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Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

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Abstract

We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data \(\nu \). We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For \(C^{1, \alpha }\)-domains with Dirichlet condition, the initial data \(\nu \) is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to \(|\nu |\). As an application, we show that the solution is fully intermittent for sufficiently high level \(\lambda \) of noise under the Dirichlet condition, and for all \(\lambda > 0\) under the Neumann condition.

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Notes

  1. The explicit form of the fundamental solution can be found, e.g., in [36, Section 4.1.2 on p. 418].

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Acknowledgements

C. Y. Lee is partially supported by Taiwan’s National Science and Technology Council grant NSTC111-2115-M-007-015-MY2. L. Chen is partially supported by NSF DMS-2246850 and a collaboration grant from Simons Foundation (959981). The authors would also like to thank the anonymous referee for the careful reading and helpful suggestions, which have greatly improved the quality of the paper.

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  1. Institut de Mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    David Candil

  2. Department of Mathematics and Statistics, Auburn University, 203 Parker Hall, Auburn, Alabama, 36849, United States

    Le Chen

  3. Department of Mathematics, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, 300044, Hsinchu, Taiwan

    Cheuk Yin Lee

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Candil, D., Chen, L. & Lee, C.Y. Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds. Stoch PDE: Anal Comp (2023). https://doi.org/10.1007/s40072-023-00310-z

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