Uniqueness of Nelson's diffusions
The paper is a continuation of the previous work [J. Funct. Anal. 153, No. 2, 276-319 (1998; Zbl 0913.35035)] on uniqueness of Schrödinger operators restricted to the domain. The present paper addresses the problem whether a generator of the Nelson diffusion, defined on \(C^\infty_0 (D)\) where \(D\subset R^d\) is an open domain, generates a unique semigroup in \(L^1 (D,\varphi^2 dx)\). The central result of the paper is Theorem 2.1 which sets a sufficient and necessary condition for the Nelson generator to be closable. That implies that its extension generates a unique semigroup of class \(C_0\) on \(L^1\). The nonuniqueness of the semigroup in \(L^2(D, \varphi^2dx)\), in the presence of nodal sets (unattainable barriers), is established in Section 4.
scientific article; zbMATH DE number 1339334