Stochastic bundles, new classes of Gaussian processes and white noise-space analysis indexed by measures

Abstract

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.