Wednesday January 31, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Intro to Brownian MotionAfter a quick introduction, we'll define and discuss what we need to know about Gaussian random variables. From there, we will define what Brownian motion is, and give a construction of Brownian motion on an interval.

Wednesday February 07, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Construction of Brownian MotionWe will start by addressing some of the questions asked in the last talk. After that and a quick review, we will construct Brownian motion on an interval.

Wednesday February 14, 2018 4:00 PM Math Tower 5127
 Matthew Dannenberg, Stony Brook University
Basic Properties of Brownian MotionWe discuss the basic properties of Brownian motion, including scaling and inversion relationships, Holder continuity, and nowhere differentiability.

Wednesday February 21, 2018 4:00 PM Math Tower 5127
 Ben Sokolowsky, Stony Brook University
Reflection and Conformal InvarianceWe will discuss stopping times and the strong Markov property of Brownian motion, proving a reflection principle along the way. Time permitting we will also discuss the conformal invariance of Brownian motion.

Wednesday February 28, 2018 4:00 PM Math Tower 5127
 Silvia Ghinassi, Stony Brook University
Dimension Results for Brownian MotionAfter recalling a few definitions and basic facts regarding Hausdorff and Minkowski dimensions, we state and prove dimension results for the graph and the image of Brownian motion with the help of Frostman's lemma. We will state and understand dimension doubling results for ddimensional Brownian motion, for $d ≥ 2$. Finally, if time allows, we will discuss in detail area of planar Brownian motion.

Wednesday March 07, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
The Zeros of Brownian MotionWe show that the zero set of Brownian motion almost surely has dimension equal to 1/2

Wednesday April 04, 2018 4:00 PM Math Tower 5127
 Matt Dannenberg, Stony Brook University
The Universality of Brownian MotionWe will discuss sections 7.27.4 in BishopPeres, including material on the Law of the Iterated Logarithm, Skorokhod's representation of random variables by Brownian Motion with a certain stopping time, and Donsker's invariance principle, which allows us to view Brownian motion as the limit of many simple, discrete random walks.

Wednesday April 11, 2018 4:00 PM Math Tower 5127
 Ben Sokolowsky and Jack Burkart, Stony Brook University
Probabilistic View of Harmonic FunctionsFollowing 7.57.7 in BishopPeres, we will develop the theory of harmonic functions in Euclidean space using the ideas using Brownian motion. Applications include the transience of Brownian motion in dimensions 3 and higher and the solution of the Dirichlet problem for domains satisfying the Poincare cone condition.

Wednesday April 18, 2018 4:00 PM Math Tower 5127
 Mu Zhao, Stony Brook University
MartingalesWe'll define and construct the conditional expectation of a random variable, and discuss its basic features. Then we'll discuss and define martingales.

Wednesday April 25, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Kakutani's Theorem on Harmonic MeasureGiven a domain in the complex plane and a sufficiently nice subset of its boundary, the harmonic measure is the solution of the Dirichlet problem with boundary data given by the indicator function. It's a central topic in analysis that touches complex analysis, geometric measure theory, and harmonic analysis.
In this lecture, we'll carefully define harmonic measure and prove Kakutani's theorem, which says that the harmonic measure of a subset evaluated at a point coincides with the probability that Brownian motion starting at that point first hits the boundary at this subset.

Monday May 07, 2018 2:30 PM Math Tower 5127
 Matt Dannenberg, Stony Brook University
Intro to Stochastic CalculusWe'll define what it means to integrate with respect to a stochastic process, such as Brownian motion. This gives us a formal way of understanding systems that behave randomly. After that, we will state a few of the main theorems and applications of this idea.

