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WEAK APPROXIMATIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL NOISE

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Journal of Computational mathematics 2024年第3期42卷 735-754页

作者： Meng Cai Siqing Gan Xiaojie Wang school of mathematics and statistics HNP-LAMACentral South UniversityChangshaChina

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations(SPDEs)driven by additive fractional Brownian motions with the Hurst parameter H∈(1/2,1).The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method.As far as we know,the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature.A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations.In the present work,a novel and efficient approach is presented to carry out the weak error analysis for the approximations,which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus.To the best of our knowledge,the rates of weak convergence,shown to be higher than the strong convergence rates,are revealed in the fractional noise driven SPDE setting for the first time.Numerical examples corroborate the claimed weak orders of convergence.

关键词： Parabolic SPDEs Fractional Brownian motion Weak convergence rates Spec-tral Galerkin method Exponential Euler method Malliavin calculus

Local Regularity Criteria for the 3D Co-rotational Beris-Edwards System in Lorentz Spaces

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Acta Mathematicae Applicatae Sinica 2024年

作者： Zhong-bao ZUO school of mathematics and statistics Central South University

In this paper, we present some new regularity criteria for suitable weak solutions to the 3D corotational Beris-Edwards system. First, we prove that suitable weak solutions are regular if the scaled Lp;q-norm of the velocity field or gradient of velocity is small with 2/p+3/q= 2, 1 < p ≤ ∞. Next, we give ε-regularity criteria in terms of velocity field u and director field Q in Lorentz spaces, which extends the results obtained by Wang et al(J. Evol. Equ. 21:1627-1650, 2021) for Navier-Stokes equations.

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Locally and globally uniform approximations for ruin probabilities of a nonstandard bidimensional risk model with subexponential claims

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Applied mathematics(A Journal of Chinese Universities) 2024年第1期39卷 98-113页

作者： LIU Zai-ming GENG Bing-zhen WANG Shi-jie school of mathematics and statistics Central South UniversityChangsha 410083China school of Big Data and statistics Anhui UniversityHefei 230601China

Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.

关键词： bidimensional risk model asymptotic formula subexponential distribution consistently varying tail ruin probability

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

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Acta Mathematicae Applicatae Sinica 2023年第1期39卷 95-108页

作者： Xiao-chun WU school of mathematics and statistics&HNP-LAMA Central South UniversityChangsha 410083China

In 2003, Gasser-Hsiao-Li [JDE(2003)] showed that the solution to the bipolar hydrodynamic model for semiconductors(HD model) without doping function time-asymptotically converges to the diffusion wave of the porous media equation(PME) for the switch-off case. Motivated by the work of Huang-Wu[arXiv:2210.13157], we will confirm that the time-asymptotic expansion proposed by Geng-Huang-Jin-Wu [arXiv:2202.13385] around the diffusion wave is a better asymptotic profile for the HD model in this paper, where we mainly adopt the approximate Green function method and the energy method.

关键词： time asymptotic expansion bipolar hydrodynamic model for semiconductors switch-off approximate Green function

Sum formulas for various kinds of cyclotomic multiple zeta values

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Acta Mathematica Sinica,English Series 2024年

作者： Jiangtao LI school of mathematics and statistics HNP-LAMA Central South University

Cyclotomic multiple zeta values are generalizations of multiple zeta values. In this paper,we establish sum formulas for various kinds of cyclotomic multiple zeta values. As an interesting application, we show that the Q-algebra generated by Riemann zeta values are contained in the Q-algebra generated by unit cyclotomic multiple zeta values of level N for any N ≥ 2.

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On the convergence of tracking differentiator with multiple stochastic disturbances

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Science China(Information Sciences) 2024年第2期67卷 157-169页

作者： Zehao WU Huacheng ZHOU Baozhu GUO Feiqi DENG school of mathematics and Big Data Foshan University school of mathematics and statistics Central South University Academy of mathematics and Systems Science Chinese Academy of Sciences Systems Engineering Institute South China University of Technology

This paper investigates the convergence, noise-tolerance, and filtering performance of a tracking differentiator in the presence of multiple stochastic disturbances for the first time. We consider a general case wherein the input signal is corrupted by additive colored noise, and the tracking differentiator is disturbed by additive colored noise and white noise. The tracking differentiator is shown to track the input signal and its generalized derivatives in the mean square sense. Further, the almost sure convergence can be achieved when the stochastic noise affecting the input signal is vanishing. Herein, numerical simulations are performed to validate the theoretical results.

关键词： tracking differentiator convergence noise-tolerance filtering performance multiple stochastic disturbances

Non-Nehari Manifold Method for Periodic Discrete Superlinear Schr(o|¨)dinger Equation

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Acta Mathematica Sinica,English Series 2016年第4期32卷 463-473页

作者： Xian Hua TANG school of mathematics and statistics Central South University

We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.

关键词： Discrete nonlinear Schrodinger equation non-Nehari manifold method superlinear ground state solutions of Nehari-Pankov type

Bubbling solutions of fourth order mean field equations on S4

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Science China mathematics 2023年第6期66卷 1217-1236页

作者： Changfeng Gui Yeyao Hu Weihong Xie Department of mathematics The University of Texas at San Antonio school of mathematics and statistics Hunan Key Laboratory of Analytical Mathematics and ApplicationsCentral South University

In this paper, we first establish the existence of blow-up solutions with two antipodal points of the fourth order mean field equations on S4. Moreover, we construct non-axially symmetric solutions with blow-up points at the vertices of regular configurations, i.e., equilateral triangles on a great circle, regular tetrahedrons,cubes, octahedrons, icosahedrons and dodecahedrons. The bubbling rates of these blow-up solutions rely on various bubbling configurations.

关键词： Paneitz operator bubbling solution axially symmetric solution non-axially symmetric solution regular polyhedra

Wavelet Characterizations of Variable Anisotropic Hardy Spaces

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Acta Mathematica Sinica,English Series 2024年

作者： Yao HE Yong JIAO Jun LIU school of mathematics JCAMChina University of Mining and Technology school of mathematics and statistics Central South University

Let p(·):Rn→(0,∞] be a variable exponent function satisfying the globally log-Holder continuous condition and A a general expansive matrix on Rn.Let HA~(p(·))(Rn) be the variable anisotropic Hardy space associated with A.In this paper,via first establishing a criterion for affirming some functions being in the space HA~(p(·))(Rn),the authors obtain several equivalent characterizations of HA~(p(·))(Rn)in terms of the so-called tight frame multiwavelets,which extend the well-known wavelet characterizations of classical Hardy spaces.In particular,these wavelet characterizations are shown without the help of Peetre maximal operators.

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