Lec 1 |
04 Aug |
Overview |
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Lec 2 |
06 Aug |
Sec. 1 Gaussian random variables. Sec. 2 Definition of Brownian motion and Wiener measure, the space C[0,1]
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Lec 3 |
11 Aug |
Sec. 3 Levy's construction of Brownian motion |
Wiener's original paper1 available online for the first time! |
Lec 4 |
13 Aug |
Sec 4 Continuity properites I - Nowhere Hölder ½+ε (proof of Erdös-Taylor-Kakutani) |
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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 L2 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] dimε≤dimH≤dimM. 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 Rd , 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 Rd, 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 |