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Selected Topics in Partial Differential Equations I

Instructor: Jan Maas

Teaching Assistant: Giovanni Zanco

Description

 

Stochastic partial differential equations (SPDEs) are used to model a wide variety of physical systems subject to randomness or noise. In many interesting situations, solutions to SPDEs are very irregular, which makes their analysis particularly challenging. In the last couple of years, spectacular progress has been obtained through the use of "rough paths theory" and the development of the theory of "regularity structures", for which M. Hairer has been awarded a Fields Medal. In this course we aim to give a gentle introduction to SPDEs with a special emphasis on these recent developments.

Prerequisites

Background knowledge in analysis and/or probability theory will be helpful but is not required. 

Literature

* P. Friz and M. Hairer, A course on rough paths
* M. Hairer, An introduction to stochastic PDEs
* M. Hairer, Introduction to Regularity Structures
* M. Hairer, Regularity structures and the dynamical Φ_4^3 model


Requirements/Exams

There will be homework exercises during the semester and an oral exam at the end of the course.

Credits

 3


Final Grade

The final grade is based on homework (50%) and the oral exam (50%).

 

 

Schedule (subject to change)

Date Topic
Oct 6 Introduction
Oct 8 Young integration
Oct 13 Solving Young ODEs; definition of a rough path
Oct 15 Controlled rough paths
Oct 20 Sewing lemma; integration of controlled rough paths
Oct 22 Fundamental theorem of calculus, Itô-Stratonovich correction, and Itô formula for rough integration
Nov 3 Brownian motion; Kolmogorov continuity criterion
Nov 5 Brownian motion as a rough path; stochastic differential equations
Nov 10 Stochastic heat equation

Homework

File Due Date
Sheet 1 October 27, 2015
Sheet 2 November 10, 2015
Sheet 3 December 15, 2015

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