Syllabus and Course Details
APMA 2812B - Fall 2019
This page constitutes the official course syllabus and may change throughout the course.
Course Information
Time: Fri 11:30am - 2:00pm
Room: Room 118, 170 Hope St.
Description
An introduction to the basic theory of stochastic partial differential equations (SPDE). Topics will likely include (time permitting) Gaussian measure theory, stochastic integration, stochastic convolutions, stochastic evolution equations in Hilbert spaces, Ito's formula, local well-posedness for semi-linear SPDE with additive noise, weak Martingale solutions to 3D Navier-Stokes, Markov processes on Polish spaces, the Krylov–Bogolyubov theorem, the Doob-Khasminskii theorem, and Bismut-Elworthy-Li formula for a class of non-degenerate SPDE.
The intention is to introduce students to the field of stochastic partial differential equations. The focus will be on rigorous mathematical frameworks to describe such equations and the hope is to give students some insight and perspective on what research in this field might involve.
For a more detailed outline of the course topics, see the Course Schedule page.
Prerequisites
The presentation will be largely self contained, but will assume some fundamental knowledge in measure theory, functional analysis, and probability theory at the graduate level. Some familiarity with SDE and PDE is also very helpful, but will not be required.
You are expected to have taken (or an equivalent of):
- Probability (APMA 2630 and APMA2640)
- Functional Analysis or PDE (MATH 1270 or APMA 2230)
- Analysis (MATH 1010 and/or MATH 2210)
Required material and notes
Suggested
There is no official or required textbook for the course. I will be writing and distributing notes as the course progresses and assigning readings from other notes and papers. The latest version can be found posted on the course home page and updated as the course progresses.
In general I will be following the general approach of DaPrato and Zabczyk. I will also be leaning on material in the set of notes by Martin Hairer (which are heavily basic on Da Prato/ Zabczyk):- Stochastic Equation in Infinite Dimensions (2nd Edition) - Guiseppe Da Prato , Jerzy Zabczyk
- An Introduction to Stochastic PDEs - Martin Hairer.
Supplementary
There are also a number of other great resources that I may pull from (or I suggest checking out), but go a bit beyond the scope of the course:
- Mathematics of Two-Dimensional Turbulence - Sergei Kuksin, Armen Shirikyan (mainly focused on stochastic Navier-Stokes)
- Stochastic Partial Differential Equations - Sergey Lototsky, Boris Rozovsky (New book by our esteemed professor. Available for free on Springer Link)
- Gaussian Measures - Vladimir I. Bogachev (Extremely broad monograph on Gaussian measure theory)
I will also be using several supplementary notes from a variety of sources. These will also be listed on the notes page when needed.
Objectives and Expectations
Objectives
By the end of the course students will:
- have developed a foundation in stochastic differential equations and partial differential equations as well as stochastic partial differential equations and be equipped with the tools necessary to begin reading research papers and conducting basic research in the field.
- understand the theory of Gaussian measures on Banach spaces and how it applies to gaussian processes
- understand the well-posedness theory for mild solutions of semi-linear dissipative SPDEs
- understand the stationary distrubutions and the properties that lead to existence and uniqueness
Expectations
The course will have weekly assigned readings from books, notes and fundamental papers. Students will be expected to show up at all lectures and actively participate. There will also be 3 assigned problem sets used to supplement the readings and give students feedback on potential misunderstandings. These assignments are expected to be turned in on time and students are expected to work independently. Near the end of the course there will be a final project that will involve reading and presenting a research paper in the field. There will be no exams.
Grades
Homework
There will be 3 homework problems sets worth 100 pts each. They will be assigned on the website on the assignments page and due dates will also appear on the course schedule page.
The assignments are due in class on the due date listed. Late assignments will not be accepted.
Final Project
There will be a final project that will involve reading or presenting a research paper in the field (among a specified list). Details will be presented at the end of the semester.
Breakdown
| Points | Percentage | |
|---|---|---|
| Homeworks | 300 (3x100) | 75% |
| Final Project | 100 | 25% |
| Total | 400 | 100% |
Grades will be tentatively be assigned based on the following cutoffs. These cutoffs may be subject to change as the course progresses.
| Percentage | Grade |
|---|---|
| 90%-100% | A |
| 80%-89% | B |
| 70%-79% | C |
| 60%-69% | D |
| < 60% | F |
Grade modifiers (+-) may be added to grades within ~1% of the boundary points. Eg: 89% = B+. I reserve the right to assign grade modifiers to student grades as I see fit.
Course Hours
The hours expected to be spent on the course are roughly as follows:
| Activity | Time spent | Total hours |
|---|---|---|
| Course meetings | (13 weeks)*2.5 hrs | 32.5 hrs |
| Reading and study | (13 weeks)*10 hrs | 130 hrs |
| Problem sets | (5 sets)*8 hrs | 40 hrs |
| Final project | (1 project)*10.5 hrs | 10.5 hrs |
| Total | 213 hrs |
Students with special needs
Brown University is committed to full inclusion of all students. Students who, by nature of a documented disability, require academic accommodations should contact the professor during office hours. Students may also speak with Student and Employee Accessibility Services at 401- 863-9588 to discuss the process for requesting accommodations.
Diversity Statement
This course is designed to support an inclusive learning environment where diverse perspectives are recognized, respected and seen as a source of strength. It is our intent to provide materials and activities that are respectful of various levels of diversity: mathematical background, gender, sexuality, disability, age, socioeconomic status, ethnicity, race, and culture.
The University of Maryland provides upon request appropriate academic accommodations for qualified students with disabilities. Please notify me immediately if you require such accommodations (do not wait until the day before the first exam).
English Language Learners
Brown University welcomes students from around the world, and the unique perspectives international students bring enrich the campus community. To empower students whose first language is not English, an array of ELL support is available on campus including language and culture workshops and individual appointments. For more information about English Language Learning at Brown, contact the ELL Specialists at ellwriting@brown.edu.
- Sam Punshon-Smith 2019