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TCC Course: Stochastic PDEs

This course is administered by TCC.

Instructors: Cyril Labbe and Weijun Xu.

Meetings: Tuesdays 2-4pm, from 28th April.

The class on May 12th will be 2-3pm, one hour only.

The class on 16th June (Tuesday) will be moved to Wednesday 17th, 2-4pm.

Prerequisites: Familiarity with probability (with measures) and Brownian motion.

Syllabus:

We give an introduction to the techniques and results in the field of stochastic partial differential equations. The course will consist of two parts.

1. We cover basics of SPDEs, including some motivating examples, Gaussian measures in infinite dimensions, linear (and some semi-linear) SPDEs, space-time regularity of the solutions, etc.

2. In the second part, we focus on the study of some SPDEs that are of particular interest, and reveal some interesting mathematical/physical phenomena behind these equations. Examples include: (1) multiplicative SHE and the positivity of its solution; (2) Approximations of some singular SPDEs converging to the 'wrong' limit; (3) Convergence of some discrete probability models to SPDEs.

Assessment:

The marks for this course will be recorded as Pass, Fail or (in very rare situations) Distinction. Students who wish to take it for credit should write a short essay (5-10 pages) on a topic related to the course. Please contact us to decide the actual topic if you are interested. Please also send an email to graduate dot studies at maths dot ox dot ac dot uk to register.

Notes:

Lecture 1, Lecture 2, Lecture 3, Lecture 4a, Lecture 4b, Lecture 5, Lecture 6

We will discuss this article in some detail in the next two classes (June 2nd and 9th).

References:

We will mainly follow Martin Hairer's notes, which can be found here: http://www.hairer.org/notes/SPDEs.pdf

The following two books are also useful:
Stochastic Equations in Infinite Dimensions, by Da Prato and Zabczyk;
An Introduction to Stochastic Partial Differential Equations, by John Walsh.