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On a class of stochastic partial differential equations

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  • Song, Jian

Abstract

This paper concerns the stochastic partial differential equation with multiplicative noise ∂u∂t=Lu+uẆ, where L is the generator of a symmetric Lévy process X, Ẇ is a Gaussian noise and uẆ is understood both in the senses of Stratonovich and Skorohod. The Feynman–Kac type of representations for the solutions and the moments of the solutions are obtained, and the Hölder continuity of the solutions is also studied. As a byproduct, when γ(x) is a nonnegative and nonnegative-definite function, a sufficient and necessary condition for ∫0t∫0t|r−s|−β0γ(Xr−Xs)drds to be exponentially integrable is obtained.

Suggested Citation

  • Song, Jian, 2017. "On a class of stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 37-79.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:1:p:37-79
    DOI: 10.1016/j.spa.2016.05.008
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    References listed on IDEAS

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    1. Balan, Raluca M. & Conus, Daniel, 2014. "A note on intermittency for the fractional heat equation," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 6-14.
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    Cited by:

    1. Han, Yuecai & Wu, Guanyu, 2024. "Hölder continuity of stochastic heat equation with rough Gaussian noise," Statistics & Probability Letters, Elsevier, vol. 210(C).
    2. Rang, Guanglin, 2020. "From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3408-3444.
    3. Li, Kexue, 2017. "Hölder continuity for stochastic fractional heat equation with colored noise," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 34-41.

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