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


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


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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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