Sobolev-Mercer展开及其在随机过程中的应用
Sobolev-Mercer Expansions and Applications to Stochastic Processes
AI总结：
通过引入作用于Sobolev空间的高阶核算子，建立了Mercer定理的推广，得到了在Sobolev空间中最优的Mercer型展开，并应用于弱可微随机场的Karhunen-Loève展开，实现过程及其导数均方最优近似。
摘要：
我们通过引入一类作用于Sobolev空间$H^k(\Theta)$的高阶核算子，建立了Mercer著名定理的基本推广，其中$\Theta \subset \mathbb{R}^d$是有界域，$k\in\mathbb{N}_0$对应弱可微性的阶数。这些算子的谱分解产生了在$H^k(\Theta\times\Theta)$中最优的Mercer型展开。值得注意的是，我们从Sobolev空间的嵌入性质推导出，当$k>d$时，这些展开也一致收敛，而不需要核是正定的。对于正定核，我们确认了这些高阶算子的核性，并建立了Mercer定理的重要改进。这些结果导致了RKHS的新谱表示，并对随机分析有微妙的影响。应用于弱可微随机场的协方差核，我们的理论提供了改进的Karhunen-Loève展开，促进了过程及其导数的同时均方最优近似。
英文摘要：
We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(Θ)$, where $Θ\subset \mathbb{R}^d$ is a bounded domain and $k\in\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(Θ\timesΘ)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite. For positive definite kernels, we confirm the nuclearity of these higher-order operators and establish a significant refinement of Mercer's Theorem. These results lead to novel spectral representations of RKHS and have subtle implications for stochastic analysis. Applied to the covariance kernels of weakly differentiable random fields, our theory provides refined Karhunen-Loeve expansions that facilitate the simultaneous mean-square optimal approximation of both the process and its derivatives.
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