Abstract
Diffusion models have recently emerged as a powerful class of generative models, achieving state-of-the-art performance in various domains such as image and audio synthesis. While most existing work focuses on finite-dimensional data, there is growing interest in extending diffusion models to infinite-dimensional function spaces. This survey provides a comprehensive overview of the theoretical foundations and practical applications of diffusion models in infinite dimensions. We review the necessary background on stochastic differential equations in Hilbert spaces, and then discuss different approaches to define generative models rooted in such formalism. Finally, we survey recent applications of infinite-dimensional diffusion models in areas such as generative modelling for function spaces, conditional generation of functional data and solving inverse problems. Throughout the survey, we highlight the connections between different approaches and discuss open problems and future research directions.This article is part of the theme issue 'Generative modelling meets Bayesian inference: a new paradigm for inverse problems'.