Lec 1 | 04 Aug | Overview | |

Lec 2 | 06 Aug | Sec. 1 Gaussian random variables. Sec. 2 Definition of Brownian motion and Wiener measure, the space C[0,1] | |

Lec 3 | 11 Aug | Sec. 3 Levy's construction of Brownian motion | Wiener's original paper^{1} available online for the first time! |

Lec 4 | 13 Aug | Sec 4 Continuity properites I - Nowhere Hölder ½+ε (proof of Erdös-Taylor-Kakutani) | |

Lec 5 | 18 Aug | Sec. 4 [cont'd] Dvoretsky's theorem and hence a second proof of Paley-Wiener-Zygmund via branching processes | Resurrected original proof of Paley-Wiener-Zygmund! ^{2} |

Lec 6 | 20 Aug | Sec. 5 BM is Hölder ½-ε (original proof of P-W-Z). | Handwritten notes |

Lec 7 | 25 Aug | Sec. 6 Completion and Joint measurability. Sec. 7 Invariance properites of BM. (scaling, time revesal, rotation) | |

Lec 8 | 27 Aug | Sec. 8 Filtrations, Stopping times and Markov property. | |

Lec 9 | 01 Sep | Sec. 8 [cont'd] Markov property w.r.t. enlarged filtrations. Sec. 9 Applications of Markov property - Blumenthal's and Kolmogorov's 0-1 laws, Oscillations of BM near t=0 and t=∞ | |

Lec 10 | 03 Sep | Sec. 9 [cont'd] Local maxima/minima of BM. Sec. 10 Stopping times and strong Markov property. | |

Lec 11 | 08 Sep | Sec. 10 [cont'd] Strong Markov property. | A note on measurability |

Lec 12 | 10 Sep | Application of SMP. Sec. 11 Gambler's ruin. Sec. 12 Dirichlet problem. | A note |

Lec 13 | 15 Sep | Sec. 13 [cont'd] Dirichlet problem and Brownian motion Sec. 14 Annular region - recurrence and transience. | |

Lec 14 | 17 Sep | Sec. 15 Reflection principle, running maximum and First passage times. | |

Lec 15 | 22 Sep | Sec. 16 Levy's identity: M-B has the same law as |B| | Beautifully written Lecture notes 1-15 by Aatira!^{3} |

Lec 16 | 24 Sep | Sec. 17 Stochastic integration of L^{2} functions and "What is a Gaussian Hilbert space?" | |

Lec 17 | 29 Sep | Fractals. Sec. 18 Minkowski and Hausdorff dimensions, Energy of a measure. Examples. | Sierpinski gasket, Sierpinski carpet |

Lec 18 | 01 Oct | Sec. 18[Cont'd] Energy of a measure and Capacity. | A long list of examples |

Lec 19 | 06 Oct | Sec. 18[Cont'd] dim_{ε}≤dim_{H}≤dim_{M}. Examples (Cantor sets, Numbers with given frequency of digits) | Aatira's notes for Lectures 16-19 |

Lec 20 | 08 Oct | Sec. 18[Cont'd] Examples. Numbers with given frequency of digits. | Completed proof. |

Lec 21 | 13 Oct | Sec. 19 Random fractals - graph, image and zero set of BM | |

Lec 22 | 15 Oct | Sec. 19 [cont'd] Zero set of 1-dim BM. Sec. 20 Recap of Martingales | Sumary of essential results on Martingales |

Lec 23 | 20 Oct | Sec. 21 Exponential martingale and exit time of an interval by 1-dim BM. | Aatira's notes for Lectures 20-22 |

Lec 24 | 22 Oct | Sec. 22 Skorokhod embedding. | Short exposition of Dubin's embedding |

Lec 25 | 27 Oct | Sec. 23 Donsker's invariance principle. | A short note on weak convergence (Typos uncorrected) |

Lec 26 | 29 Oct | Sec. 24 Applications of Donsker's theorem | Correction to proof in class (as discussed with Tamal and Subhamay) |

Lec 27 | 03 Nov | Sec. 25 Law of iterated logarithm for BM and random walks. | LIL from BM to random walks |

Lec 28 | 05 Nov | Sec. 25 [Cont'd] LIL for BM. Sec. 26 Cameron-Martin | |

Lec 29 | 10 Nov | Sec. 26 [Cont'd] Cameron-Martin theorem. Brownian bridge. | Proof of Cameron Martin and a note on Brownian bridge |

Lec 30 | 12 Nov | Sec. 27 Martingales for higher dimensional BM. Sec. 28 Polar sets for BM in R^{d} , d≥3. | A page from Rogers and Williams about martingales for BM. |

Lec 31 | 17 Nov | Sec. 28 [cont'd] Polar sets for BM in R^{d}, d≥3. | Proof of lower bound for Markov chains Original paper - see last page for remarks on usefulness of the capacity criterion. |

Lec 32 | 19 Nov | Sec. 29 Karhunen-Loeve expansion for BM. What is Gaussian free field? | Proof of uniform convergence of the series |

01 Dec | Presentation by Saptak Banerjee: BM has no points of increase | A discussion by Krzysztof Burdzy about points of increase | |

03 Dec | Last update | Notes for Lectures 23-33 by Rajesh Sundaresan |