This semester, we will spend time defining and developing the basic properties of Brownian motion, and then move on to its connections to harmonic analysis (i.e solving the Dirichlet problem), harmonic measure, and develop some stocahstic calculus at the end time permitting.

- [BP] "Fractals in Probability and Analysis", by Chris Bishop and Yuval Peres, our primary reference. Link
- [GM] "Harmonic Measure", by John Garnett and Donald Marshall.
- [La1] "Notes on Probability", by Greg Lawler. A quick intro to probability theory. The first three sections are recommended for those not familiar with probability, and we will draw more from it when needed. Link
- [La2] "Stochastic Calculus: An Introduction with Applications", by Greg Lawler, for the last week or two. Link

Date |
Speaker |
Topic |
Reading |
---|---|---|---|

Jan. 31 |
Jack Burkart |
Introduction and Definition |
[BP] 6.1-6.2 |

Feb. 7 |
Jack Burkart |
Levy's Construction of Brownian Motion |
[BP] 6.2 |

Feb. 14 |
Matt Dannenberg |
Scaling Relations, Nowhere Differentiability, Holder Continuity |
[BP]6.3 |

Feb. 21 |
Ben Sokolowsky |
Reflection, Conformal Invariance, The Strong Markov Property |
[BP] 6.6,7.9 |

Feb. 28 |
Silvia Ghinassi |
Dimension Results |
[BP]6.4, 7.1 |

Mar. 7 |
Jack Burkart |
Zeros of Brownian Motion |
[BP] 6.10 |

Mar. 14 |
Spring Break | ||

Mar. 21 |
TBA |
Law of the Iterated Logarithm |
[BP] 7.2 |

Mar. 28 |
TBA |
Connections to Harmonic Functions pt. 1 |
[BP] |

Apr. 4 |
TBA |
Connections to Harmonic Functions pt. 2 |
[BP] 7.5-7.7, 7.10 |

Apr. 11 |
TBA |
Martingales and Conditional Probability |
[La1] |

Apr. 18 |
TBA |
Harmonic Measure and Kakutani's Theorem |
[GM], [BP] 7.9 |

Apr. 25 |
TBA |
Intro to Stochastic Calculus pt. 1 |
[La2] |

May 1 |
TBA |
Intro to Stochastic Calculus pt. 2 |
[La2] |