18.366 Random Walks and Diffusion

2005 Lecture Notes

Martin Z. Bazant

The notes below were written by student "scribes" as homework assignments. In most cases, Prof. Bazant has reviewed the notes and made revisions or extersions of the text, but beware many unedited portions still exist.

Note that solutions to the problem sets and exams also contain detailed discussions (also mostly edited by Prof. Bazant) of many topics not appearing in the lectures below.

I. Normal Diffusion

I.A. Fundamental Theory

1. Introduction. History; simple analysis of the isotropic random walk in d dimensions, using the continuum limit; Bachelier and diffusion equations; normal versus anomalous diffusion.

2. Moments, Cumulants, and Scaling. Markov chain for the position (in d dimensions), exact solution by Fourier transform, moment and cumulant tensors, additivity of cumulants, "square-root scaling" of normal diffusion.

3. The Central Limit Theorem. Multi-dimensional CLT for sums of IID random vectors (derived by Laplace's method of asymptotic expansion), Edgeworth expansion for convergence to the CLT with finite moments.

4. Asymptotics Inside the Central Region . Gram-Charlier expansions for random walks, Berry-Esseen theorem, width of the "central region", "fat" power-law tails.

5. Asymptotics with Fat Tails. Singular characteristic functions, generalized Gram-Charlier expansions, Dawson's integral, edge of the central region, additivity of power-law tails.

6. Asymptotics Outside the Central Region. Additivity of power-law tails: intuitive explanation, "high-order" Tauberian theorem for the Fourier transform; Laplace's method and saddle-point method, uniformly valid asymptotics for random walks.

7. Approximations of the Bernoulli Random Walk. Example of saddle-point asymptotics for a symmetric random walk on the integers, detailed comparison with Gram-Charlier expansion and the exact combinatorial solution.

8. The Continuum Limit. Application of the Bernoulli walk to percentile order statistics; Kramers-Moyall expansion from Bachelier's equation for isotropic walks, scaling analysis, continuum derivation of the CLT via the diffusion equation.

9. Kramers-Moyall Cumulant Expansion . Recursive substitution in Kramers-Moyall moment expansion to obtain modified coefficients in terms of cumulants, continuum derivation of Gram-Charlier expansion as the Green function for the Kramers-Moyall cumulant expansion.

   I.B. Some Finance

10. Applications in Finance . Models for financial time series, additive and multiplicative noise, derivative securities, Bachelier's fair-game price.

11. Pricing and Hedging Derivative Securities . Static hedge to minimize risk, optimal trading by linear regression of the random payoff, quadratic risk minimization, riskless hedge for a binomial process. (Additional notes)

12. Black-Scholes and Beyond . Riskless hedging and pricing on a binomial tree, Black-Scholes equation in the continuum limit, risk neutral valuation. (Additional notes on "Gram-Charlier" corrections for residual risk in Bouchaud-Sornette theory; see also Problem Set 3.)

13. Discrete versus Continuous Stochastic Processes . Discrete Markov processes in the continuum limit, Chapman-Kolomogorov equation, Kramers-Moyall moment expansion, Fokker Planck equation. Continuous Wiener processes, stochastic differential equations, Ito calculus, applications in finance.

    I.C. Some Physics

14. Applications in Statistical Mechanics. Random walk in an external force field, Einstein relation, Boltzmann equilibrium, Ornstein-Uhlenbeck process, Ehrenfest model.

15. Brownian Motion in Energy Landscapes. Kramers escape rate from a trap, periodic potentials, asymmetric structures, Brownian ratchets and molecular motors. (Guest lecture by Armand Ajdari.)

    I.D. First Passage

16. First Passage in the Continuum Limit. General formula for the first passage time PDF, Smirnov density in one dimension, first passage to boundaries by general stochastic processes.

17. Return and First Passage on a Lattice. Return probability in one dimension, generating functions, first passage and return on a lattice, return of a biased Bernoulli walk, reflection principle. (Guest lecture by Chris Rycroft.)

18. First Passage in Higher Dimensions. Return and first passage on a lattice, Polya's theorem, continuous first passage in in complicated geometries, moments of the time and the location of first passage, electrostatic analogy.

    I.E. Correlations

19. Polymer models: Persistence and Self-Avoidance. Random walk models of polymers, radius of gyration, persistent random walk, self-avoiding walk, Flory's scaling theory.

20. (Physical) Brownian Motion I . Ballistic to diffusive transition, correlated steps, Green-Kubo relation, Taylor's effective diffusivity, telegrapher's equation as the continuum limit of the persistent random walk.

21. (Physical) Brownian Motion II. Langevin equations, Stratonivich vs. Ito stochastic differentials, multi-dimensional Fokker-Planck equation, Kramers equation (vector Ornstein-Uhlenbeck process) for the velocity and position, breakdown of normal diffusion at low Knudsen number, Levy flight for a particle between rough parallel plates (lecture on 4/28/05)

    II. Anomalous Diffusion

22. Levy Flights. Steps with infinite variance, Levy stability, Levy distributions, generalized central limit theorems. (lecture on 4/21/05 by Chris Rycroft)

23. Continuous-Time Random Walks. Random waiting time between steps, Montroll-Weiss theory of separable CTRW, formulation in terms of random number of steps, Tauberian theorems for the Laplace transform and long-time asymptotics.

24. Fractional Diffusion Equations. Continuum limits of CTRW; normal diffusion equation for finite mean waiting time and finite step variance, exponential relaxation of Fourier modes; fractional diffusion equations for super-diffusion (Riesz fractional derivative) and sub-diffusion (Riemann-Liouville fractional derivative); Mittag-Leffler power-law relaxation of Fourier modes.

25. Large Jumps and Long Waiting Times . CTRW steps with infinite variance and infinite mean waiting time, "phase diagram" for anomalous diffusion, polymer surface adsorption (random walk near a wall), multidimensional Levy stable laws.

26. Leapers and Creepers. Hughes' formulation of non-separable CTRW, leapers: Cauchy-Smirnov non-separable CTRW for polymer surface adsorption, creepers: Levy walks for tracer dispersion in homogenous turbulence.

bazant@mit.edu