Characterization Theorems for Differential Operators on White Noise Spaces

We characterize through their action on stochastic exponentials the class of white noise operators which are derivations with respect to both the point-wise and Wick products. We define the class of second order differential operators and second order Wick differential operators and we characterize the white noise operators belonging to both classes. We find that the intersection of these two classes, in the first and second order cases, is identified by a skewness condition on the coefficients of the differential operator. Our technique relies on simple algebraic properties of commutators and on the Gaussian structure of our white noise space. Our approach is suitable to study differential operators of any order