Compactifications of Banach spaces and construction of diffusion processes
Classical Dirichlet forms \(({\mathcal E},D({\mathcal E}))\) on topological vector spaces \(E\) in the sense of \textit{S. Albeverio} and \textit{M. Röckner} [J. Funct. Anal. 88, No. 2, 395-436 (1990; Zbl 0737.46036)] are studied. Here \(E\) is a (not necessarily separable) Banach space which is the topological dual of a separable normed space. The main result states that there exists a Hunt process on \(E\) associated with \(({\mathcal E},D({\mathcal E}))\). Lemma 10 which claims (cf. also Theorem 7) that the \(\sigma\)- algebras \(B_ 2\) and \(B_ 1\) generated by the norm topology respectively the \(\text{weak}^*\)-topology on \(E\) coincide, is however incorrect since this is known to be false for \(E = l^ \infty\) (= space of bounded sequences) which is a possible choice for \(E\) according to the above assumptions. Since Lemma 10 is used to prove the main result, an additional assumption like \(E\) separable, appears to be necessary. In the case of arbitrary separable Banach spaces (not necessarily duals), however, the existence of even a diffusion process associated with \(({\mathcal E},D({\mathcal E}))\) has been proved before by \textit{B. Schmuland} [Stochastics Stochastics Rep. 33, No. 1/2, 75-90 (1990; Zbl 0726.31008)]. On the other hand the methods are somewhat different and of their own interest. Surprisingly Schmuland's work, which is well-known among people working in this field, is not quoted in this paper.
scientific article; zbMATH DE number 713754