Starting from a fixed measure space \((X, {\mathcal {F}}, \mu )\) , with \(\mu \) a positive sigma-finite measure defined on the sigma-algebra \({\mathcal {F}}\) , we continue here our study of a generalization \(W^{(\mu )}\) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized Brownian motion is a centered Gaussian process \(W^{(\mu )}\) , indexed by the elements A in \({\mathcal {F}}\) of finite \(\mu \) measure, and with covariance function \(\mu (A\cap B)\) . The purpose of our present paper is to make precise and study the corresponding white-noise process, i.e., a point-wise process which is indexed by X, and which arises as a generalized \(\mu \) derivative of \(W^{(\mu )}\) . A key tool in our definition and analysis of this pair is a construction of three operators between the underlying Hilbert spaces. One of these operators is a stochastic integral, the second is a gradient associated with the measure \(\mu \) , and the third is a mathematical expectation in the underlying probability space. We show that, with the setting of families of processes indexed by sets of measures \(\mu \) , our results lead to new stochastic bundles. They serve in turn to extend the tool set for stochastic calculus.
