18.325 Topics in Applied Mathematics

Lecture Notes for Spring 2001

Martin Z. Bazant

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Lecture Topics

  1. Tu Feb 6: Historical Introduction to Random Walks and Diffusion. Pearson, Rayleigh, Bachelier, Fick, Fourier, Einstein. Handouts: Bachelier, Ito/McKean, Hughes, Crank. (Notes by B. White: ps)

  2. Th Feb 8: Continuum Approximations of General Random Walks with Drift. Bachelier/Chapman-Kolmogorov equation, Kramers-Moyal expansion, Fokker-Planck equation, Einstein relation. Handout: Sarkar. (Notes by S. Yang: ps

  3. Tu Feb 13: Long-Time Behavior of Isotropic Random Walks in d>1 Diffusive mixing, Green function of the diffusion equation, Fourier transforms, leading order approximations for Bernoulli (d=1), Pearson (d=2), Rayleigh (d=3), and Polya (integer lattice) walks. (Notes by J. Levitan: ps)

  4. Th Feb 15: Intermediate Asymptotics of Diffusion. "First kind" regular perturbations due to initial conditions, regular perturbations due to discreteness (via higher-order terms in the Kramers-Moyal expansion). Handouts: Goldenfeld, Hughes (Notes by G.-S. Kim: ps)

  5. Th Feb 22: Intermediate Asymptotics of Random Walks I . Approach to self-similarity, moment-based corrections to the Gaussian limiting distribution due to discreteness, regular perturbations of the similarity function. (Notes by Y. H. Lee: ps)

  6. Tu Feb 27: Intermediate Asymptotics of Random Walks II . Kurtosis-based corrections to Gaussian asymptotic behavior for symmetric random walks. Width of the "central region" when the skewness is zero and the kurtotsis is non-zero. (Notes by Y. H. Lee: ps)

  7. Th Mar 1: Intermediate Asymptotics of Random Walks (Flights) III . Gram-Charlier expansions from the dimensionless Kramers-Moyal expansion, continuum methods using Fourier transforms, additivity of cumulants, width of the central region. (Notes by F. Blanchette: ps)

  8. Tu Mar 6: Exact Solutions for Random Walks . Discrete methods using Fourier transforms (characteristic functions), moments and cumulants, convolution theorem, additivity of cumulants, Gaussian random walk. (Notes by F. Blanchette: ps)

  9. Th Mar 8: Examples of Random Walks . 1. Simple random walk on the integers (negative kurtosis), 2. p(x) = A/(a+cosh(x)) (positive kurtosis, all moments finite), 3. p(x) = A/(1+x^4) (postive kurtosis, divergent moments >2). (Notes by E. Silva: ps).

  10. Tu Mar 13: Central Limit Theorems. Chebyshev expansion, Berry-Eseen theorem, Dawson-integral corrections for diverging third moment, e.g. 1/(1+x^4), width of the "central region". (Notes by E. Silva: ps).

  11. Th Mar 15: Strong Central Limit Theorems. Berry-Esseen theorems, "slowly" diverging variance, e.g. 1/(1+|x|^3), non-identical steps. (Notes by B. Thurber: ps).

  12. Tu Mar 20: Anomalous Diffusion due to Non-indentical Steps. Power-law growing or decaying steps, exponentially growing or decaying steps. (Notes by J. Choi: ps).

  13. Th Mar 22: Correlation Functions. Correlations between steps, Green-Kubo formula relating the diffusion coefficient and the velocity autocorrelation function, anomalous diffusion for long-range correlations, exponentially decaying correlations, renormalization-group scaling analysis for the ballistic to diffusive transition. Handout G. I. Taylor. (Notes by J. Choi: ps).

  14. Tu Apr 3: Exact Solutions for Correlated Random Walks. Explicit construction of exponentially correlated walks from uncorrelated walks, Markov-chain difference equations for persistent random walks. (Notes by B. Thurber: ps)

  15. Th Apr 5: Persistent Random Walks . Continuum limits: diffusion equation at the diffusive time scale, Telegrapher's equation at the ballistic time scale (when correlation time >> step time), exact and asymptotic solution of the Markov chain equations using Fourier series. (Notes by K. Dorfman: ps)

  16. Tu Apr 10: Saddle-Point Asymptotics for Random Walks . Laplace integrals, steepest descent contours, saddle point asymptotics, global approximations (valid in the "tail" and in the "central region"), application to the Bernoulli random walk. (Notes by K. Gosier: ps, supplementary notes by D. Margetis: ps)

  17. Th Apr 12: Steepest-Descent Asymptotics for Fat-Tails. Asymptotics of the position distribution of the "student-t walk", p(x) = A/(1+x^2)^2, with a dominant power-law tail from the end-point at the origin and a central Gaussian from a nonzero saddle point in the integrand of the inverse Fourier transform. (Notes by K. Gosier: ps)

  18. Th Apr 19: Levy Flights and Power-Law Tails. Symmetric Levy distributions, "stability" of Levy flights, addivitity of power-law tail amplitudes for general random walks. (Notes by D. Harmon: ps)

  19. Tu Apr 24: Levy Flights and Extreme Events. Finish derivation of additivity of power-law tail amplitudes, asymptotic expansions and Laurent series for symmetric Levy densities, statistics of the largest step size (out of n steps) for exponential and power-law tails, Fisher-Tippett distribution, connection between extreme events and divergent moments. (Notes by D. Harmon: ps)

  20. Th Apr 26: Levy Stable Laws and Renormalization. Finish discussion of extremes -- power-law tails and the Frechet distribution, divergent moments. Applications of Levy flights to financial time series (truncated Levy flights) and polymer surface adsorption. Convergence to stable laws, basins of attraction. (Notes by G.-S. Kim: ps)

  21. Tu May 1: Continuous-Time Random Walks. Random waiting times (renewal theory for birth processes), Laplace tansform analysis, one-sided stable laws, Smirnov density, random number of steps, Poisson process, Montroll-Weiss equation, Central Limit Theorem for continuous-time random walks. (Notes by B. Thurber: ps)

  22. Th May 3: Continuous-Time Random Walks and Anomalous Diffusion. Fourier-Laplace analysis for moments, random sums of random variables, anomalous subdiffusion, inverse Levy transforms, random walks on comb structures, application to diffusion in random media. (Notes by J. Levitan: ps)

  23. Tu May 8: Introduction to Percolation . Percolation (and invasion percolation). The topological "phase transition" and critical exponents. Applications: polymer gelation, oil recovery, epidemics, Internet stability. Exact solution for a Cayley tree (p_c and strength of infinite cluster), approximate solutions for periodic lattices using small-cell real-space renormalization group methods. (Notes by F. Blanchette: ps.)

  24. Th May 10: Brownian Motion in Fluid Mechanics. (Guest lecture by Prof. Michael Brenner.) Taylor dispersion, motion in a potential field. (Notes by K. Dorfman, ps)

  25. Tu May 15: Finite-Size Effects and Renormalization in Percolation. Scaling and shape of the largest cluster distribution. Fisher-Tippett theory of limiting distributions for extremes of independent random variables, Central Limit Theorem for supercritical percolation, self-similar "fractal CLT" at the critical point, saddle-point approximation, RG prediction of crossover and the strength of the infinite cluster in the supercritical regime.

  26. Th May 17: Student final project presentations.

A related set of lectures on applications of random walks in finance were by given Jean-Philippe Bouchaud in 4-163 from 4:30-5:30pm on April 10 (Real World Colloquium), April 12, April 19, and May 3. Scribe reports by 18.325 students are also available for these lectures.

  1. Apr 10: Theory of Financial Risks: A Physicist's Perspective. (Notes by H. Lee: ps)

  2. Apr 12: Central Limit Theorem and Discrete Models for Financial Time Series. (Notes by S. Yang: ps)

  3. Apr 19: Extreme Events and Options Valuation (Notes by K. Gosier: ps)