Topics in Gaussian Wiener chaos expansion

TL;DR

This paper provides an introductory overview of Wiener chaos decomposition, Gaussian fields, and applications to the $\

Contribution

It offers a focused lecture-based introduction to Gaussian Wiener chaos expansion and its applications, excluding stochastic calculus topics.

Findings

Introduction to Wiener chaos decomposition in finite dimension

Construction of Gaussian fields on the torus including white noise and Gaussian free field

Application to the $\

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.