Advertisement

Large Deviations for Quasilinear Parabolic Stochastic Partial Differential Equations

  • Zhao Dong
  • Rangrang ZhangEmail author
  • Tusheng Zhang
Article
  • 8 Downloads

Abstract

In this paper, we establish the Freidlin-Wentzell’s large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The proof is based on the weak convergence approach.

Keywords

Freidlin-Wentzell’s large deviations Quailinear stochastic partial differential equations Weak convergence approach 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to the anonymous referees for comments and suggestions. This work is partly supported by National Natural Science Foundation of China (No.11371041, 11671372, 11431014, 11401557, 11801032). Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (No. 2008DP173182). China Postdoctoral Science Foundation funded project (No. 2018M641204).

References

  1. 1.
    Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20, 39–61 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36(4), 1390–1420 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. 44(3), 1916–1955 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Jones and Bartlett, Boston (1993)zbMATHGoogle Scholar
  6. 6.
    Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields 102(3), 367–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gagneux, G., Madaune-Tort, M.: Analyse Mathématique de Modéles Non Linéaires de L’ingénierie Pétroliére. Springer, Berlin (1996)zbMATHGoogle Scholar
  8. 8.
    Hofmanová, M., Zhang, T.S.: Quasilinear parabolic stochastic differential equations: existence, uniqueness. Stoch. Process. Appl. 127(10), 3354–3371 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, W., Röckner, M.: SPDEs in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259, 2902–2922 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.RCSDS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  4. 4.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations