Closure of the set of diffusion functionals -- the one-dimensional case
The authors investigate the closure with respect to Mosco or \(\Gamma\)-convergence of the set of diffusion functionals in the one dimension case. In the one dimensional case it is a fact commonly accepted that any limit remains local. In article under review this fact is rigorously established. The authors consider the different notions of locality and prove that none of them are closed for the Mosco-convergence. A new definition of locality is introduced and a representation theorem for any local Dirichlet form is obtained. The representation theorem plays a crucial role for the proof of the closure theorem. It states that the closure of the set of diffusion functional coincides with the set of local Dirichlet forms.
scientific article; zbMATH DE number 5294535