TuTh 2:00-3:15pm — MONT 214
Jan 17, 2023 – Apr 28, 2023
The topic of the first part of the course will be the relationship between random walks and the heat equation. The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation by considering the individual random particles one gains further insight into the problem. We will discuss the discrete case, random walk, and the heat equation on the integer lattice, the continuous case, Brownian motion, and the usual heat equation. Solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion can be introduced and developed from the first principles. We also will discuss martingales and fractal dimensions. The second part of the course will be devoted to a broader range of topics, selected according to the mutual interests of students.
Textbook: Random Walk and the Heat Equation by Gregory F. Lawler, University of Chicago (draft)
Extra meeting (during the final exam week):
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5/1/2023, Monday
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3:30PM – 5:30PM
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MONT 214
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