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Selected Topics in Partial Differential Equations II

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

Part I of this course is a prerequisite for Part II.

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 Time Topic
Nov 25 8:45 - 10:00 Background material in probability theory (Giovanni Zanco)
Nov 27 10:15 - 11:30 Space-time regularity of the stochastic heat equation
Nov 30 14:00 - 15:15 Stochastic reaction-diffusion equations
Dec 4 10:15 - 11:30 Stochastic Burgers equations
Dec 7 14:00 - 15:15 Stochastic heat equation as a rough path
Dec 9 10:15 - 11:30 Wiener chaos, hypercontractivity, and equivalence of moments I
Dec 18 14:00 - 15:15 Wiener chaos, hypercontractivity, and equivalence of moments II
Jan 11 10:15 - 11:30 Rough stochastic Burgers equations
Jan 13 10:15 - 11:30 Introduction to regularity structures
Jan 18 10:15 - 11:30 Regularity structures, models, and modelled distributions
Jan 20 10:15 - 11:30 The Reconstruction Theorem and its consequences
Jan 25 10:15 - 11:30 Schauder estimates
Jan 27 10:15 - 11:30 KPZ equation

Homework

File Due Date
Sheet 4 January 25, 2015
Sheet 5 February 3, 2015

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