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Hölder Continuity for the Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise

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Abstract

In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang’s condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in Lp (Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in [6, 10] under more restrictive conditions, and with sub-optimal exponents for Hölder continuity.

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References

  1. Balan R M, Le C. Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial conditions. J Theor Probab, 2018, 31: 2216–2265. Preprint available on arXiv:1606.08875

    Article  MathSciNet  MATH  Google Scholar 

  2. Balan R M, Song J. Hyperbolic Anderson Model with space-time homogeneous Gaussian noise. Latin Amer J Probab Math Stat, 2017, 14: 799–849

    MathSciNet  MATH  Google Scholar 

  3. Chen L, Huang J. Comparison principle for stochastic heat equation on ℝd. Ann Probab, 2019, 43: 989–1035. Preprint available on arXiv:1607.03998

    Article  MathSciNet  Google Scholar 

  4. Conus D, Dalang R C. The non-linear stochastic wave equation in high dimensions. Electr J Probab, 2009, 22: 629–670

    MathSciNet  MATH  Google Scholar 

  5. Dalang R C. Extending martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electr J Probab, 1999, 4

  6. Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electr J Probab, 2015, 20(55): 1–50

    MATH  Google Scholar 

  7. Hu Y, Lê K. Holder continuity of Parabolic Anderson Model. Preprint available on arXiv:1809.10096, 2018

  8. Nualart D. The Malliavin Calculus and Related Topics. Second Edition. Berlin: Springer-Verlag, 2006

    MATH  Google Scholar 

  9. Sanz-Solé M, Sarra M. Hölder continuity for the stochastic heat equation with spatially correlated noise. Progress in Probab, 2002, 52: 259–268

    Google Scholar 

  10. Song J. On a class of stochastic partial differential equations. Stoch Proc Appl, 2017, 127: 37–79

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Raluca M Balan, Lluís Quer-Sardanyons or Jian Song.

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The first author is supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the second author is supported by the grant MTM2015-67802P.

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Balan, R.M., Quer-Sardanyons, L. & Song, J. Hölder Continuity for the Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise. Acta Math Sci 39, 717–730 (2019). https://doi.org/10.1007/s10473-019-0306-3

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  • DOI: https://doi.org/10.1007/s10473-019-0306-3

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