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A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

Author

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  • Hu, Yaozhong
  • Nualart, David
  • Song, Jian

Abstract

In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.

Suggested Citation

  • Hu, Yaozhong & Nualart, David & Song, Jian, 2013. "A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1083-1103.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:1083-1103
    DOI: 10.1016/j.spa.2012.11.004
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    Cited by:

    1. Hu, Yaozhong & Li, Juan & Mi, Chao, 2023. "BSDEs generated by fractional space-time noise and related SPDEs," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    2. Zhang, Bin & Yao, Zhigang & Liu, Junfeng, 2023. "On a class of mixed stochastic heat equations driven by spatially homogeneous Gaussian noise," Statistics & Probability Letters, Elsevier, vol. 196(C).
    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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