Admissible vectors and their associated Dirichlet forms
Let \(E\) be a locally convex topological vector space and \(\mu\) a probability measure on \(E\). Let \(K=\{k_ n\}\) be a sequence of linearly independent admissible vectors in \(E\). Then a symmetric form \[ {\mathcal E}(u,v)=\int_ E \sum_ n {{\partial u} \over {\partial k_ n}}(\omega) {{\partial v} \over {\partial k_ n}} (\omega)\mu(d\omega) \] is well-defined on the space of equivalence classes in \(L^ 2(E,\mu)\) of functions representable as \(u(\omega)=f(l_ 1(\omega), l_ 2(\omega),\dots,l_ n(\omega))\) where \(f\in C_ b^ \infty(\mathbb{R}^ n)\) and \(l_ i\in E^*\). This form is closable in \(L^ 2(E,\mu)\) and its closure is a Dirichlet form denoted by \(({\mathcal F},{\mathcal E})\). Moreover, the form admits a square operator of the field \(\Gamma\). The main result of the paper is the following energy-image density property: Let \(u_ i\in{\mathcal F}\), \(i=1,2,\dots,m\). Then the image measure by \(u=(u_ 1,u_ 2,\dots,u_ m)\) of the measure \(\text{det}(\Gamma(u_ i,u_ j))\cdot\mu\) is absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}^ m\). Steps in proving this result are a disintegration formula for the expectation of nonnegative function on \(E\) using admissible vectors \(\{k_ n\}\), and formulation of the proper concept of differentiability of vectors in \({\mathcal F}\) in terms of the approximate differentiability.
Dirichlet forms on topological vector-spaces
scientific article; zbMATH DE number 109446