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 Ẇ 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.
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Acknowledgements
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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Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; K. Lê is supported by Martin Hairer’s Leverhulme Trust leadership award.
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Hu, Y., Lê, K. Joint Hölder Continuity of Parabolic Anderson Model. Acta Math Sci 39, 764–780 (2019). https://doi.org/10.1007/s10473-019-0309-0
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DOI: https://doi.org/10.1007/s10473-019-0309-0
Key words
- Gaussian process
- stochastic heat equation
- parabolic Anderson model
- multiplicative noise
- chaos expansion
- hypercontractivity
- Hölder continuity
- joint Hölder continuity