We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy 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 Lévy 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 Lévy 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.

中文翻译：

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各向异性分数Lévy随机场驱动的非守恒随机偏微分方程

我们研究了由多参数各向异性分数 Lévy 噪声 (AFLN) 在不同初始和/或边界条件下驱动的非守恒二阶随机偏微分方程 (SPDE)。它包括分数噪声下的时变线性热方程和拟线性热方程作为特例。证明并构造了方程解的唯一存在性和表达式。AFLN 在某种意义上被定义为各向异性分数列维随机场 (AFLRF) 的导数。与现有的噪声系统相比，我们的非高斯分数噪声基本上是从系统加速度的随机扰动而不是系统运动速度的扰动中观察到的。在证明我们的主张的过程中，有三个方面。第一的，我们将 AFLRF 视为纯跳跃 Lévy 过程的样本路径的广义泛函。其次，我们通过白噪声方法构建关于 AFLN 的 Skorohod 集成。第三，通过将这种噪声分析方法与传统的 PDE 求解技术相结合，我们为我们的主张提供了可靠的证据。