Displacement Operator and Generalization of Cameron-Martin-Girsanov Theorem (Mathematical Aspects of Quantum Fields and Related Topics)

Abstract

We study the displacement operators within the framework of quantum white noise calculus. The displacement operators are characterized by implementation problems which are equivalent to linear differential equations associated with the quantum white noise derivatives for white noise operators. Then the displacement operators are applied to study a generalization of the Cameron-Martin-Girsanov theorem. More precisely, we prove that the affine transform, with an isometric dilation and a regular drift, of a Brownian motion is again a Brownian motion with respect to a new probability measure which is derived explicitly in terms of the displacement operators.