UC Davis Math 235C
Probability Theory

Basic information

CRN code:	83924(MAT 235 B) and 84827 (STA 235 B)
Room/Time:	10:30am-11:50am TR online via Canvas Conference
Instructor:	Alexander Soshnikov
Office:	3140 Math Sci Building
Office Hours:	online via Canvas Conference F 2:10pm-4:00 pm plus by appointment.
E-mail:	soshniko@math.ucdavis.edu

Webpage:	www.math.ucdavis.edu/~soshniko/235c

Topical Outline

-Brownian Motion

-Ergodic Theory

Grading:

A final project, which will take the form of reading a research paper or section of a book and writing about it/presenting it during an office hour or in class.

Recommended reading:

Probability: Theory and Examples (fifth edition) by Rick Durrett. The textbook may be downloaded as a PDF from the author's website

Introduction to Ergodic Theory by Yakov Sinai

Notes by Dan Romik

Ergodic Theory and Information by Patrick Billingsley

Lectures

Lecture 1 (March 31): Introduction. Definition of Brownian Motion (Section 7.1 of Durrett).

Lecture 2 (April 2): Construction of Brownian Motion (Section 7.1).

Lecture 3 (April 7): Construction of Brownian Motion. Kolmogorov Continuity Theorem (Section 7.1).

Lecture 4 (April 9): Nondifferentiability of Brownian Paths (Paley-Wiener-Zygmund theorem).

Lecture 5 (Spril 14): Quadratic Variation. Paley-Wiener Integrals. Applications (non-canonocal representation of BM, fractional BM).

Lecture 6 (April 16) Markov Property (Section 7.2).

Lecture 7 (April 21): Levy's Construction of Brownian Motion (using Paley-Wiener integrals and Haar basis).

Lecture 8 (April 23): Levy's Modulus of Continuity Theorem. Stopping Times.

Lecture 9 (April 28): Stopping Times. Strong Markov Property (Section 7.3). Paths Properties (Section 7.4).

Lecture 10 (April 30): Martingales. More on Hitting Times (Section 7.5).

Lecture 11 (May 5): Maximum Process of Brownian Motion. Markov Processes Derived from Brownian Motion.

Lecture 12 (May 7): Brownian Bridge. Ito's Formula (Section 7.6).

Lecture 13 (May 12): Ito's Formula (Section 7.6).

Lecture 14

Lecture 15

Lecture 16

Lecture 17

Lecture 18

Lecture 19

Lecture 20