Parabolic Anderson Model with Rough Dependence in Space

  • Yaozhong HuEmail author
  • Jingyu Huang
  • Khoa Lê
  • David Nualart
  • Samy Tindel
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter \(H \in (\frac {1}{4}, \frac {1}{2})\) in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the nth moment of the solution.



We thank the referees for their useful comments which improved the presentation of the paper.


  1. 1.
    Alberts, T., Khanin, K., Quastel, J.: The continuum directed random polymer. J. Stat. Phys. 154(1–2), 305–326 (2014).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balan, R., Jolis, M., Quer-Sardanyons, L.: SPDEs with fractional noise in space with index H < 1∕2. Electron. J. Probab. 20(54), 36 (2015)zbMATHGoogle Scholar
  3. 3.
    Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman- Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bezerra, S., Tindel, S., Viens, F.: Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36(5), 1642–1675 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, X.: Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann. Probab. 44(2), 1535–1598 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, Y.: Analysis on Gaussian Space. World Scientific, Singapore (2017)Google Scholar
  8. 8.
    Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143(1–2), 285–328 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hu, Y., Huang, J., Lê, K., Nualart, D., Tindel, S.: Stochastic heat equation with rough dependence in space. Ann. Probab. 45, 4561–4616 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hu, Y., Huang, J., Nualart, D., Tindel, S.: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20(55), 50 (2015)zbMATHGoogle Scholar
  11. 11.
    Hu, Y., Nualart, D., Song, J.: Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab. 30, 291–326 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang, J., Lê, K., Nualart, D.: Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann. Inst. H. Poincaré 53, 1305–1340 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khoshnevisan, D.: Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 119, pp. viii+116. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (2014)Google Scholar
  14. 14.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)CrossRefGoogle Scholar
  15. 15.
    Li, W.V., Shao, Q.-M.: Gaussian processes: inequalities, small ball probabilities and applications. In: Stochastic Processes: Theory Methods, Handbook of Statistics, vol. 19. North-Holland, Amsterdam (2001)Google Scholar
  16. 16.
    Nualart, D.: The Malliavin Calculus and Related Topics. 2nd edn. Probability and its Applications (New York), pp. xiv+382. Springer, Berlin (2006)Google Scholar
  17. 17.
    Pipiras, V., Taqqu, M.: Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118(2), 251–291 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Strichartz, R.S.: A guide to distribution theory and Fourier transforms. World Scientific Publishing Co., Inc., River Edge (2003)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Yaozhong Hu
    • 1
    Email author
  • Jingyu Huang
    • 2
  • Khoa Lê
    • 3
  • David Nualart
    • 4
  • Samy Tindel
    • 5
  1. 1.Department of Mathematical and Statistical SciencesUniversity of Alberta at EdmontonEdmontonCanada
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK
  3. 3.Department of MathematicsImperial College LondonLondonUK
  4. 4.Department of MathematicsUniversity of KansasLawrenceUSA
  5. 5.Department of MathematicsPurdue UniversityWest LafayetteUSA

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