Asymptotic Behavior for High Moments of the Fractional Heat Equation with Fractional Noise

  • Litan Yan
  • Xianye YuEmail author


In this paper, we investigate the large time behavior of the solution to the fractional heat equation
$$\begin{aligned} \frac{\partial u}{\partial t}(t,x)=-(-\Delta )^{\beta /2}u(t,x)+u(t,x)\frac{\partial ^{d+1} W}{\partial t\partial x_1\cdots \partial x_d},\quad t>0,\quad x\in \mathbb {R}^d, \end{aligned}$$
where \(\beta \in (0,2)\) and the noise W(tx) is a fractional Brownian sheet with indexes \(H_0, H_1,\ldots ,H_d\in (\frac{1}{2},1)\). By using large deviation techniques and variational method, we find a constant \(M_1\) such that for any integer \(p\ge 1\) and \(\alpha _0\beta +\alpha <\beta ,\)
$$\begin{aligned} \lim _{t\rightarrow \infty }t^{-\frac{2\beta -\beta \alpha _0-\alpha }{\beta -\alpha }} \log Eu(t,x)^p=p^{\frac{2\beta -\alpha }{\beta -\alpha }}\left( \frac{\alpha _H}{2} \right) ^{\frac{\beta }{\beta -\alpha }}M_1, \end{aligned}$$
where \(\alpha _0=2-2H_0\), \(\alpha =\sum \nolimits _{j=1}^d(2-2H_j)\) and \(\alpha _H=\prod \nolimits _{i=0}^dH_i(2H_i-1)\).


Fractional heat equation Fractional Brownian sheet Asymptotic behavior 

Mathematics Subject Classification (2010)

60H15 35B40 



The first author was sponsored by National Natural Science Foundation of China (No. 11571071) and the Innovation Program of Shanghai Municipal Education Commission (12ZZ063). The second author was supported by National Natural Science Foundation of China (No. 11701589).


  1. 1.
    Balan, R., Conus, D.: A note on intermittency for the fractional heat equation. Stat. Probab. Lett. 95, 6–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bass, R., Chen, X., Rosen, J.: Large deviations for Riesz potentials of additive processes. Ann. Inst. H. Poincaré Probab. Stat. 45, 626–666 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bingham, N.H.: Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 26, 273–296 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertini, L., Giacomin, G.: On the long time behavior of the stochastic heat equation. Probab. Theory Relat. Fields 114, 279–289 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for fBm and Applications. Probability and Its Application. Springer, Berlin (2008)Google Scholar
  6. 6.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, X.: Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs, vol. 157. Amer. Math. Soc. Providence, RI (2010)Google Scholar
  8. 8.
    Chen, X.: Precise intermittency for the parabolic Anderson equation with an \((1 + 1)\)-dimensional time–space white noise. Ann. Inst. H. Poincaré Probab. Stat. 51, 1486–1499 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, X., Hu, Y., and Song, J.: Feynman–Kac formula for fractional heat equations driven by fractional white noises. PreprintGoogle Scholar
  10. 10.
    Chen, X., Hu, Y., Song, J., Xing, F.: Exponential asymptotics for time–space Hamiltonians. Ann. Inst. H. Poincaré Probab. Stat. 51, 1529–1561 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X., Rosen, J.: Exponential asymptotics for intersection local times of stable processes and random walks. Ann. Inst. H. Poincaré Probab. Stat. 41, 901–928 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, X., Rosen, J.: Large deviations and renormalization for Riesz potentials of stable intersection measures. Stochastic Process. Appl. 120, 1837–1878 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  15. 15.
    Donoghue, W.: Distributions and Fourier Transforms. Academic Press, New York (1969)zbMATHGoogle Scholar
  16. 16.
    Foondun, M., Liu, W., Omaba, M.: Moment bounds for a class of fractional stochastic eat equations. Ann. Probab. 45, 2131–2153 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Giné, E., Marcus, M.B.: The central limit theorem for stochastic integrals with respect to Lévy processes. Ann. Probab. 11, 58–77 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hu, Y.: heat equation driven by fractional white noise potentials. Appl. Math. Optim. 43, 221–243 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hu, Y.Z., Huang, J., Nualart, D., Tindel, S.: Stochastic heat equations with general multiplicative Gaussian noise: Hölder continuity and intermittency. Electron. J. Probab. 20, 1–50 (2015)zbMATHGoogle Scholar
  20. 20.
    Hu, Y.Z., Nualart, D., Song, J.: Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39, 291–326 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Le Gall, J.-F.: Exponential Moments for the Renormalized Self-Intersection Local Time of Planar Brownian Motion, Séminaire de Probabilités XXVIII. Lecture Notes in Math., vol. 1583, pp. 172–180. Springer, Berlin (1994)Google Scholar
  22. 22.
    Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lect. Notes in Math. vol. 1929 (2008)Google Scholar
  23. 23.
    Rosinski, J., Woyczynski, W.A.: On Itô stochastic integration with respect to \(p\)-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14, 271–286 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Song, J.: Asymptotic behavior of the solution of heat equation driven by fractional white noise. Stat. Probab. Lett. 82, 614–620 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Cambridge (1970)zbMATHGoogle Scholar
  26. 26.
    Viens, F.G., Zhang, T.: Almost sure exponential behavior of a directed polymer in a fractional Brownian environment. J. Funct. Anal. 255, 2810–2860 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yan, L., Yu, X., Sun, X.: Asymptotic behavior of the solution of the fractional heat equation. Stat. Probab. Lett. 117, 54–61 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of ScienceDonghua UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouPeople’s Republic of China

Personalised recommendations