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A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise

Author

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  • Ruinan Li

    (Shanghai University of International Business and Economics)

  • Ran Wang

    (Wuhan University)

  • Beibei Zhang

    (Wuhan University)

Abstract

We study the Freidlin–Wentzell large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise: $$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$ ∂ u ε ( t , x ) ∂ t = ∂ 2 u ε ( t , x ) ∂ x 2 + ε σ ( t , x , u ε ( t , x ) ) W ˙ ( t , x ) , t > 0 , x ∈ R , where $$\dot{W}$$ W ˙ is white in time and fractional in space with Hurst parameter $$H\in \left( \frac{1}{4},\frac{1}{2}\right) $$ H ∈ 1 4 , 1 2 . Recently, Hu and Wang (Ann Inst Henri Poincaré Probab Stat 58(1):379–423, 2022) have studied the well-posedness of this equation without the technical condition of $$\sigma (0)=0$$ σ ( 0 ) = 0 which was previously assumed in Hu et al. (Ann Probab 45(6):4561–4616, 2017). We adopt a new sufficient condition proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) for the weak convergence criterion of the large deviation principle.

Suggested Citation

  • Ruinan Li & Ran Wang & Beibei Zhang, 2024. "A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise," Journal of Theoretical Probability, Springer, vol. 37(1), pages 251-306, March.
  • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-022-01228-3
    DOI: 10.1007/s10959-022-01228-3
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    References listed on IDEAS

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    1. Balan, Raluca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2016. "SPDEs with rough noise in space: Hölder continuity of the solution," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 310-316.
    2. Peszat, Szymon & Zabczyk, Jerzy, 1997. "Stochastic evolution equations with a spatially homogeneous Wiener process," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 187-204, December.
    3. Xu, Tiange & Zhang, Tusheng, 2009. "White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3453-3470, October.
    4. Budhiraja, Amarjit & Chen, Jiang & Dupuis, Paul, 2013. "Large deviations for stochastic partial differential equations driven by a Poisson random measure," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 523-560.
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