论文发表信息

X. Lu and W. Dai, "A Non-Conservation Stochastic Partial Differential Equation driven by Anisotropic Fractional Levy Random Field", Stochastics and Dynamics, Vol. 21, No. 5, 2150028 (20 pages), 2021 (SCI (Q3)).

英文摘要

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Levy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Levy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Levy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

中文简介

针对粒子运动中量子纠缠与聚变裂变可能出现的随机跳跃及粒子加速可能出现的随机干扰以及区块链长历史数据决策分析中出现的难点，我们研究了 由多参数各向异性分数Levy噪声驱使的具有初值与边值条件的二阶非守衡随机偏微分方程，该方程包含具有分数噪声的时间相依线性热方程与拟线性 热方程作为特例。我们证明了该方程解的唯一存在性并构造出其显式表达式，其中，各向异性分数Levy噪声由相应的各向异性分数Levy随机场定义。 与现存的噪声系统相比，我们的非高斯分数噪声来自于系统加速度的随机干扰，而不是来自于通常运动速度上的随机干扰。我们论断的证明包含了三 个部分：首先，我们考虑了各向异性分数Levy随机场为纯跳Levy过程样本轨道的广义泛函；其次，我们构造了关于各向异性分数Levy噪声的Skorohod 积分；再者，通过组合新发展的噪声分析方法与常规偏微分方程求解技术，我们完成了主要论断的证明。

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