Some calculations on the conditional densities of well-admissible measures on linear spaces
The paper deals with so-called well-admissible measures on a topological vector space \(E\). The notion of well-admissibility was first introduced in [\textit{S. Albeverio}, \textit{S. Kusuoka} and \textit{M. Röckner}, J. Lond. Math. Soc. 42, 122-136 (1990; Zbl 0667.31010)] and has applications to many models of Dirichlet forms; however its definition is too long to be given here. It is known that when a probability measure on \(E\) is well- admissible in a direction \(k \in E\), the conditional law in the direction \(k\) given the other directions is absolutely continuous with respect to the Lebesgue measure. The author proves that its density function is differentiable and calculates its derivatives. The results are applied to compute the capacity of the zero set of the conditional density function, and sufficient conditions are obtained ensuring that the space of bounded smooth functions is dense in a Dirichlet space.
zero set of the conditional density function
scientific article; zbMATH DE number 554339
scientific article