Mathematics > Probability
arXiv:2605.14630 (math)
[Submitted on 14 May 2026 (v1), last revised 30 Jul 2026 (this version, v4)]
Abstract:These notes have been written for a series of lectures to be given at the 44th Finnish Summer School on Probability and Statistics in Lammi, Finland, from 25th to 29th May, 2026. They contain an introduction to Wiener chaos decomposition in finite dimension, a construction of Gaussian fields on the torus, including white noise and the Gaussian free field, and applications to the $\Phi^4$ model. They do not cover other important aspects of the topic, such as stochastic integration, stochastic PDEs and Malliavin calculus.
| Comments: | 79 pages, 6 figures. Some minor typos corrected |
| Subjects: | Probability (math.PR); Mathematical Physics (math-ph); History and Overview (math.HO) |
| MSC classes: | 60-01, 60G15 (primary), 81S20, 82C28 (secondary) |
| Cite as: | arXiv:2605.14630 [math.PR] |
| (or arXiv:2605.14630v4 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.14630 arXiv-issued DOI via DataCite |
Submission history
From: Nils Berglund [view email]
[v1]
Thu, 14 May 2026 09:41:50 UTC (176 KB)
[v2]
Sun, 17 May 2026 16:50:50 UTC (176 KB)
[v3]
Wed, 20 May 2026 20:52:52 UTC (176 KB)
[v4]
Thu, 30 Jul 2026 09:33:10 UTC (176 KB)
Current browse context:
math.PR
Bookmark
Bibliographic Tools
Bibliographic and Citation Tools
Bibliographic Explorer Toggle
Code, Data, Media
Code, Data and Media Associated with this Article
Demos
Demos
Related Papers
Recommenders and Search Tools
About arXivLabs
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.