A Seminar on Brownian Motion

organizer : Luokai Li

date : July, 2024.

time : 18:30-21:00, 20 minutes for breaks.

prerequisites : real analysis, probability.

main reference : [PY] Peter Morters and Yuval Peres, Brownian Motion.

location : 2-A312

Content: [Time]topic.ref.speaker.

Part I: Preliminary Learning for Brownian Motion.
[7.1] Construction of Brownian motion. Introduction. [PY]1.1. Luokai Li.
[7.2] Regularity of B.M.'s path: continuity and nondifferentiability.[PY]1.2-1.3. Siyu Chen.
[7.3] Brownian motion as a Markov process.[PY]2.1-2.2. Luokai Li.
[7.4] Brownian motion and martingales.[PY]2.3-2.4. Luokai Li.

Part II: Potential Theory in Probability.
[7.7] Harmonic functions. Recurrence and transience of Brownian motion. [PY]3.1-3.2. Chenhao Zhao.
[7.8] Green's functions and the Dirichlet problem.[PY]3.3-3.4&8.1. Junjie Cao.
[7.9] Boundary regularity.[PY]8.2-8.4. Junjie Cao.

Part III: Several Related Topics.
[7.11] The law of the iterated logarithm and the arcsine law for random walk and Brownian motion. [PY]5.1&5.4. Jingzhe Yang & Luokai Li.
[7.12] Skorohod embeding and Donsker's invariance principle.[PY]5.3, Siyu Chen.
[7.13] Ito integration and its application.[PY]7.1&7.4, Junjie Cao.

Part IV: Hausdorff Dimension of Random Fatals.
[7.16] Hausdorff dimension and the mass distribution principle.[PY]4.1-4.2. Chenhao Zhao.
[7.17] The energy method and Frostman's lemma.[PY]4.3-4.4. Chenhao Zhao.
[7.18] Hausdorff dimension of intersection of paths.[PY]9.1 Yukun Lin.
[7.19] Hausdorff dimension of level sets of Brownian motion. [PY]9.3-9.4 Yukun Lin.

Part V: A Brief of Some Advanced Topics.
[7.22] 2-dimensional Brownian motion and Shramnn Lowewner evolution(SLE).[PY]7.2,11.1-11.4. Luokai Li.
[7.23] Fractional Brownian Motion. Luokai Li.
[7.24] Directed polynomial model and stochastic heat equation. Junjie Cao.
[7.25] Ergodicity of Lorentz system. Luokai Li.