Some of future directions in white noise analysis (Q2735184)

White noise analysis is a basic field of stochastic analysis and serves as an important tool of both the classical and the quantum information theory. White noise can be regarded as a time derivative of a standard Brownian motion, but the mathematical setup is the following: if \(E\) is a nuclear space of test functions, and \(E^*\) its dual space, then from Bochner-Minlos theorem it follows that for a continuous positive definite functional \(C(\xi)= \exp[-1{1\over 2}\|\xi\|^2]\), on \(E\), there exists a measure \(\mu\) on \(E^*\). The measure space \((E^*,\mu)\) is called the white noise. Then the complex Hilbert space \((L^2)= L^2(E^*,\mu)\) is called Fock space.NEWLINENEWLINENEWLINEAn infinite-dimensional analogue of the Schwartz space of generalized functions can be formed by starting from the Hilbert space \((L^2)\). The analogue is a triple \((S)\subset (L^2)\subset (S)^*\). The spaces \((S)\) and \((S)^*\) are called the space of test functionals and the space of generalized functionals of white noise, respectively. The author proposes two important directions in the investigations of generalized stochastic processes in \((S)^*\): 1. An investigation of the complexity of random phenomena expressible as functionals of white noise. 2. The analysis of random fields indexed by a multidimensional parameter or a certain kind of manifold in a space. Some results in these directions are given.NEWLINENEWLINEFor the entire collection see [Zbl 0924.00055].