A Gaussian (Hilbert) space is a vectorial space composed of random variables built from centred Gaussian. They have a rich structure and compose the core of various probabilistic theories, like the stochatic integration (e.g. Itô integration). These spaces have a probabilistic but also analytic aspect throught Hibert space theory. there is a transcription of Gaussian space properties into Hilbert space properties and inversely, this gives at the original perpective of Gaussian spaces non-trivial impacts in analysis. There is also a between Gaussian spaces and objects of quantum field theory, for example Fock spaces.
In this EA, we will begin with some reviews of measure theory. We will then define and give the first Gaussian spaces to quickly reach the Wiener chaos decomposition theorem, and then to the Wick product. We will present the connection between Gaussian spaces and Hilbert symmetric products of spaces. Finally, we will cover applications of Gaussian space notion, like stochastic integration.
Required level
Elements of Hilbertian analysis and probality theory will help understanging the covered concepts
Bibliograph
Svante Janson "Gaussian Hilbert Spaces"
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Course language: French or English according the request
- Teaching coordinator: Dubach Guillaume