The Annals of Applied Probability

Moderate deviation for random elliptic PDE with small noise

Xiaoou Li, Jingchen Liu, Jianfeng Lu, and Xiang Zhou

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


Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such elliptic PDEs with random inputs. In particular, we consider the asymptotic regime that the noise level converges to zero suggesting that the system uncertainty is low, but does exist. We develop sharp approximations of the probability of a large class of rare events.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2781-2813.

Received: October 2016
Revised: November 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60Z05
Secondary: 60G15: Gaussian processes

Random partial differential equation rare event moderate deviation


Li, Xiaoou; Liu, Jingchen; Lu, Jianfeng; Zhou, Xiang. Moderate deviation for random elliptic PDE with small noise. Ann. Appl. Probab. 28 (2018), no. 5, 2781--2813. doi:10.1214/17-AAP1373.

Export citation


  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • Adler, R. J., Blanchet, J. H. and Liu, J. C. (2008). Efficient simulation for tail probabilities of Gaussian random fields. In Proceeding of Winter Simulation Conference.
  • Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Probab. 22 1167–1214.
  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Armstrong, S., Kuusi, T. and Mourrat, J.-C. (2017). Quantitative stochastic homogenization and large-scale regularity. ArXiv Preprint ArXiv:1705.05300.
  • Azaï s, J.-M. and Wschebor, M. (2008). A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stochastic Process. Appl. 118 1190–1218.
  • Azaïs, J. M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ.
  • Berglund, N., Di Gesu, G. and Weber, H. (2017). An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two. Electron. J. Probab. 22.
  • Berman, S. M. (1985). An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Probab. 22 454–460.
  • Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • Borell, C. (2003). The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337 663–666.
  • Charbeneau, R. J. (2000). Groundwater Hydraulics and Pollutant Transport. Prentice Hall.
  • Cirel’son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan–USSR Symposium on Probability Theory (G. Maruyama and J. V. Prokhorov, eds.) 20–41. Springer, Berlin.
  • Cirel’son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan–USSR Symposium on Probability Theory (Tashkent, 1975) Lecture Notes in Math. 550 20–41. Springer, Berlin.
  • Freeze, R. A. (1975). A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res. 11.
  • Gilbarg, D. and Trudinger, N. S. (2015). Elliptic Partial Differential Equations of Second Order. Springer.
  • Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhya A 32 369–378.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete 3. Folge 23. Springer, Berlin.
  • Li, X., Liu, J. and Xu, G. (2016). On the tail probabilities of aggregated lognormal random fields with small noise. Math. Oper. Res. 41 236–246.
  • Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Probab. 40 1069–1104.
  • Liu, J., Lu, J. and Zhou, X. (2015). Efficient rare event simulation for failure problems in random media. SIAM J. Sci. Comput. 37 A609–A624.
  • Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40 262–293.
  • Liu, J. and Zhou, X. (2013). On the failure probability of one dimensional random material under delta external force. Commun. Math. Sci. 11 499–521.
  • Liu, J. and Zhou, X. (2014). Extreme analysis of a random ordinary differential equation. J. Appl. Probab. 51 1021–1036.
  • Marcus, M. B. and Shepp, L. A. (1970). Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151 377–391.
  • Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Amer. Math. Soc., Providence, RI.
  • Sudakov, V. N. and Tsirelson, B. S. (1974). Extremal properties of half spaces for spherically invariant measures. Zap. Nauchn. Sem. LOMI 45 75–82.
  • Sun, J. Y. (1993). Tail probabilities of the maxima of Gaussian random-fields. Ann. Probab. 21 34–71.
  • Talagrand, M. (1996). Majorizing measures: The generic chaining. Ann. Probab. 24 1049–1103.
  • Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533–563.
  • Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • Xu, G., Lin, G. and Liu, J. (2014). Rare-event simulation for the stochastic Korteweg–de Vries equation. SIAM/ASA J. Uncertain. Quantificat. 2 698–716.