Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Brownian Motion (Fall 2009)

Tue-Thu 2:30 - 4:00, in LH III

  • Some references
    • Peter Mörters and Yuval Peres: Brownian motion (the earlier link pointed to an older version of this book). We shall follow this book to a large extent.
    • Rogers and Williams: Diffusions, Markov processes and Martingales - vol I (second ed.)
    • Karatzas and Shreve: Brownian motion and Stoshastic Calculus.
    • Durrett : Probability: Theory and examples.
    • Kallenberg: Foundations of modern Probability.
    • Revuz and Yor: Continuous Martingales and Brownian Motion.

    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 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)
    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 1210 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≤dimHdimM. 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

  • 1 Wiener's essential idea is to define finite dimensional distirbutions, and thus define integrals of functionals that depend on finitely many co-ordinates, extend the integration to uniformly continuous functionals L:C[0,1]-->R, and invoke Daniell's theory to extend integration to an even larger class of functions. Daniell's extension theorem is the same as (but preceded by more than a decade) what is called Kolmogorov's extension today. Read this very well written article on the intricate history of integration and measure and probability before Kolmogorov. In his paper, Wiener calculates the measure of some subsets of functions that are not cylinders, for example, the space of continuous functions carries the full measure, but the space of differentiable fuctions has zero measure.

    2 See Theorem VII on page 666 in the original paper of Paley Wiener and Zygmund where they do not mention branching processes and estimate probabilities in a rather complicated way, but with simple modifications it immediately leads to our proof of Dvoretskys theorem. Also recommended is chapter IX of the book 'Fourier transforms in the complex domain' by Paley and Wiener which gives the feeling of great depth. Note from Subhamay Saha From this proof one can obtain the constant π½/8 while the actual truth is 1.

    3 Thanks to Aatira (and Sandeep) for the notes. More will be added irregularly as the course progresses.