Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 3, pp 629–644 | Cite as

Precise Moment Asymptotics for the Stochastic Heat Equation of a Time-Derivative Gaussian Noise

  • Heyu LiEmail author
  • Xia ChenEmail author
Article
  • 44 Downloads

Abstract

This article establishes the precise asymptotics
$$\mathbb{E}u^m(t,x)\;\;\;\;(t\rightarrow\infty\;{\rm{or}}\;m\rightarrow\infty)$$
for the stochastic heat equation
$$\frac{\partial{u}}{\partial{t}}(t,x)=\frac{1}{2}\Delta{u}(t,x)+u(t,x)\frac{\partial{W}}{\partial{t}}(t,x)$$
with the time-derivative Gaussian noise \({{\partial W} \over {\partial t}}(t,x)\) that is fractional in time and homogeneous in space.

Key words

Stochastic heat equation time-derivative Gaussian noise Brownian motion Feynman-Kac representation Schilder’s large deviation 

2010 MR Subject Classification

60J65 60K37 60H15 60G60 60F10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Chen L, Hu Y Z, Kalbasi K, et al. Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise. Probab Theor Rel Fields, 2018, 171(1/2): 431–457MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chen X. Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann Probab, 2016, 44(2): 1535–1598MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Chen X. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: in Sko-rokhod regime. Ann Inst Henri Poincaré Probab Stat, 2017, 53(2): 819–841MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Dembo A, Zeitouni O. Large Deviations Techniques and Applications[M]. 2nd ed. New York: Springer, 1998CrossRefzbMATHGoogle Scholar
  5. [5]
    Hu Y Z, Lu F, Nualart D. Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2. Ann Probab, 2012, 40(3): 1041–1068MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Kalbasi K and Mountford T S. Feynman-Kac representation for the parabolic Anderson model driven by fractional noise. J Funct Anal, 2015, 269(5): 1234–1263MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Strassen V. An invariance principle for the law of the iterated logarithm. Z Wahrsch Verw Gebiete, 1964, 3(3): 211–226MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Hu Y Z. A random transport-diffusion equation. Acta Mathematica Scientia, 2010, 30B(6): 2033–2050MathSciNetzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of MathematicsJilin UniversityChangchunChina
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleTNUSA

Personalised recommendations