Instructor: Prof. Robert (Bob) Haslhofer
Contact Information: roberth(at)math(dot)toronto(dot)edu, BA6208
Lectures: Monday 10--12 and Wednesday 11--12 in BA6180
Office Hours: Monday 9--10 in BA6208 (or by appointment)
Course Description: There has been a long-standing interaction between probability and geometry, witnessed most strikingly by random motion on manifolds. Classical examples for this include the probabilistic proof of the Atiyah-Singer index theorem and Driver's integration by parts formula on path space. More recently, it has been discovered that Brownian motion can be used to characterize solutions of the Einstein equations and of Hamilton's Ricci flow via certain sharp estimates on path space.
In this course we will provide a general introduction to Brownian motion on manifolds and some of its applications. In the first half of the course we will develop the classical theory of Brownian motion in Euclidean space and on manifolds. In the second half of the course, we will consider applications, including spectral gap estimates via coupling, a probabilistic proof of the Chern-Gauss-Bonnet theorem, integration by parts on path space, and the recent characterizations of solutions of the Einstein equations.
Prerequisites: Some basic background in probability and geometry is recommended, i.e. it is helpful to be (somewhat) familiar with fundamental notions such as Markov processes, martingales, Riemannian manifolds, and curvature.
Grading Scheme: attendance and participation 20%, homework 30%, final exam 50%
M. do Carmo: Riemannian geometry, Birkhauser, 1992
B. Driver: A Primer in Riemannian Geometry and Stochastic Analysis on Path Spaces, FIM notes, 1995
P. Morters, Y. Peres: Brownian motion, CUP, 2010
R. Haslhofer, A. Naber: Ricci Curvature and Bochner Formulas for Martingales, CPAM, 2018
E. Hsu: Stochastic Analysis on Manifolds, AMS, 2002
S. Lalley: Lecture notes on Stochastic differential equations, 2016
L. Rogers, D. Williams: Diffusions, Markov Processes and Martingales, CUP, 1994
D. Stroock: An Introduction to the Analysis on Paths on a Riemannian Manifold, AMS, 2000
Handwritten Lecture Notes: Lecture Notes
Topics covered:Brownian motion in Euclidean space: Levy's construction (Peres 1.1), Continuity properties (Peres 1.2), Markov property (Peres 2.1), Martingale property (Peres 2.4), Harmonic functions (Peres 3.1), Stochastic integrals (Peres 7.1), Dirichlet problem (Peres 8.1), Stochastic differential equations (Lalley). Brownian motion on manifolds: Review of basic concepts from Riemannian geometry (do Carmo), Frame bundle (Hsu 2.1), Eells-Elworthy-Malliavin construction (Hsu 3.2), Heat kernel (Hsu 4.1), Feynman-Kac formula (Hsu 7.2). Applications: Spectral gap via coupling (Hsu 6.7), Probabilistic proof of the Chern-Gauss-Bonnet theorem (Hsu 7.3), Driver's integration by parts formula on path-space (Hsu 8.4), Bochner formula for martingales on path-space (Haslhofer-Naber), Characterizations of solutions of the Einstein equations and the Ricci flow (Haslhofer-Naber)