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Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α-stable Processes

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Abstract

The irreducibility, moderate deviation principle and ψ-uniformly exponential ergodicity with ψ(x) := 1 + ∥x∥0 are proved for stochastic Burgers equation driven by the α-stable processes for α ∈ (1, 2), where the first two are new for the present model, and the last strengthens the exponential ergodicity under total variational norm derived in Dong et al. (J. Stat. Phys. 154:929–949, 2014).

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References

  1. Applebaum, D.: Lévy processes and stochastic calculus. Cambridge Studies in Advance Mathematics, 116, 2nd. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. Brzeźniak, Z., Goldys, B., Le Gia, Q.T.: Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere. J. Math. Fluid Mech. 20(1), 227–253 (2018)

    Article  MathSciNet  Google Scholar 

  3. Budhiraja, A., Friedlander, E.: Diffusion approximations for controlled weakly interacting large finite state systems with simultaneous jumps. Ann. Appl. Probab. 28(1), 204–249 (2018)

    Article  MathSciNet  Google Scholar 

  4. Chen, Z.Q., Wang, J.M.: Perturbation by non-local operators. Ann. Inst. Henri Poincar Probab. Stat. 54(2), 606–639 (2018)

    Article  MathSciNet  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series 229. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  6. Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer, New York (1998)

    Book  Google Scholar 

  7. Dong, Z., Xu, L., Zhang, X.: Exponential ergodicity of stochastic Burgers equations driven by α-stable processes. J. Stat. Phys. 154(4), 929–949 (2014)

    Article  MathSciNet  Google Scholar 

  8. Dong, Z., Xu, T., Zhang, T.: Invariant measures for stochastic evolution equations of pure jump type. Stochastic process. Appl. 119, 410–427 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Down, D., Meyn, S.P., Tweedie, R.L.: Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23, 1671–1691 (1995)

    Article  MathSciNet  Google Scholar 

  10. Funaki, T., Xie, B.: A stochastic heat equation with the distributions of Lévy processes as its invariant measures. Stochastic process. Appl. 119, 307–326 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Gourcy, M.: A large deviation principle for 2D stochastic Navier–Stokes equation. Stochastic process. Appl. 117(7), 904–927 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Gourcy, M.: Large deviation principle of occupation measure for stochastic Burgers equation. Ann. Inst. H. Poincaré, Statist. Probab. 43(4), 441–459 (2007)

    Article  MathSciNet  Google Scholar 

  13. Jaksic, V., Nersesyan, V., Pillet, C., Shirikyan, A.: Large deviaitons from a stationary measure for a class of dissipative PDE’s with random kicks. Comm. Pure Appl. Math. 68(12), 2108–2143 (2015)

    Article  MathSciNet  Google Scholar 

  14. Jaksic, V., Nersesyan, V., Pillet C., Shirikyan, A.: Large deviaitons and Gallavotti-Cohen principle for dissipative PDE’s with rough noise. Comm. Math. Phys. 336(1), 131–170 (2015)

    Article  MathSciNet  Google Scholar 

  15. Masuda, H.: Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps. Stochastic process. Appl. 117, 35–56 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Lévy Noise. An evolution equation approach. Encyclopedia of Mathematics and its Applications 113. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  17. Priola, E., Shirikyan, A., Xu, L., Zabczyk, J.: Exponential ergodicity and regularity for equations with Lévy noise. Stochastic process. Appl. 122, 106–133 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Priola, E., Xu, L., Zabczyk, J.: Exponential mixing for some SPDEs with Lévy noise. Stoch. Dyn. 11, 521–534 (2011)

    Article  MathSciNet  Google Scholar 

  19. Priola, E., Zabczyk, J.: Structural properties of semilinear SPDEs driven by cylindrical stable processes. Probab. Theory Relat. Fields 149, 97–137 (2011)

    Article  MathSciNet  Google Scholar 

  20. Reznikoff, M.G., Vanden-Eijnden, E.: Invariant measures of stochastic partial differential equations and conditioned diffusions. C. R. Math. Acad. Sci. Paris 340(4), 305–308 (2005)

    Article  MathSciNet  Google Scholar 

  21. Sato, K.: Lévy processes and infinite divisible distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  22. Riedle, M.: Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes. Potential Anal. 42, 809–838 (2015)

    Article  MathSciNet  Google Scholar 

  23. Temam, R.: Navier-Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, 2nd edn., vol. 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995)

  24. Wang, F.Y.: Gradient estimate for Ornstein-Uhlenbeck jump processes. Stoch. Proc. Appl. 121(3), 466–478 (2011)

    Article  MathSciNet  Google Scholar 

  25. Wang, F.Y., Wang, J.: Coupling and strong Feller for jump processes on Banach spaces. Stochastic process. Appl. 123(5), 1588–1615 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Wang, F.Y., Wang, J.: Functional inequalities for stable-like Dirichlet forms. J. Theoret. Probab. 28(2), 423–448 (2015)

    Article  MathSciNet  Google Scholar 

  27. Wang, F.Y., Xu, L., Zhang, X.: Gradient estimates for SDEs driven by multiplicative Lévy noise. J. Funct. Anal. 269(10), 3195–3219 (2015)

    Article  MathSciNet  Google Scholar 

  28. Wang, R., Xiong, J., Xu, L.: Irreducibility of stochastic real Ginzburg-Landau equation driven by α-stable noises and applications. Bernoulli 23(2), 1179–1201 (2017)

    Article  MathSciNet  Google Scholar 

  29. Wu, L.: Moderate deviations of dependent random variables related to CLT. The Annals of Probability 23(1), 420–445 (1995)

    Article  MathSciNet  Google Scholar 

  30. Wu, L.: Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic process. Appl. 91(2), 205–238 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Xu, L.: Ergodicity of stochastic real Ginzburg-Landau equation driven by α-stable noises. Stochastic process. Appl. 123(10), 3710–3736 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Zhang, T.: White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal. 33, 137–151 (2010)

    Article  MathSciNet  Google Scholar 

  33. Zhai, J., Zhang, T.: Large deviations for stochastic models of two-dimensional second grade fluids. Appl. Math. Optim. 75(3), 471–498 (2017)

    Article  MathSciNet  Google Scholar 

  34. Zhang, X.: Derivative formula and gradient estimate for SDEs driven by α-stable processes. Stoch. Proc. Appl. 123, 422(4), 1213–1228 (2013)

    Article  MathSciNet  Google Scholar 

  35. Zhang, X.: Densities for SDEs driven by degenerate α-stable processes. Ann. Probab. 42(5), 1885–1910 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

ZD is supported by NSFC 11431014, FW is supported by NNSFC (11771326, 11431014). and LX is supported by the following grants: Macao S.A.R. (FDCT 038/2017/A1, FDCT 030/2016/A1, FDCT 025/2016/A1), NNSFC 11571390, University of Macau MYRG (2016-00025-FST, 2018-00133-FST).

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Authors and Affiliations

  1. Key Laboratory of Random Complex Structures and Data Science, RCSDS? Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Zhao Dong

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Feng-Yu Wang

  3. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

    Feng-Yu Wang

  4. Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau, China

    Lihu Xu

  5. UM Zhuhai Research Institute, Zhuhai, China

    Lihu Xu

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  1. Zhao Dong
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  2. Feng-Yu Wang
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  3. Lihu Xu
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Correspondence to Lihu Xu.

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Dong, Z., Wang, FY. & Xu, L. Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α-stable Processes. Potential Anal 52, 371–392 (2020). https://doi.org/10.1007/s11118-018-9736-0

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  • DOI: https://doi.org/10.1007/s11118-018-9736-0

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