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Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise

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  • Xichao Sun
  • Junfeng Liu

Abstract

We consider a class of stochastic fractional equations driven by fractional noise on t,x∈0,T×0,1 ∂u/∂t=Dδαu+ft,x,u+∂2BHt,x/∂t ∂x, with Dirichlet boundary conditions. We formally replace the random perturbation by a family of sequences based on Kac‐Stroock processes in the plane, which approximate the fractional noise in some sense. Under some conditions, we show that the real‐valued mild solution of the stochastic fractional heat equation perturbed by this family of noises converges in law, in the space 𝒞([0, T] × [0,1]) of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.

Suggested Citation

  • Xichao Sun & Junfeng Liu, 2014. "Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise," Advances in Mathematical Physics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlamp:v:2014:y:2014:i:1:n:479873
    DOI: 10.1155/2014/479873
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    References listed on IDEAS

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    1. Li, Ming & Zhao, Wei, 2012. "Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6268-6278.
    2. Nguyen, Dung Tien, 2012. "Mackey–Glass equation driven by fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5465-5472.
    3. Wu, Dongsheng, 2011. "On the solution process for a stochastic fractional partial differential equation driven by space-time white noise," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1161-1172, August.
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