2017 Fall: Calculus 4 (MATH V1202) Section 002,
Time: Tu, Th, 2:40pm-3:55pm. Location: 203 Mathematics Building.
My Office: Math 614.
Topics course taught in 2017 Spring:
Topics in Stochastic PDE (MATSGR 8260)
Time: MW 2:40pm-3:55pm.
1st meeting (Jan 18): Overview of the course. Slides
Jan 23 and 25: Review of Ito Calculus, Introduction to Rough paths.
Jan 30 and Feb 1: Rough paths. (Reading: Chapter 4 and 8 of book by Friz and Hairer)
Feb 6 and Feb 8: Weak solutions to SDEs and Stroock-Varadhan’s martingale problems.
(Reading: Section 5.3 and 5.4 of book by Karatzas-Shreve.)
Feb 13 and Feb 15: White noise, stochastic heat equation (SHE) with additive noise.
Feb 20 and Feb 22: SHE with additive noise (cont.), introduction to semigroup method
Feb 27 and March 1: Semigroup method, examples: Phi^4 in 1D, stochastic Navier-Stokes in 2D with colored noise (for more details about semigroup method please read these lecture notes by M.Hairer)
Mar 6 and 8: stochastic heat equation (SHE) with multiplicative noise: mild solution and chaos solution ( based on Section 1,2,10 of MSRI lecture notes by Corwin (and TA: Kuan))
Mar 20 and 22: Weakly asymmetric simple exclusion process scales to SHE ( Original paper by Bertini and Giacomin)
April 3 and 5: Perturbation theory and Feynman diagrams in quantum field theory and SPDE
April 10: Da Prato - Debussche method for Phi4 equation in two dimensions ( Original paper ; For a very nicely written review: Lecture 2, 3 in the notes written by Chandra and Weber )
April 12: Regularity structures, models, modeled distributions. (Suggested reading for regularity structures: Chapter 13,14 of book by Friz and Hairer)
April 17: Examples: polynomial regularity structures and Holder spaces; rough paths as regularity structures and controlled rough paths as modeled distributions.
April 19: Reconstruction theorem. Products on modeled distributions.
April 24: Schauder estimates on modeled distributions. Building a regularity structure for a subcritical SPDE (Phi^4 in 3D)
April 26: Fixed point in space of modeled distributions; canonical model and renormalized model.
Stochastic Integrals and solutions to SDE via Rough Paths: Chapter 4,8 of book by Friz and Hairer
``An Introduction to Stochastic PDEs” by Hairer download
Stochastic Heat Equation with Multiplicative Noise: Section 1,2,10 of MSRI lecture notes by Corwin (and TA: Kuan)
Previous courses at Columbia:
2015 Fall: Calculus 1 (MATH 1101) Sections 3 and 5.
2016 Spring: Calculus 4 (MATH V1202) Section 2.
2016 Fall: Calculus 4 (MATH V1202) Section 1 and 2.