Introduction to White Noise Analysis (in Russian)
Topics Covered
- white noise analysis
- Gaussian measures
- infinite-dimensional analysis
White noise theory is a small but rich (in terms of the objects it contains) island within the theory of Gaussian measures on infinite-dimensional spaces. It starts from the Bochner-Minlos theorem, which allows one to define the standard Gaussian measure on the space of generalized functions $S'(R)$ via its Fourier transform. With respect to this measure, one can develop a special calculus called white noise analysis, as well as introduce classes of infinite-dimensional generalized functions (Hida distributions). In Hida spaces, Brownian motion coexists with integrals over it, its derivatives of all orders, infinite-dimensional delta functions, and many other interesting objects. Moreover, white noise analysis allows one to define a multiplication operation on Hida generalized functions, which is related to what physicists call renormalization.
In these lectures, I briefly cover the main constructions of white noise theory. All the facts I discuss can be considered standard (if not classical), and we do not touch on any subtle issues of the theory. We start by discussing Gaussian measures on nuclear spaces, explain why Hilbert spaces are too small to support a standard Gaussian measure and what to do about it, and show how to equip $S'(R)$ with a topology that makes it nuclear. Topics that remain largely untouched include infinite-dimensional Malliavin calculus (although we do introduce some concepts from this field), fractional white noise theory, applications of white noise theory to Feynman integrals and SDEs, as well as numerous probabilistic applications.