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Bismut formula for a stochastic heat equation with fractional noise

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  • Yan, Litan
  • Yin, Xiuwei

Abstract

In this note, we establish the Bismut formula for stochastic heat equation ∂∂tu(t,x)=Δu(t,x)+ẆH(t,x),t≥0,x∈[0,1],∂∂xu(t,x)|x=0=∂∂xu(t,x)|x=1=0,t≥0,u(0,x)=f(x),x∈[0,1],where f(x)∈H≔L2([0,1]) and WH is the fractional noise with Hurst index H∈(12,1). As an application, we also introduce the Harnack inequality.

Suggested Citation

  • Yan, Litan & Yin, Xiuwei, 2018. "Bismut formula for a stochastic heat equation with fractional noise," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 165-172.
    • Handle: RePEc:eee:stapro:v:137:y:2018:i:c:p:165-172
    DOI: 10.1016/j.spl.2018.01.018
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    References listed on IDEAS

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    1. Zhang, Xicheng, 2010. "Stochastic flows and Bismut formulas for stochastic Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1929-1949, September.
    2. Zhang, Xicheng, 2013. "Derivative formulas and gradient estimates for SDEs driven by α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1213-1228.
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