Acta Applicandae Mathematicae

, Volume 139, Issue 1, pp 59–80 | Cite as

Moderate Deviations for a Stochastic Heat Equation with Spatially Correlated Noise



In this paper, we proved a central limit theorem and established a moderate deviation principle for a perturbed stochastic heat equations defined on [0,T]×[0,1]d. This equation is driven by a Gaussian noise, white in time and correlated in space.


Stochastic heat equation Freidlin-Wentzell’s large deviation Moderate deviations Central limit theorem 

Mathematics Subject Classification

60H15 60F05 60F10 


  1. 1.
    De Acosta, A.: Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Am. Math. Soc. 329, 357–375 (1992) MATHCrossRefGoogle Scholar
  2. 2.
    Bally, V., Millet, A., Sanz-Solé, M.: Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23, 178–222 (1995) MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Budhiraja, A., Dupuis, P., Ganguly, A.: Moderate deviation principles for stochastic differential equations with jumps (2014). arXiv:1401.7316v1
  4. 4.
    Cerrai, S., Röckner, M.: Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32, 1100–1139 (2004) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, X.: The moderate deviations of independent random vectors in a Banach space. Chinese J. Appl. Probab. Statist. 7, 24–33 (1991) MATHMathSciNetGoogle Scholar
  6. 6.
    Chenal, F., Millet, A.: Uniform large deviations for parabolic SPDEs and applications. Stoch. Process. Appl. 72, 161–186 (1997) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Conus, D., Dalang, R.: The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13, 629–670 (2008) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dalang, R.: Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E’s. Electron. J. Probab. 4, 1–29 (1999) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dalang, R., Frangos, N.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26, 187–212 (1998) MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) MATHCrossRefGoogle Scholar
  11. 11.
    Deuschel, J., Stroock, D.: Large Deviations. Pure and Applied Mathematics. Academic Press, Boston (1989) MATHGoogle Scholar
  12. 12.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics, vol. 38. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  13. 13.
    Dupuis, P., Ellis, R.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (1997) MATHCrossRefGoogle Scholar
  14. 14.
    Ermakov, M.: The sharp lower bound of asymptotic efficiency of estimators in the zone of moderate deviation probabilities. Electron. J. Stat. 6, 2150–2184 (2012) MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Franzova, N.: Long time existence for the heat equation with a spatially correlated noise term. Stoch. Anal. Appl. 17, 169–190 (1999) MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Freidlin, M.I.: Random perturbations of reaction-diffusion equations: the quasi-deterministic approach. Trans. Am. Math. Soc. 305, 665–697 (1988) MATHMathSciNetGoogle Scholar
  17. 17.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbation of Dynamical Systems. Translated by J. Szuc. Springer, Berlin (1984) CrossRefGoogle Scholar
  18. 18.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, New York (1964) MATHGoogle Scholar
  19. 19.
    Foondun, M., Khoshnevisan, D.: On the stochastic heat equation with spatially-colored random forcing. Trans. Am. Math. Soc. 365, 409–458 (2013) MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Gao, F.Q., Jiang, H.: Moderate deviations for squared Ornstein-Uhlenbeck process. Stat. Probab. Lett. 79(11), 1378–1386 (2009) MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gao, F.Q., Zhao, X.Q.: Delta method in large deviations and moderate deviations for estimators. Ann. Stat. 39, 1211–1240 (2011) MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Guillin, A., Liptser, R.: Examples of moderate deviation principle for diffusion processes. Discrete Contin. Dyn. Syst., Ser. B 6, 803–828 (2006) MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Inglot, T., Kallenberg, W.: Moderate deviations of minimum contrast estimators under contamination. Ann. Stat. 31, 852–879 (2003) MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Kallenberg, W.: On moderate deviation theory in estimation. Ann. Stat. 11, 498–504 (1983) MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Ledoux, M.: Sur les deviations modérées des sommes de variables aléatoires vectorielles independantes de même loi. Ann. Henri Poincaré 28, 267–280 (1992) MATHMathSciNetGoogle Scholar
  26. 26.
    Márquez-Carreras, D., Sarrà, M.: Large deviation principle for a stochastic heat equation with spatially correlated noise. Electron. J. Probab. 8, 1–39 (2003) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Millet, A., Sanz-Solé, M.: A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27(2), 803–844 (1999) MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Nualart, D., Quer-Sardanyons, L.: Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27, 281–299 (2007) MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Nualart, E., Quer-Sardanyons, L.: Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension. Stoch. Process. Appl. 122, 418–447 (2012) MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Peszat, S., Zabczyk, J.: Stochastic evolution equations with a spatially homogeneous Wiener process. Stoch. Process. Appl. 72, 187–204 (1997) MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Sowers, R.: Large deviations for a reaction-diffusion equation with non-Gaussian perturbation. Ann. Probab. 20, 504–537 (1992) MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Walsh, J.: An introduction to stochastic partial differential equations. In: École d’été de Probabilités St Flour XIV. Lect. Notes Math., vol. 1180. Springer, Berlin (1986) Google Scholar
  33. 33.
    Wang, R., Zhang, T.S.: Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Potential Anal. (2014). doi:10.1007/s11118-014-9425-6 Google Scholar
  34. 34.
    Wu, L.: Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23, 420–445 (1995) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations