**Brownian Motion (0366-4887-01)**

Spring 2014, Tel Aviv University

Location: Shenkar Physics 105, Sundays 16-19

Instructor: Ron Peled

The course is an introduction to the mathematical theory of Brownian motion.

__Syllabus__

Brownian motion is an important stochastic process featuring prominently in mathematics, physics, finance and other sciences. It is named after the botanist Robert Brown who noted in 1827 the erratic movement of pollen particles in water. Its theory was developed, among others, by Albert Einstein in 1905. From a physical point of view, Brownian motion models the movement of a particle with "full randomness" in choosing direction and speed at every time instance. From a mathematical point of view, Brownian motion is characterized by being a continuous stochastic process with stationary independent increments. It arises as the scaling limit of random walks. Brownian motion is a Markov process, a Gaussian process and a martingale and is a central example in each of these theories.

In this course we will follow the book "Brownian motion" by Yuval Peres and Peter Mörters available at the authors' websites.

As time allows, we will consider the following topics:

1) Brownian motion definition and the basic properties of its sample paths.

2) Brownian motion as a Markov process and martingale.

3) Brownian motion and potential theory - Recurrence/transience, Green functions and harmonic measure.

4) Hausdorff dimension and its uses for Brownian motion.

5) Brownian motion as a scaling limit of random walks.

6) The local time of Brownian motion.

7) Stochastic integrals with respect to Brownian motion, conformal invariance and related theorems.

8) Relations between Brownian motion and differential equations.

Besides the book of Peres and Mörters we will make occasional use of the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve. A good reference on martingales in discrete time is the lecture notes of Peter Mörters. An interesting book explaining applications of Brownian motion to analysis is Richard Durrett's "Brownian motion and martingales in analysis".

Required prerequisites: The courses Probability for Mathematicians or Probability for Sciences or Advanced Probability Theory.

Recommended prerequisite: Functions of a Real Variable.

Tasks: There will be weekly homework assignments throughout the course.

__Lectures so far__

__Lecture 1 (16.2)__: Introduction. Definition and construction of Brownian motion.

As further background material, some notes on sigma algebras related to Brownian motion are provided.

Assignment 1.

__Lecture 2 (23.2)__: Uniqueness of the distribution of Brownian motion. Symmetry properties - scaling and time reversal. Modulus of continuity and proof of non-existence of monotonic segment.

Assignment 2.

__Lecture 3 (2.3)__: Behavior of B(t) / sqrt(t) as t tends to 0. Non-differentiability everywhere of Brownian motion. Multidimensional Brownian motion and the Markov property. Filtrations associated to the Brownian motion. Tail events and extrema of Brownian motion.

For the third assignment, solve exercises 1.13, 1.16, 2.1 and 2.2 from the Peres-Mörters book.

__Lecture 4 (9.3)__: Stopping times. Reflection principle and area of two-dimensional Brownian motion.

Assignment 4.

__Lecture 5 (23.3)__: Planar Brownian motion does not hit points. Brownian motion and harmonic functions. Use of Brownian motion to solve the Dirichlet problem.

Assignment 5.

__Lecture 6 (30.3)__: Bounded harmonic functions in R^d are constant. Radial harmonic functions and recurrence / transience of Brownian motion. Proof of the Dvoretzky-Erdős theorem.

Assignment 6.

__Lecture 7 (6.4)__: End of proof of the Dvoretzky-Erdős theorem. Primer on martingales in discrete time (without proofs).

Assignment 7 - reviewing martingales in discrete time.

__Lecture 8 (27.4)__: Martingales in continuous time. Wald's lemmas for Brownian motion. The Skorohod embedding problem and Dubins' proof of it. Introduction to the Donsker invariance principle.

Assignment 8.

__Lecture 9 (4.5 - only two hours)__: The Donsker invariance principle. Application to the asymptotic law of the maximum of a random walk. Law of the iterated logarithm for Brownian motion (without full details in the proof of the lower bound).

Assignment 9.

__Lecture 10 (11.5)__: Discrete analog of stochastic integral. Construction of the L^2-stochastic integral with respect to Brownian motion.

Assignment 10.

__Lecture 11 (18.5)__: Itô's formula. Harmonic functions applied to Brownian motion produce local martingales. Introduction to conformal invariance and beginning of proof.

Assignment 11.

__Lecture 12 (25.5)__: End of proof of conformal invariance. Application to probability of leaving cone through curved part. Skew-product representation of planar Brownian motion and application to Spitzer's law for the asymptotic distribution of the winding number. Introduction to Brownian motion local time.

Assignment 12.

__Lecture 13 (1.6)__: Brownian motion local time treatment based on the Karatzas and Shreve book. Definition and construction via Tanaka's formula. Joint continuity of the local time. Extension of Itô's formula to non-twice differentiable functions following Peres and Mörters book. Proof of Lévy's distributional identity for the joint distribution of Brownian motion local time at a fixed level and the absolute value of Brownian motion. Statement without proof of the Ray-Knight distributional identity for the local time process at the first time Brownian motion hits 0.

Assignment 13.

__Lecture 14 (8.6)__: Survey of further topics not treated in our course: More general stochastic integrals. The martingale representation theorem and the Dambis, Dubins-Scwartz theorem on representing martingales as time-changed Brownian motion. Primer on the Black-Scholes option pricing formula (based on the Karatzas-Shreve book). Idea of Davis' proof of Picard's little theorem in complex analysis via Brownian motion (based on Durrett's "Brownian motion and martingales in analysis"). Primer on Hausdorff dimensions and the Hausdorff dimension of the zero set of Brownian motion. Self-intersections of Brownian motion. A brief introduction to Schramm-Lowener evolution.