En introduktion till stokastiska partiella differentialekvationer (7,5 hp)
Lecturer: Eric Hall, ejhall@kth.se
Place: 3424, Lindstedsvägen 25 ← note new place
Time: Tuesdays 10:00-12:00 ← note new time
We will follow the lecture notes A Primer on Stochastic Partial Differential Equations by D. Khoshnevisan, given at summer school at the University of Utah in 2006, which are loosely based on the chapters 1-3 of the monograph An Introduction to Stochastic Partial Differential Equations by J. Walsh. The lecture notes are concise and touch on interesting topics at the very heart of the study of SPDEs.
The course will give existence and uniqueness results for the mild solution to a (nonlinear) stochastic heat equation and will explore regularity properties of the solution. If time permits, an example of a (deterministic) heat equation with random initial data will also be considered. Background concepts will be covered as needed.
First lecture will be held in room 3733 (Lindstedsvägen 25, 7th Floor, KTH Mathematics) on Friday 2014-01-31 from 15:00–17:00.
The notes by Khoshnevisan contain a number of very good exercises. As these form an integral part of the notes, the expectation is to work through the problems listed below as `due' during the associated lecture. Please prepare your solutions (individually or in small groups) to hand in before we work through them together. Exercises in bold will count towards the assessment grade. A tentative schedule is as follows.
| Date | Content | Exercises due |
|---|---|---|
| 01/31 | Introduction, motivating examples | --- |
| 02/07 | Gaussian random vectors, Gaussian processes | 2.1, 3.1, 3.2, 3.3, 3.4, 3.6, 3.8 |
| 02/25 | Gaussian processes cont., regularity of processes | 3.12, 3.14, 3.15, 3.16, 3.18, 3.19, 3.20 |
| 03/11 | Regularity of processes, Kolmogorov Continuity Theorem | 4.2, 4.5, 4.7, 4.8 |
| 03/25 | Regularity of processes cont. | 4.9, 4.11, 4.12 |
| 04/01 | Martingales, martingale measures (white noise example) | 5.2, 5.4, 5.5 |
| 04/08 | Martingale measures | 5.7, 5.8, 5.9, 5.11, 5.12, 5.13 |
| 04/15 | Martingale measures cont. | 5.16, 5.17, 5.22, 5.24, 5.28, 5.29 |
| 04/22 | Nonlinear heat equation | 6.1, 6.2, 6.9, 6.10 |
| 04/29 | PDE with random data, wrapping up | 7.2, 7.5, 7.6, 7.7, 7.8 |
This document was last modified on:
.