Gaussian fluctuations for the stochastic heat equation with colored noise

Abstract

In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein’s method. We also provide a functional central limit theorem.

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References

  1. 1.

    Chen, L., Hu, Y., Nualart, D.: Regularity and strict positivity of densities for the nonlinear stochastic heat equation. To appear in Memoirs of the American Mathematical Society (2018)

  2. 2.

    Chen, L., Huang, J.: Comparison principle for stochastic heat equation on \({\mathbb{R}}^{d}\). To appear in Annals of Probability

  3. 3.

    Chen, L., Huang, J.: Regularity and strict positivity of densities for the stochastic heat equation on \({\mathbb{R}}^d\). arXiv preprint: arxiv:1902.02382 (2019)

  4. 4.

    Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDEs. Electron. J. Probab. 4(6), 29 (1999)

    MATH  Google Scholar 

  5. 5.

    Delgado-Vences, F., Nualart, D., Zheng, G.: A central limit theorem for the stochastic wave equation with fractional noise. arXiv preprint: arxiv:1812.05019 (2018)

  6. 6.

    Gaveau, B., Trauber, P.: L’intégrale stochastique comme opérateur de divergence dans l’espace founctionnel. J. Funct. Anal. 46, 230–238 (1982)

    MATH  Google Scholar 

  7. 7.

    Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33, 948–983 (2005)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Huang, J., Nualart, D., Viitasaari, L.: A central limit theorem for the stochastic heat equation. arXiv preprint: arxiv:1810.09492 (2018)

  9. 9.

    Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145(1), 75–118 (2009)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus. From Stein’s Method to Universality. Cambridge Tracts in Mathematics, 192. Cambridge University Press, Cambridge, xiv+239 pp (2012)

  11. 11.

    Nualart, D.: The Malliavin Calculus and Related Topics. Second edition. Probability and its Applications (New York). Springer, Berlin, xiv+382 pp (2006)

  12. 12.

    Nualart, D., Nualart, E.: Introduction to Malliavin Calculus. IMS Textbooks, Cambridge University Press, Cambridge (2018)

  13. 13.

    Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields 78, 535–581 (1988)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Nualart, D., Quer-Sardanyons, L.: Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27, 281–299 (2007)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Nualart, D., Zhou, H.: Total variation estimates in the Breuer-Major theorem. arXiv preprint: arxiv:1807.09707 (2018)

  17. 17.

    Sanz-Solé, M., Sarrà, M.: Hölder continuity for the stochastic heat equation with spatially correlated noise. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications, pp. 259–268. Birkhäuser, Basel (2002)

    Google Scholar 

  18. 18.

    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, pp. 265–439. Lecture Notes in Mathematics 1180. Springer, Berlin (1986)

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Acknowledgements

The authors thank the anonymous referee for many constructive advices that improved this paper.

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Correspondence to Guangqu Zheng.

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David Nualart is supported by NSF Grant DMS 1811181.

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Huang, J., Nualart, D., Viitasaari, L. et al. Gaussian fluctuations for the stochastic heat equation with colored noise. Stoch PDE: Anal Comp 8, 402–421 (2020). https://doi.org/10.1007/s40072-019-00149-3

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Keywords

  • Stochastic heat equation
  • Central limit theorem
  • Malliavin calculus
  • Stein’s method

Mathematics Subject Classification

  • 60H15
  • 60H07
  • 60G15
  • 60F05