Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2842-2874.

Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise

Jie Xiong and Jianliang Zhai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by Lévy noise. The weak convergence method plays an important role.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2842-2874.

Dates
Received: August 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051227

Digital Object Identifier
doi:10.3150/17-BEJ947

Mathematical Reviews number (MathSciNet)
MR3779704

Zentralblatt MATH identifier
06853267

Keywords
Freidlin–Wentzell type large deviation principle Levy processes locally monotone coefficients stochastic partial differential equations

Citation

Xiong, Jie; Zhai, Jianliang. Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise. Bernoulli 24 (2018), no. 4A, 2842--2874. doi:10.3150/17-BEJ947. https://projecteuclid.org/euclid.bj/1522051227


Export citation

References

  • [1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge Univ. Press.
  • [2] Bao, J. and Yuan, C. (2015). Large deviations for neutral functional SDEs with jumps. Stochastics 87 48–70.
  • [3] Bessaih, H. and Millet, A. (2009). Large deviation principle and inviscid shell models. Electron. J. Probab. 14 2551–2579.
  • [4] Brzeźniak, Z., Liu, W. and Zhu, J. (2014). Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise. Nonlinear Anal. Real World Appl. 17 283–310.
  • [5] Budhiraja, A., Chen, J. and Dupuis, P. (2013). Large deviations for stochastic partial differential equations driven by a Poisson random measure. Stochastic Process. Appl. 123 523–560.
  • [6] Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 20 39–61.
  • [7] Budhiraja, A., Dupuis, P. and Maroulas, V. (2008). Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36 1390–1420.
  • [8] Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representations for continuous time processes. Ann. Inst. Henri Poincaré Probab. Stat. 47 725–747.
  • [9] Cardon-Weber, C. (1999). Large deviations for a Burgers’-type SPDE. Stochastic Process. Appl. 84 53–70.
  • [10] Cerrai, S. and Röckner, M. (2004). Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32 1100–1139.
  • [11] Chenal, F. and Millet, A. (1997). Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72 161–186.
  • [12] Chow, P.L. (1992). Large deviation problem for some parabolic Itô equations. Comm. Pure Appl. Math. 45 97–120.
  • [13] de Acosta, A. (1994). Large deviations for vector-valued Lévy processes. Stochastic Process. Appl. 51 75–115.
  • [14] de Acosta, A. (2000). A general non-convex large deviation result with applications to stochastic equations. Probab. Theory Related Fields 118 483–521.
  • [15] Duan, J. and Millet, A. (2009). Large deviations for the Boussinesq equations under random influences. Stochastic Process. Appl. 119 2052–2081.
  • [16] Dupuis, P. and Ellis, R.S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley. A Wiley-Interscience Publication.
  • [17] Feng, J. and Kurtz, T.G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. Providence, RI: Amer. Math. Soc.
  • [18] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. Amsterdam–New York: North-Holland; Tokyo: Kodansha, Ltd.
  • [19] Kallianpur, G. and Xiong, J. (1996). Large deviations for a class of stochastic partial differential equations. Ann. Probab. 24 320–345.
  • [20] Liu, W. (2010). Large deviations for stochastic evolution equations with small multiplicative noise. Appl. Math. Optim. 61 27–56.
  • [21] Liu, W. and Röckner, M. (2010). SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259 2902–2922.
  • [22] Manna, U., Sritharan, S.S. and Sundar, P. (2009). Large deviations for the stochastic shell model of turbulence. NoDEA Nonlinear Differential Equations Appl. 16 493–521.
  • [23] Ren, J. and Zhang, X. (2008). Freidlin–Wentzell’s large deviations for stochastic evolution equations. J. Funct. Anal. 254 3148–3172.
  • [24] Röckner, M. and Zhang, T. (2007). Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 26 255–279.
  • [25] Röckner, M., Zhang, T. and Zhang, X. (2010). Large deviations for stochastic tamed 3D Navier–Stokes equations. Appl. Math. Optim. 61 267–285.
  • [26] Sowers, R.B. (1992). Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann. Probab. 20 504–537.
  • [27] Świȩch, A. and Zabczyk, J. (2011). Large deviations for stochastic PDE with Lévy noise. J. Funct. Anal. 260 674–723.
  • [28] Xu, T. and Zhang, T. (2009). Large deviation principles for 2-D stochastic Navier–Stokes equations driven by Lévy processes. J. Funct. Anal. 257 1519–1545.
  • [29] Yang, X., Zhai, J. and Zhang, T. (2015). Large deviations for SPDEs of jump type. Stoch. Dyn. 15 1550026, 30 pp.
  • [30] Zhai, J. and Zhang, T. (2015). Large deviations for 2-D stochastic Navier–Stokes equations driven by multiplicative Lévy noises. Bernoulli 21 2351–2392.
  • [31] Zhang, T.S. (2000). On the small time asymptotics of diffusion processes on Hilbert spaces. Ann. Probab. 28 537–557.
  • [32] Zhang, X. (2010). Stochastic Volterra equations in Banach spaces and stochastic partial differential equation. J. Funct. Anal. 258 1361–1425.