AI 中文总结
本文研究含Caputo时间分数阶导数、分数阶Laplacian及白噪声的Burgers型方程，通过Da Prato-Debussche分解与粗糙路径理论及Schauder估计，证明解的存在唯一性。
AI 中文摘要
我们建立了$(1+1)$维分数阶随机Burgers型方程解的存在唯一性，该方程包含Caputo分数阶时间导数、分数阶Laplacian以及由Riemann-Liouville分数阶积分修正的空间-时间白噪声强迫项。从物理角度看，此类方程描述了具有长程时间和空间依赖性的非马尔可夫动力学。主要困难在于线性化方程的解具有较低的空间Hölder正则性，使得奇异非线性项$G(u)\partial_x u$在经典意义下无定义。证明依赖于两个主要要素：首先，将Da Prato-Debussche分解与粗糙路径理论相结合，以严格解释非线性乘积；其次，推导以缓慢衰减的Mittag-Leffler函数表示的分数阶热核的Schauder型估计。分数阶时间导数的非局部性要求对初始条件施加强假设。
英文摘要
We establish the existence and uniqueness of solutions to a $(1+1)$-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equations of this type describe non-Markovian dynamics with long-range temporal and spatial dependencies. The main difficulty is that the solution to the linearized equation has low spatial Hölder regularity, rendering the singular nonlinearity $G(u)\partial_x u$ classically ill-defined. The proof relies on two main ingredients: first, combining the Da Prato-Debussche decomposition with rough path theory to rigorously interpret the nonlinear product; and second, deriving Schauder-type estimates for fractional heat kernels given in terms of slowly decaying Mittag-Leffler functions. The nonlocality of the fractional time derivative requires a strong assumption on the initial condition.
