带测度数据的泊松方程、重构公式与过程的Doob类
Poisson equation with measure data, reconstruction formula and Doob classes of processes
AI总结：
针对带对称暂态正则Dirichlet型关联算子与有界Borel测度的泊松方程Dirichlet问题，提出新函数空间区分不同测度的解，给出其解析与概率刻画，并证明解的极大值行为重构公式。
摘要：
我们研究方程的Dirichlet问题，该方程涉及与对称暂态正则Dirichlet型相关的一般算子，且方程右端为有界Borel测度。我们引入了一种依赖于该Dirichlet型的新函数空间，可区分带漫测度与带一般Borel测度的解。该新空间既可通过与底层算子相关的Poisson核进行解析刻画，也可借助与该算子自然关联的过程的Doob类（D类）的概念从概率角度刻画。我们还证明了一个重构公式，利用与底层Dirichlet型相关的carré du champ算子和跳测度，描述了解在其取值极大的集合上的行为。
英文摘要：
We consider the Dirichlet problem for equation involving a general operator associated with a symmetric transient regular Dirichlet form and bounded Borel measure on the right-hand side of the equation. We introduce a new function space (depending on the form) which allows us to distinguish between solutions with diffuse measure and with general Borel measure. This new space can by characterized analytically in terms of the Poisson kernel associated with the underlying operator or probabilistically by using the notion of Doob class (D) of processes naturally associated with the operator. We also prove a reconstruction formula describing, in terms of the carré du champ operator and jump measure associated with the underlying form, the behaviour of the solution on the set where it is very large.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
