Generalised Stochastic Processes and Their Distributions

As was announced in §1.3, (ii), this chapter is devoted to a study of the distributions of generalised stochastic processes. In the finite-dimensional case Bochner’s theorem establishes a one-to-one correspondence between distributions and characteristic functions, and this result has a counterpart in the present context, where the Bochner-Minlos theorem (§3.2) links probability measures on the space of generalised functions with characteristic functionals on the space of test functions. A secondary aim of this chapter is to provide a discussion of the generalised stochastic process white noise from several points of view (§§3.3, 3.4), this serving as background to much of what will be considered in the next chapter.