鞅维数与解析维数
Martingale and analytic dimensions
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中文总结 AI 辅助
本文定义了鞅维数与解析维数，证明在双侧高斯热核估计下二者一致，并引入能量像密度性质，最后指出行走维数大于2的次高斯扩散空间非利普希茨可微。
中文摘要 AI 辅助
这些笔记引入了强局部正则狄利克雷形式的鞅维数以及利普希茨可微性空间的解析维数。我们解释了在双侧高斯热核估计下这些维数为何一致，并讨论了它们与其他维数概念的关系。我们还引入了能量像密度性质、其最近的解决进展及其在鞅维数上的应用。最后，我们证明了具有行走维数大于2的双侧次高斯热核估计的扩散空间不是利普希茨可微性空间。
英文摘要
These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these dimensions coincide under two-sided Gaussian heat kernel estimates and discuss their relationships with other notions of dimension. We also introduce the energy image density property, its recent resolution, and its applications to martingale dimension. Finally, we show that spaces carrying diffusions with two-sided sub-Gaussian heat kernel estimates of walk dimension greater than two are not Lipschitz differentiability spaces.
