发表机构
Michigan State University; The Chinese University of Hong Kong (Shenzhen)（密歇根州立大学; 香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有界域上带分数阶空间相关噪声的随机热方程，证明温和解存在唯一性的充要条件，并给出最优Holder正则性、连续模、重对数律及尖锐小球概率估计。
AI 中文摘要
我们考虑有界Lipschitz域上具有零Dirichlet边界条件和零初始条件的随机热方程$\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$，其中$\dot{W}_\alpha$是时间上为白噪声的高斯噪声，其空间协方差为$(-\Delta)^{-\alpha}$的核，且$\alpha>0$。我们证明，当且仅当$\alpha>d/2-1$时，存在唯一的逐点定义的温和解。在这种情况下，若域额外为$C^2$，我们还建立了解的空间和时间Holder正则性。当$d/2-1<\alpha<d/2$时，我们证明Holder指数是最优的，并获得解的精确局部和一致连续模、Chung型重对数律以及尖锐的小球概率估计。
英文摘要
We consider the stochastic heat equation $\partial_t u(t,x) = Δu(t,x) + \dot{W}_α(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_α$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-Δ)^{-α}$ with $α>0$. We prove that a unique pointwise defined mild solution exists if and only if $α>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<α<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.
Comments30 pages
