We introduce and analyze a nonlocal generalization of Whittle--Matérn Gaussian fields in which the smoothness parameter varies in space through the fractional order, $s=s(x)\in[\underline{s}\,,\bar{s}]\subset(0,1)$. The model is defined via an integral-form operator whose kernel is constructed from the modified Bessel function of the second kind and whose local singularity is governed by the symmetric exponent $β(x,y)=(s(x)+s(y))/2$. This variable-order nonlocal formulation departs from the classical constant-order pseudodifferential setting and raises new analytic and numerical challenges. We develop a novel variational framework adapted to the kernel, prove existence and uniqueness of weak solutions on truncated bounded domains, and derive Sobolev regularity of the Gaussian (spectral) solution controlled by the minimal local order: realizations lie in $H^r(G)$ for every $r<2\underline{s}-\tfrac{d}{2}$ (here $H^r(G)$ denotes the Sobolev space on the bounded domain $G$), hence in $L_2(G)$ when $\underline s>d/4$. We also present a finite-element sampling method for the integral model, derive error estimates, and provide numerical experiments in one dimension that illustrate the impact of spatially varying smoothness on samples covariances. Computational aspects and directions for scalable implementations are discussed.

翻译：本文引入并分析了一类Whittle-Matérn高斯场的非局部推广模型，其平滑度参数通过分数阶在空间中变化，即$s=s(x)\in[\underline{s}\,,\bar{s}]\subset(0,1)$。该模型通过积分形式的算子定义，其核由第二类修正贝塞尔函数构造，其局部奇异性由对称指数$β(x,y)=(s(x)+s(y))/2$控制。这种变阶非局部表述有别于经典的常阶伪微分设定，并带来了新的解析与数值挑战。我们针对该核函数发展了一套新的变分框架，证明了在截断有界域上弱解的存在唯一性，并推导了由最小局部阶控制的高斯（谱）解的Sobolev正则性：对于任意$r<2\underline{s}-\tfrac{d}{2}$，其实现属于$H^r(G)$（此处$H^r(G)$表示有界域$G$上的Sobolev空间），因此当$\underline s>d/4$时属于$L_2(G)$。我们还提出了该积分模型的有限元采样方法，推导了误差估计，并在一维情形下提供了数值实验，以说明空间变化的平滑度对样本协方差的影响。文中还讨论了计算方面的考量以及可扩展实现的方向。