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Stability theory of stochastic evolution equations with multiplicative fractional Brownian motions in Hilbert spaces

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  • Ding, Xiao-Li
  • Wang, Dehua

Abstract

Exponential stability is often used to barricade stochastic disturbance in some practical problems. However, the stability theory of mild solution of infinite-dimensional stochastic differential equations (SDEs) with multiplicative fractional Brownian motions (fBms) is still an unsolved problem of great concern until now. In this paper, we try to address this problem by considering a class of semilinear stochastic evolution equations with multiplicative fBms for H∈(1/2,1). Firstly, we impose some natural assumptions on the nonlinear term multiplied by fBms, and then use the assumptions to obtain two crucial estimates of pth mean of stochastic integral for general integrand. The proof of the estimates is technical and delicate. With the help of the obtained estimates of pth mean of the stochastic integral, we give the sufficient conditions for the exponentially asymptotic stability and almost sure asymptotic stability of the mild solution. The obtained results are new and innovative in this field. Finally, we give numerical simulations to verify the theoretical results.

Suggested Citation

  • Ding, Xiao-Li & Wang, Dehua, 2024. "Stability theory of stochastic evolution equations with multiplicative fractional Brownian motions in Hilbert spaces," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:chsofr:v:187:y:2024:i:c:s0960077924009871
    DOI: 10.1016/j.chaos.2024.115435
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    References listed on IDEAS

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    1. Liu, Kai, 1997. "On stability for a class of semilinear stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 219-241, October.
    2. Neuenkirch, Andreas, 2008. "Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2294-2333, December.
    3. Liu, Kai & Mao, Xuerong, 1998. "Exponential stability of non-linear stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 78(2), pages 173-193, November.
    4. Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
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