Singular stochastic partial differential equations (SPDEs) arise naturally in the study of random interfaces, quantum field theory, and non-equilibrium statistical mechanics. Over the past decade, a series of breakthroughs has led to a robust solution theory for a large class of such equations. In particular, the introduction of regularity structures by Martin Hairer has provided systematic frameworks to interpret and solve equations that were previously beyond reach such as the Phi4_3 equation. In this talk, I will present an introduction to the analysis of invariant measures for stochastic dynamics, starting from classical examples such as Brownian motion and stochastic differential equations. I will then discuss how these ideas extend to infinite-dimensional settings, and introduce singular SPDEs arising from quantum field theory as an example.
