BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space (Q611159)

The authors introduce the notion of a BV function in a Gelfand triple which extends earlier definitions, e.g., by \textit{L. Ambrosio} et al.\ [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 21, No. 4, 405--414 (2010; Zbl 1206.26014)] or \textit{M. Fukushima} et al.\ [J. Funct. Anal. 174, No.1, 227--249 (2000; Zbl 0978.60088); J. Funct. Anal. 183, No. 1, 245--268 (2001; Zbl 0993.60049)]. The main tool is the theory of quasi-regular (symmetric) Dirichlet forms. The new definition allows to consider the stochastic reflection problem associated with a selfadjoint operator and a cylindrical Wiener process on a convex set \(\Gamma\). In particular, the existence and uniqueness of a strong solution of this problem is established -- at least for convex regular sets \(\Gamma\). Extensions to the non-symmetric case and to the case where \(\Gamma=\big\{f\in L^2(0,1)\;:\;f\geq-\alpha\big\}\), \(\alpha\geq 0\), are given.