### MA 367: Brownian Motion

#### Credits: 3:0

• Levy’s construction of Brownian motion.
• Invariance properties : Under scaling, rotation, time-reversal, conformal maps (dim=2), shifts (Markov property), random shifts (strong Markov property).
• Blumenthal’s and Kolmogorov’s zero-one law, Law of large numbers, Strassen’s law of iterated logarithm.
• Continuity properties: law of iterated logarithm, Levy’s theorem on modulus of Continuity of BM, Nowhere Holder continuity of order greater than 1/2.
• Hausdorff and Minkowski dimensions. Dimension computation of certain random fractals derived from Brownian motion (range, graph and zero set).
• Random walks and discrete harmonic functions. Skorokhod and Dubins embedding of random walks in Brownian motion, Donsker’s invariance principle. Brownian motion and harmonic functions.
• Recurrence and transience. What sets does Brownian motion hit? (Polar sets and Capacity).
• Stochastic integral and Ito’s formula. Martingales. Levy’s characterization of Brownian motion. Tanaka’s formula for Local time.
• Brownian motion in the plane : Conformal invariance, Winding number. Davis’ proof of Picard’s theorem for entire functions using Brownian motion. Distribution of the filling of Brownian motion in a simply connected domain (Virag’s lemma).
• Gaussian free field : Definition and basic properties. A synopsis of some recent advances due to Scott Sheffield and others involving the GFF.

#### Suggested books :

1. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus ,Springer, 1991.
2. A. Kallenberg, Foundation of Modern Prability Theory ,Second Edition, Springer.

#### All Courses

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E-mail: chairman.math[at]iisc[dot]ac[dot]in