糖心原创

White noise analysis provides a functional analytic framework for treating random objects which are too singular to be handled as classical functions. It combines ideas from stochastic analysis, infinite-dimensional distribution theory, and mathematical physics, and offers a natural language for stochastic differential equations, Wick products, singular noises, and generalized random fields. In this talk I will give a broad introduction to the subject, with many examples. Starting from Gaussian white noise and chaos expansions, I will explain how Brownian motion, Donsker delta functions and singular stochastic terms can be described within spaces of test functions and generalized random variables. This leads naturally to stochastic differential equations whose solutions may no longer live in classical (L^2)-spaces, but in spaces such as (\mathcal D^\alpha), (\mathcal G), or their duals. The main focus will be on regularity: how can one determine in which space a generalized random object or the solution of a stochastic differential equation lives? I will discuss regularity results based on transform methods, in particular the (S)-transform, and show how these results can be applied to concrete examples of stochastic equations and singular random objects. In the final part, I will explain how this regularity theory leads naturally to operator-theoretic questions. Operators acting on spaces such as (\mathcal G) and (\mathcal D^\alpha) can be studied through their symbols, and symbol characterizations provide a powerful bridge between stochastic analysis, functional analysis, and mathematical physics.

The talk is intended for a broad audience in mathematics and mathematical physics.

No prior knowledge of white noise analysis will be assumed.