On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Tuan, H. T.

doi:10.3934/dcdsb.2020318

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…In recent years, stability of stochastic ordinary and stochastic partial differential equations, providing relevant information on the long time behavior of the solutions of such equations, has received much attention (see, e.g., 28,34,35,37,46,51,55,58,61 ). Hölder continuous paths approach has been used in 13,14,25 to study the exponential stability of a ordinary or partial differential equation driven by fractional Brownian motion with Hurst parameter H ∈ (1/2, 1).…”

Section: Introductionmentioning

confidence: 99%

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

2022

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We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order [Formula: see text] driven by a tempered fractional Brownian motion (TFBM) [Formula: see text] with [Formula: see text] and [Formula: see text]. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of [Formula: see text]th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of [Formula: see text], [Formula: see text] or [Formula: see text], [Formula: see text]. In particular, we obtain [Formula: see text]th moment Hölder regularity in time and [Formula: see text]th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.

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Section: Introductionmentioning

confidence: 99%

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

2022

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“…On the other hand, in order to capture ubiquitous noise factors in the actual situation, stochastic integro-differential equations emerge in anomalous diffusion [29], stochastic feedback system [35] and option pricing [8]. Nowadays, more and more scholars focus on stochastic fractional equations [2,12,37,41] since it can be applied to explore the memory, hereditary and hidden properties of some noise systems in physics [24], mathematical finance [13] and ecological epidemiology [34,42], etc.…”

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confidence: 99%

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

Dai

Xiao

2022

DCDS-B

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<p style='text-indent:20px;'>This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when <inline-formula><tex-math id="M1">\begin{document}$ \alpha = 1 $\end{document}</tex-math></inline-formula>. Finally, several numerical tests are reported for verification of the theoretical findings.</p>

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On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Cited by 2 publications

References 18 publications

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

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