Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy

Manna, Utpal; Mukherjee, Debopriya; Panda, Akash Ashirbad

doi:10.1016/j.jmaa.2019.123384

Cited by 9 publications

(4 citation statements)

References 12 publications

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“…In [8,28], the authors used the Wong-Zakai approximation to obtain a solution and the convergence of (63). In this section, we show the same results using the results from the previous section and by taking as geometric rough path the rough path generated by the Brownian motion where the stochastic integral is interpreted in the Stratonovich sense.…”

Section: Application To Stochastic Landau-lifshitz-gilbert Equations (Sllges)mentioning

confidence: 99%

“…Brzeźniak et al have proved in [6,7] existence of weak martingale solution for SLLGEs in three dimensions perturbed by jump noise in the Marcus canonical form with non-zero anisotropic energy E an , see [6] and non-zero exchange energy E ex only, see [7]. Recently, in [8,28], the authors have employed Wong-Zakai approximation technique to obtain the solvability and convergence of the time dependent transformed PDEs. The open questions framed in [8] have motivated us, as a first step to adapt Lyons' rough paths theory to study LLGEs driven by geometric rough paths in one dimension.…”

Section: Introductionmentioning

confidence: 99%

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Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

2021

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We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

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Section: Application To Stochastic Landau-lifshitz-gilbert Equations (Sllges)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

2021

Self Cite

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“…where (−∆) α (0 < α < 1) is the fractional Laplace operator. The Wong-Zakai process W λ (t) (introduced in [40], see also [2,4,15,27]) is the λ-difference of a scalar Brownian motion W (t) on the Wiener quadruple (Ω, F, P, θ), more precisely, W λ (t, ω) = (W (t + λ, ω) − W (t, ω))/λ, ∀λ > 0, t ∈ R, ω ∈ Ω.…”

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confidence: 99%

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Yang

Long

2022

DCDS-B

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<p style='text-indent:20px;'>We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset <inline-formula><tex-math id="M1">\begin{document}$ \Lambda^* $\end{document}</tex-math></inline-formula> of the parameterized space such that the binary map <inline-formula><tex-math id="M2">\begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document}</tex-math></inline-formula> is continuous at all points of <inline-formula><tex-math id="M3">\begin{document}$ \Lambda^*\times \mathbb{R} $\end{document}</tex-math></inline-formula> with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space <inline-formula><tex-math id="M4">\begin{document}$ (0, \infty] $\end{document}</tex-math></inline-formula> and the infinity of noise means that the equation is deterministic.</p>

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“…So far, there has been a rich literature about the Wong-Zakai approximations, and we only mention some recent work related to our topic. By means of the Wong-Zakai approximations, Brzeźniak et al [3] and Manna et al [22] proved the existence and uniqueness of solution of stochastic Landau-Lifshitz-Gilbert equations with different energy. Lv and Wang et al [21,29] and Shen et al [24] studied the approximations of random attractors and invariant manifolds for stochastic partial differential equations.…”

mentioning

confidence: 99%

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on <inline-formula><tex-math id="M1">$ {\mathbb{R}}^N $</tex-math></inline-formula>

Liu

Zhao

2021

era

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<p style='text-indent:20px;'>In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of <inline-formula><tex-math id="M2">\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> space. We establish the Wong-Zakai approximations of solutions in <inline-formula><tex-math id="M3">\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> for arbitrary <inline-formula><tex-math id="M4">\begin{document}$ l\geq q $\end{document}</tex-math></inline-formula> in the sense of upper semi-continuity of their random attractors, where <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the growth exponent of the nonlinearity. The <inline-formula><tex-math id="M6">\begin{document}$ L^l $\end{document}</tex-math></inline-formula>-pre-compactness of attractors is proved by using the truncation estimate in <inline-formula><tex-math id="M7">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and the higher-order bound of solutions.</p>

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Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy

Cited by 9 publications

References 12 publications

Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on <inline-formula><tex-math id="M1">$ {\mathbb{R}}^N $</tex-math></inline-formula>

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