Theory Related Fields 140 (2008), no. 1–2, 41–82.
  • [16] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach, Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, 2007.
  • [17] E. Priola, A. Shirikyan, L. Xu, and J. Zabczyk, Exponential ergodicity and regularity for equations with Lévy noise, Stochastic Process. Appl. 122 (2012), no. 1, 106–133.
  • [18] E. Priola, L. Xu, and J. Zabczyk, Exponential mixing for some SPDEs with Lévy noise, Stoch. Dyn. 11 (2011), 521–534.
  • [19] E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields 149 (2011), no. 1–2, 97–137.
  • [20] L. Xu, Exponential mixing for SDEs forced by degenerate Lévy noises, J. Evol. Equ. 14 (2014), no. 2, 249–272.
  • [21] L. Xu and B. Zegarliński, Ergodicity of the finite and infinite dimensional $\alpha $-stable systems, Stoch. Anal. Appl. 27 (2009), no. 4, 797–824.