Locality of the number of particles operator

We view the number of particles operator N as the infinitesimal generator of the Ornstein-Uhlenbeck semigroup in an abstract Wiener setting. It is shown that if two functions ƒ, g in the domain of N agree a.e. on an open set O then Nƒ = Ng on O. The restriction of A to a large core acts as an infinite dimensional partial differential operator L, and it is shown that L may be defined locally in an L²_loc setting.