Abstract
Complex Gaussian systems are the most important families of complex-valued random variables, and this chapter begins by presenting the general background to such systems. We then observe that complex white noise, the white noise of Chapter 3 complexified, is a complex Gaussian system. Functionals of complex white noise may also be viewed as functionals of complex Brownian motion and the analysis of such functionals is not only useful in the study of stochastic processes, but is also widely used in applications. Consequently it is an important problem to express these in a concrete form and to develop ways of analysing them (§§6.2–6.3). On the other hand, the infinite dimensional unitary group arises naturally in the study of the probability measure determined on the complex-valued (generalised) function space by complex white noise. This unitary group plays the same role here as the infinite-dimensional rotation group did in describing properties of white noise (§7.1). For added interest, this group is intimately related to aspects of the theory of differential equations and quantum mechanics (§7.5–7.6).
Published Version
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