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具有通用粗糙噪声的随机分数热方程

Stochastic fractional heat equation with general rough noise

Suzhou University of Technology（苏州科技大学）；Wuhan University（武汉大学）

Bin Qian, Ran Wang

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28 pages
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AI总结：

本文研究了非线性一维随机分数热方程在α∈(1,2)时的well-posed性，无需假设σ(0)=0，利用精确的热核估计。

摘要：

考虑以下非线性一维随机分数热方程$$\frac{\partial }{\partial t}u(t, x)= -(-Δ)^{α/2}u(t, x) +σ(t,x,u(t,x)) \dot{W}(t, x), $$其中$-(-Δ)^{α/2}$是$\mathbb R$上的分数拉普拉斯算子，$1<α<2$，$\dot{W}$是时间白噪声，空间行为为 Hurst 指数 $H$ 满足 $\frac{3-α}{4}<H<\frac12$ 的分数布朗运动。当 $α=2$ 时，Hu 和 Wang (Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022) 379-423) 研究了解的well-posed性及其Hölder连续性，去除了之前假设的 $σ(0)=0$ 条件。他们的方法依赖于在加权空间中使用合适的衰减函数。对于 $α∈(1,2)$ 的情况，受 Hu 和 Wang 启示，我们研究了随机分数热方程的well-posed性，无需假设 $σ(0)=0$，这在 Liu 和 Mao 的早期工作中是必需的。在我们的分析中，与分数拉普拉斯算子 $-(-Δ)^{α/2}$ 关联的热核的精确估计起着关键作用。

英文摘要：

Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-Δ)^{α/2}u(t, x) +σ(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-Δ)^{α/2}$ is the fractional Laplacian on $\mathbb R$ for $1<α<2$, and $\dot{W}$ is a Gaussian noise that is white in time and behaves in space as a fractional Brownian motion with Hurst index $H$ satisfying $\frac{3-α}{4}<H<\frac12$. When $α=2$, Hu and Wang ({\it Ann. Inst. Henri Poincaré Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of the solution and its Hölder continuity, removing the technical condition $σ(0)=0$ that was previously assumed in Hu et al. ({\it Ann. Probab.} {\bf 45} (2017) 4561-4616). Their approach relied on working in a weighted space with a suitable power decay function. For the case $α\in (1,2)$, inspired by Hu and Wang, we investigate the well-posedness of the stochastic fractional heat equation without imposing the technical condition of $σ(0)=0$, which was required in the earlier work of Liu and Mao ({\it Bull. Sci. Math.} {\bf181} (2022) 103207). In our analysis, precise estimates of the heat kernel associated with the fractional Laplacian $-(-Δ)^{α/2}$ play a crucial role.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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