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