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Parabolic Anderson Model with Rough Dependence in Space

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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Abstract

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter \(H \in (\frac {1}{4}, \frac {1}{2})\) in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the nth moment of the solution.

Y. Hu is supported by an NSERC discovery grant. D. Nualart is supported by the NSF grant DMS1512891 and the ARO grant FED0070445. S. Tindel is supported by the NSF grant DMS1613163.

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Acknowledgements

We thank the referees for their useful comments which improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta at Edmonton, Edmonton, AB, Canada

    Yaozhong Hu

  2. School of Mathematics, University of Birmingham, Birmingham, UK

    Jingyu Huang

  3. Department of Mathematics, Imperial College London, London, UK

    Khoa LĂȘ

  4. Department of Mathematics, University of Kansas, Lawrence, KS, USA

    David Nualart

  5. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Samy Tindel

Authors
  1. Yaozhong Hu
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  2. Jingyu Huang
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  3. Khoa LĂȘ
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  4. David Nualart
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  5. Samy Tindel
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Corresponding author

Correspondence to Yaozhong Hu .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Hu, Y., Huang, J., LĂȘ, K., Nualart, D., Tindel, S. (2018). Parabolic Anderson Model with Rough Dependence in Space. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_17

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