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 ErdösTaylorKakutani) 

Lec 5  18 Aug  Sec. 4 [cont'd] Dvoretsky's theorem and hence a second proof of PaleyWienerZygmund via branching processes  Resurrected original proof of PaleyWienerZygmund! ^{2} 
Lec 6  20 Aug  Sec. 5 BM is Hölder ½ε (original proof of PWZ).  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 01 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: MB has the same law as B  Beautifully written Lecture notes 115 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 1619 
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 1dim 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 1dim BM.  Aatira's notes for Lectures 2022 
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 CameronMartin  
Lec 29  10 Nov  Sec. 26 [Cont'd] CameronMartin 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 KarhunenLoeve 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 2333 by Rajesh Sundaresan 