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Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

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  • Wang, Ran
  • Xu, Lihu

Abstract

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by α-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.

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  • Wang, Ran & Xu, Lihu, 2018. "Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1772-1796.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:5:p:1772-1796
    DOI: 10.1016/j.spa.2017.08.010
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    1. Funaki, Tadahisa & Xie, Bin, 2009. "A stochastic heat equation with the distributions of Lévy processes as its invariant measures," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 307-326, February.
    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.
    3. Gourcy, Mathieu, 2007. "A large deviation principle for 2D stochastic Navier-Stokes equation," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 904-927, July.
    4. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
    5. Dong, Zhao & Xu, Tiange & Zhang, Tusheng, 2009. "Invariant measures for stochastic evolution equations of pure jump type," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 410-427, February.
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    Cited by:

    1. Xu, Lihu, 2018. "Singular integrals of stable subordinator," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 115-118.
    2. Hu, Shulan & Wang, Ran, 2020. "Asymptotics of stochastic Burgers equation with jumps," Statistics & Probability Letters, Elsevier, vol. 162(C).
    3. Liu, Xianming, 2022. "Limits of invariant measures of stochastic Burgers equations driven by two kinds of α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 1-21.
    4. Ankit Kumar & Manil T. Mohan, 2023. "Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation," Journal of Theoretical Probability, Springer, vol. 36(1), pages 661-709, March.

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