Stochastic Partial Differential Equations

News

• 25-09-25: Summaries of your given lectures can be written as a group or individually. In order to help your follow students, please send the summary to us lecturers the day before your first lecture but no later than at the second lecture of your group.
• 25-09-25: Some clarifications on the rules for missed lectures: Under normal circumstances, a student cannot miss more than two weeks during the course period. We strongly recommend to read the missed material until the next lecture and expect to receive a report within a week. The length of the handwritten report should be 2-3 pages. The report should contain the main definitions and results as well as a discussion of what was new and what was challenging for the student.
• 25-09-10: We distributed the topics and fixed the schedule during LP1. The remaining lectures are scheduled when the teaching for LP2 is distributed.
• 25-08-26: First version of the course page online, choodle sent for first meeting (reply asap, latest Friday lunch)

When and where?

• Course code: NFMV019 (GU master MMF500) with registration via FUBAS
• LP1-2 2025/26 (old course pages 2021 and 2017)
• Schedule: see below
• Place: MVL:14 and see below

Who?

• Department of Mathematical Sciences, Chalmers University of Technology & University of Gothenburg
• Teachers: David Cohen, Annika Lang (examiner), and Andrea Papini

Course description

Stochastic partial differential equations (SPDE) are considered in the sense of Itô. We extend the theory of It̂ stochastic differential equations to infinite dimensions by considering SPDE in the framework of Hilbert spaces. This requires the definition of Wiener processes in Hilbert space and the derivation of the stochastic integral in that abstract setting. We show existence and uniqueness of mild solutions to linear SPDE. Mild solutions are then simulated which requires approximation in space, time, and of the infinite-dimensional driving Wiener noise. We prove strong and weak convergence rates of the considered approximation schemes.
The course will start in September and run max until mid-January (LP1-2 2025/26). The schedule will be decided by the participants at an introductory meeting.

Examination

There will be lectures given by the students with summaries and an individual project. Participants are expected to come to all lectures. Missed lectures can be compensated by a written report on the topic of that lecture.
The grading scale comprises Fail (U), Pass (G) (only for MMF500 even (VG)) and successful completion of the course will be rewarded by 7.5 hp credit points.

Main references

• Barth and Lang: Lecture notes (LN) distributed to the participants
• Kovács, Larsson, Lindgren: Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise
• Debussche and Printems: Weak order for the discretization of the stochastic heat equation
• Wang: Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus

Lecture notes (unsorted)

• Levajkovic and Cohen: SPDEs: From Theory to Implementation distributed to the participants
• Kovács and Larsson: Introduction to stochastic partial differential equations
• Bréhier: A short introduction to Stochastic PDEs
• Jentzen: Stochastic Partial Differential Equations: Analysis and Numerical Approximations
• Punshon-Smith: An introduction to SPDE: Lecture Notes
• Hairer: An Introduction to Stochastic PDEs
• Balan: A gentle introduction to SPDEs: the random field approach
• Khoshnevisan: Stochastic Integration and Stochastic Partial Differential Equations: a Tutorial
• Khoshnevisan: Parabolic SPDEs and Intermittency
• Sanz-Solé: An Introductory Course on Stochastic Partial Differential Equations
• Scheutzow: Stochastic Partial Differential Equations
• Jaschek: Probability meets PDEs: The Numerics of Stochastic Heat Equation

Books (unsorted)

• Prévôt and Röckner: A Concise Course on Stochastic Partial Differential Equations
• Liu and Röckner: Stochastic Partial Differential Equations: An Introduction
• Da Prato and Zabczyk: Stochastic Equations in Infinite Dimensions
• Pardoux: Stochastic Partial Differential Equations: An Introduction
• Goodair, Crisan: Stochastic Calculus in Infinite Dimensions and SPDEs
• Peszat and Zabczyk: Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach
• Lord, Powell, and Shardlow: An Introduction to Computational Stochastic PDEs (last chapters)
• Kruse: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations
• Khoshnevisan: Analysis of Stochastic Partial Differential Equations
• Walsh: An introduction to stochastic partial differential equations
• Dalang, Sanz-Solé: Stochastic Partial Differential Equations, Space-time White Noise and Random Fields to appear, arXiv version