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Acta Mathematica Scientia

, Volume 39, Issue 3, pp 764–780 | Cite as

Joint Hölder Continuity of Parabolic Anderson Model

  • Yaozhong HuEmail author
  • Khoa LêEmail author
Article
  • 28 Downloads

Abstract

We obtain the Holder continuity and joint Hölder continuity in space and time for the random field solution to the parabolic Anderson equation \((\partial_t-\frac{1}{2}\Delta)u=u\diamond\dot{W}\) in d-dimensional space, where is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density µ(ξ). We assume that \(\gamma_0(t)\leq{c}|t|^{\alpha_0}\) and \(|\mu(\xi)|\leq{c}\prod_{i=1}^d|\xi_i|^{-\alpha_i}\;{\rm{or}}\;|\mu(\xi)|\leq{c}|\xi|^{-\alpha}\), where αi, i = 1, …, d (or α) can take negative value.

Key words

Gaussian process stochastic heat equation parabolic Anderson model multiplicative noise chaos expansion hypercontractivity Hölder continuity joint Hölder continuity 

2010 MR Subject Classification

60H15 35R60 60G60 

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Notes

Acknowledgements

We thank Raluca Balan and the referee for carefully reading the paper and for the many constructive comments which significantly improve this article.

References

  1. [1]
    Balan R, Quer-Sardanyons L, Song J. Holder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise. Acta Mathematica Scientia, 2019, 39B(3): 717–730. See also arXiv preprint. 1807.05420Google Scholar
  2. [2]
    Chen L, Dalang R C. Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions. Stoch Partial Differ Equ Anal Comput, 2014, 2(3): 316–352MathSciNetzbMATHGoogle Scholar
  3. [3]
    Chen L, Kalbasi K, Hu Y, Nualart D. Intermittency for the stochastic heat equation driven by time-fractional Gaussian noise with H ∈ (0, 1/2). Prob Theory and Related Fields, 2018, 171(1/2): 431–457MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Hu Y. Analysis on Gaussian space. Singapore: World Scientific, 2017Google Scholar
  5. [5]
    Hu Y, Le K. A multiparameter Garsia-Rodemich-Rumsey inequality and some applications. Stochastic Process Appl, 2013, 123(9): 3359–3377MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab Theory Related Fields, 2009, 143(1/2): 285–328MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hu Y, Huang J, Lê K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann Probab, 2017, 45(6B): 4561–4616MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Hu Y, Huang J, Lê K, Nualart D, Tindel S. Parabolic Anderson model with rough dependence in space. Computation and Combinatorics in Dynamics, Stochastics and Control, 2018: 477–498Google Scholar
  9. [9]
    Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron J Probab, 2015, 20(55): 50 ppGoogle Scholar
  10. [10]
    Hu Y, Nualart D, Song J. A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution. Stochastic Process Appl, 2013, 123(3): 1083–1103MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Hu Y, Nualart D, Song J. Feynman-Kac formula for heat equation driven by fractional white noise. Ann Probab, 2011, 30: 291–326MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Memin J, Mishura Y, Valkeila E. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist Probab Lett, 2001, 51: 197–206MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Nualart D. The Malliavin calculus and related topics. Second Edition. Probability and its Applications (New York). Berlin: Springer-Verlag, 2006. xiv+382 ppGoogle Scholar
  14. [14]
    Sanz-Solé M, Sarrà M. Hölder continuity for the stochastic heat equation with spatially correlated noise//Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), 259–268, Progr Probab, 52. Basel: Birkhäuser, 2002Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematics, South Kensington CampusImperial College LondonLondonUnited Kingdom

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