Analysis of complex Brownian motion

A theory of generalized functions based on the complex Brownian motion {Z(t) : t ∈ R}, for which each Z(t) is N(0, |t|), is established on the probability space (S′ c,B(S′ c), ν(dz)), where S′ is the dual of the Schwartz space S, S′ c, the complexification of S′, identified as the product space S′×S′, B(S′ c) the Borel field of S′ × S′ and ν(dz) denotes the product measure μ1(dx)μ1(dy). Using the representation of the complex Brownian motion Zt(x, y) = 1 √ 2 (〈x, ht〉+ i〈y, ht〉) ,