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Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

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  • Chen, Le
  • Hu, Yaozhong
  • Nualart, David

Abstract

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: ∂β+ν2(−Δ)α∕2u(t,x)=Itγρ(u(t,x))Ẇ(t,x),t>0,x∈Rd,where Ẇ is the space–time white noise, α∈(0,2], β∈(0,2), γ≥0 and ν>0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: d<2α+αβmin(2γ−1,0). In some cases, the initial data can be measures. When β∈(0,1], we prove the sample path regularity of the solution.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5073-5112
    DOI: 10.1016/j.spa.2019.01.003
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    References listed on IDEAS

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    1. Mijena, Jebessa B. & Nane, Erkan, 2015. "Space–time fractional stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3301-3326.
    2. Chen, Le & Dalang, Robert C., 2015. "Moment bounds and asymptotics for the stochastic wave equation," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1605-1628.
    3. Debbi, Latifa & Dozzi, Marco, 2005. "On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1764-1781, November.
    4. Chen, Zhen-Qing & Kim, Kyeong-Hun & Kim, Panki, 2015. "Fractional time stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1470-1499.
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    Cited by:

    1. Junmei Wang & James Hoult & Yubin Yan, 2021. "Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise," Mathematics, MDPI, vol. 9(16), pages 1-38, August.
    2. 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.

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