Advertisement

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

Article
• 32 Downloads

## Abstract

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 $$\alpha _H=\prod \nolimits _{i=0}^dH_i(2H_i-1)$$.

## Keywords

Fractional heat equation Fractional Brownian sheet Asymptotic behavior

60H15 35B40

## References

1. 1.
Balan, R., Conus, D.: A note on intermittency for the fractional heat equation. Stat. Probab. Lett. 95, 6–14 (2014)
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)
3. 3.
Bingham, N.H.: Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 26, 273–296 (1973)
4. 4.
Bertini, L., Giacomin, G.: On the long time behavior of the stochastic heat equation. Probab. Theory Relat. Fields 114, 279–289 (1999)
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)
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)
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)
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)
12. 12.
Chen, X., Rosen, J.: Large deviations and renormalization for Riesz potentials of stable intersection measures. Stochastic Process. Appl. 120, 1837–1878 (2010)
13. 13.
Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1999)
14. 14.
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)
15. 15.
Donoghue, W.: Distributions and Fourier Transforms. Academic Press, New York (1969)
16. 16.
Foondun, M., Liu, W., Omaba, M.: Moment bounds for a class of fractional stochastic eat equations. Ann. Probab. 45, 2131–2153 (2017)
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)
18. 18.
Hu, Y.: heat equation driven by fractional white noise potentials. Appl. Math. Optim. 43, 221–243 (2001)
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)
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)
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)
24. 24.
Song, J.: Asymptotic behavior of the solution of heat equation driven by fractional white noise. Stat. Probab. Lett. 82, 614–620 (2012)
25. 25.
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Cambridge (1970)
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)
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)

## 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