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SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

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  • Giordano, Luca M.
  • Jolis, Maria
  • Quer-Sardanyons, Lluís

Abstract

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H∈(14,1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.

Suggested Citation

  • Giordano, Luca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2020. "SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7396-7430.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:12:p:7396-7430
    DOI: 10.1016/j.spa.2020.08.001
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    References listed on IDEAS

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    1. Balan, Raluca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2016. "SPDEs with rough noise in space: Hölder continuity of the solution," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 310-316.
    2. Jolis, Maria & Viles, Noèlia, 2010. "Continuity in the Hurst parameter of the law of the Wiener integral with respect to the fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 566-572, April.
    3. Jolis, Maria & Viles, Noèlia, 2010. "Continuity in the Hurst parameter of the law of the symmetric integral with respect to the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1651-1679, August.
    4. Bezdek, Pavel, 2016. "On weak convergence of stochastic heat equation with colored noise," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2860-2875.
    5. Balan, Raluca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2017. "Intermittency for the Hyperbolic Anderson Model with rough noise in space," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2316-2338.
    6. Wu, Dongsheng & Xiao, Yimin, 2009. "Continuity in the Hurst index of the local times of anisotropic Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1823-1844, June.
    7. Jolis, Maria & Viles, Noèlia, 2007. "Continuity with respect to the Hurst parameter of the laws of the multiple fractional integrals," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1189-1207, September.
    8. Chen, Le & Hu, Yaozhong & Nualart, David, 2019. "Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5073-5112.
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