发表机构
Athabasca University（阿萨巴斯卡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为正则化 $p(x)$-Laplacian 构建 Itô 扩散框架，通过通量谱分析和散度展开确定协方差与对数漂移，证明系数正则性及路径唯一性，并验证轮廓重现与鞅恒等式，给出规范系数选择。
AI 中文摘要
对于给定的经典轮廓和固定的正正则化参数，我们构造了一个 Itô 扩散，其内部生成元在应用于该轮廓时重现正则化的 $p(x)$-Laplacian。我们确定了未正则化和正则化通量的精确切谱，刻画了它们在零梯度处的行为，并在有界梯度范围内建立了定量椭圆性界。完整的散度展开识别出一个协方差，等于正则化通量在轮廓梯度处的梯度导数的两倍，以及一个由变指数 $p(x)$ 的空间变化产生的对数漂移。在显式经典正则性假设下，我们建立了系数的有界性和局部 Lipschitz 连续性，并证明了在到达域边界之前的强存在性和路径唯一性。我们识别了边界停止过程的内部生成元，并验证了轮廓重现恒等式。进一步的结果包括固定全局扩展的初始点稳定性、局部鞅恒等式以及有界停时处的 Dynkin 公式。一个显式的非常数指数例子说明了在临界点处的构造，并且系数选择在基于通量的规定下被证明是规范的，而非仅由轮廓重现唯一确定。
英文摘要
For a prescribed classical profile and a fixed positive regularization parameter, we construct an Itô diffusion whose interior generator reproduces the regularized $p(x)$-Laplacian when applied to that profile. We determine the exact tangent spectra of the unregularized and regularized fluxes, characterize their behavior at zero gradient, and establish quantitative ellipticity bounds on bounded gradient ranges. The full divergence expansion identifies a covariance equal to twice the gradient derivative of the regularized flux, evaluated at the profile gradient, and a logarithmic drift generated by spatial variation of the variable exponent $p(x)$. Under explicit classical regularity assumptions, we establish boundedness and local Lipschitz continuity of the coefficients and prove strong existence and pathwise uniqueness up to domain exit. We identify the interior generator of the boundary-stopped process and verify the profile-reproduction identity. Further consequences include initial-point stability for a fixed global extension, localized martingale identities, and Dynkin's formula at bounded stopping times. An explicit nonconstant exponent example illustrates the construction at a critical point, and the coefficient selection is shown to be canonical under the flux-based prescriptions rather than determined uniquely by profile reproduction alone.
