Invitation to white noise calculus

Abstract

In recent years, starting from some early work of P. Lévy, Professor T. Hida has developed a theory of generalized Brownian functionals or, as we prefer to call it, white noise calculus ([2], [7]). Hida's main idea is to treat the derivative of the Brownian motion $\{ \dot B(t)\}$ as a complete orthonormal system (c.o.n.s.) in the Schwartz space S* and use it for the analysis of nonlinear functionals of white noise.

In this paper, I present a one-dimensional version of a reformulation of Hida's theory due to I. Kubo and myself [5]. We hope that it gives a simple description of the white noise — or causal-calculus.