Brownian Motion
- 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 abd 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.
Books
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991
- O. Kallenberg, Foundation of Modern Prability Theory, Second Edition, Springer.
. — — Brownian Motion
- 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 abd 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.
Books
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991
- O. Kallenberg, Foundation of Modern Prability Theory, Second Edition, Springer.
.