Introduction to SPDEs from Probability and PDE

TL;DR

This paper provides an introductory overview of stochastic partial differential equations (SPDEs), focusing on the variational method, functional analysis, and pathwise approach, suitable for learners and researchers new to the field.

Contribution

It offers a comprehensive exposition of the variational method for monotone and coercive SPDEs, integrating functional analysis and stochastic analysis in a pedagogical manner.

Findings

Clarifies the variational approach to SPDEs

Includes foundational functional analysis and operator theory

Discusses the pathwise approach to SPDEs

Abstract

Lecture notes accompanying an 8hr hour mini-course on SPDE given at Bo\u{g}azi\c{c}i University, Istanbul in June/July 2025. They are based on earlier notes of a shorter mini-course given at the University of Oxford in 2021. The main focus of these notes is on an exposition of the variational method for monotone and coercive SPDE. A recap of the necessary functional analysis, operator theory and stochastic analysis on Hilbert spaces is included, with additional references. The final chapter contains a discussion of the pathwise approach.