L^p-预解式的拟正则拓扑与半Dirichlet型

通过Dirichlet形式和预解式研究定常Fokker-Planck方程解的局部椭圆正则性

Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents

本文针对具一般系数的定常Fokker-Planck方程解（L²密度测度），利用Dirichlet形式和预解式，结合椭圆正则性结果，证明其密度具H¹·²正则性与Hölder连续性，突显相关理论在解正则性分析中的核心作用。

$L^p$-Kato class measures and their relations with Sobolev embedding theorems for Dirichlet spaces

In this paper, we discuss relationships between the continuous embeddings of Dirichlet spaces $(\mathcal{F}, \mathcal{E}_1)$ into Lebesgue spaces and the integrability of the associated resolvent kernel $r_α(x, y)$. For a positive measure $μ$, we consider the following two properties; the first one is that the Dirichlet space $(\mathcal{F}, \mathcal{E}_1)$ is continuously embedded into $L^{2p}(E;μ)$ (which we write as (Sob)$_p$), and the second one is that the family of 1-order resolvent kernels $\{r_1(x, y)\}_{x\in E}$ is uniformly $p$-th integrable in $y$ with respect to the measure $μ$ (whi…

道格拉斯加权狄利克雷空间：分析与概率方面

Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects

本文引入道格拉斯加权狄利克雷空间，证明其再生核是Szego核在1预解式下的像，发展相关位势理论，得到关联对称纯跳Hunt过程，经典情形下为缠绕柯西过程。

具有不连续尺度的扩散的预解式方法

Resolvent approach to diffusions with discontinuous scale

本文研究尺度函数s仅非降时算子L=1/2 D_m D_s对应的马氏过程，证明其再生核由α-调和方程的两个正单调解生成，并诱导出与半群或Dirichlet型方法相同的马氏过程。

利用Dirichlet型构造$\mathcal L^p$ -强Feller过程及其在椭圆扩散中的应用

Construction of $\mathcal L^p$-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

在局部紧可分度量空间上给出了$\mathcal L^p$ -强Feller过程的一般构造方案.从正则Dirichlet型出发，在特定的正则性假设下，构造了一个相应的半群和具有$\mathcal L^p$ -强Feller性质的核的预解式.它们允许我们构造一个过程，该过程从已知集合（即正则性假设成立的集合）中的所有起点解决相应的鞅问题。我们应用这个结果来构造椭圆扩散具有局部Lipschitz矩阵系数和奇异漂移一般开集吸收在边界上。在这个应用中，椭圆正则性结果意味着期望的正则性假设。

Closability, regularity, and approximation by graphs for separable bilinear forms

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e. energy forms on finite weighted gra…