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