18.366 | Fall 2006 | Graduate

Random Walks and Diffusion

Lecture Notes

Topics covered in lectures in 2006 are listed below. In some cases, links are given to new lecture notes by student scribes. All scribed lecture notes are used with the permission of the student named in the file. The recommended reading refers to the lectures notes and exam solutions from previous years or to the books listed below. Lecture notes from previous years are also found in the study materials section.

Hughes, B. Random Walks and Random Environments. Vol. 1. Oxford, UK: Clarendon Press, 1996. ISBN: 0198537883.

Redner, S. A Guide to First Passage Processes. Cambridge, UK: Cambridge University Press, 2001. ISBN: 0521652480.

Risken, H. The Fokker-Planck Equation. 2nd ed. New York, NY: Springer-Verlag, 1989. ISBN: 0387504982.

Further Readings

Bouchaud, J. P., and M. Potters. Theory of Financial Risks. Cambridge, UK: Cambridge University Press, 2000. ISBN: 0521782325.

Crank, J. Mathematics of Diffusion. 2nd ed. Oxford, UK: Clarendon Press, 1975. ISBN: 0198533446.

Rudnick, J., and G. Gaspari. Elements of the Random Walk. Cambridge, UK: Cambridge University Press, 2004. ISBN: 0521828910.

Spitzer, F. Principles of the Random Walk. 2nd ed. New York, NY: Springer-Verlag, 2001. ISBN: 0387951547.



History (Pearson, Rayleigh, Einstein, Bachelier)

Normal vs. Anomalous Diffusion

Mechanisms for Anomalous Diffusion


2005 Lecture 1 (PDF)


I. Normal Diffusion
I.A. Linear Diffusion

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


2005 Lecture 2 (PDF)



The Central Limit Theorem and the Diffusion Equation

Multi-dimensional CLT for Ssums of IID Random Vectors

Continuum Derivation Involving the Diffusion Equation


2005 Lecture 1 (PDF)

2005 Lecture 3 (PDF)


Asymptotic Shape of the Distribution

Berry-Esseen Theorem

Asymptotic Analysis Leading to Edgeworth Expansions, Governing Convergence to the CLT (in one Dimension), and more Generally Gram-Charlier Expansions for Random Walks

Width of the Central Region when Third and Fourth Moments Exist


2005 Lecture 3 (PDF)

2005 Lecture 4 (PDF)




Globally Valid Asymptotics

Method of Steepest Descent (Saddle-Point Method) for Asymptotic Approximation of Integrals

Application to Random Walks

Example: Asymptotics of the Bernoulli Random Walk


2005 Lecture 6 (PDF)

2005 Lecture 7 (PDF)



Power-law “Fat Tails”

Power-law Tails, Diverging Moments and Singular Characteristic Functions

Additivity of Tail Amplitudes


2005 Lecture 5 (PDF)

2005 Lecture 6 (PDF)

Bouchaud and Potters


Asymptotics with Fat Tails

Corrections to the CLT for Power-law Tails (but Finite Variance)

Parabolic Cylinder Functions and Dawson’s Integral

A Numerical Example Showing Global Accuracy and Fast Convergence of the Asymptotic Approximation

(PDF) (Courtesy of Damian Burch. Used with permission.)

Numerical Example (PDF) (Courtesy of Chris H. Rycroft. Used with permission.)

2005 Lecture 5 (PDF)

From Random Walks to Diffusion

Examples of Random Walks Modeled by Diffusion Equations

  • Flagellar Bacteria

Run and Tumble Motion, Chemotaxis

  • Financial Time Series

Additive Versus Multiplicative Processes

(PDF) (Courtesy of Daniel Rudoy. Used with permission.)

2005 Lecture 10 (PDF)

Bouchaud and Potters


Discrete Versus Continuous Stochastic Processes

Corrections to the Diffusion Equation Approximating Discrete Random Walks with IID Steps

Fat Tails and Riesz Fractional Derivatives

Stochastic Differentials, Wiener Process

(PDF) (Courtesy of Kwai Hung Henry Lam. Used with permission.)

2005 Lecture 8 (PDF)

2005 Lecture 9 (PDF)

2005 Lecture 13 (PDF)



Weakly Non-identical Steps

Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation. Probability Flux

Modified Kramers-Moyall Cumulant Expansion for Identical Steps


2005 Lecture 8 (PDF)

2005 Lecture 9 (PDF)

2005 Lecture 13 (PDF)


I.B. Nonlinear Diffusion

Nonlinear Drift

Interacting Random Walkers, Concentration-dependent Drift

Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers’ Equation

Surface Growth, Kardar-Parisi-Zhang Equation

(PDF) (Courtesy of Lou Odette. Used with permission.)  

Nonlinear Diffusion

Cole-Hopf Transformation, General Solution of Burgers Equation

Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects


Problem set 3 solutions

I.C. First Passage and Exploration

Return Probability on a Lattice

Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya’s Theorem

(PDF) (Courtesy of Chris H. Rycroft. Used with permission.)

2005 Lecture 17 (PDF)

2005 Lecture 18 (PDF)

Hughes Redne


The Arcsine Distribution

Reflection Principle and Path Counting for Lattice Random Walks, Derivation of the Discrete Arcsine Distribution for the Fraction of Time Spent on One Side of the Origin, Continuum Limit

(PDF) (Courtesy of Chris H. Rycroft. Used with permission.) Feller

First Passage in the Continuum Limit

General Formulation in One Dimension

Smirnov Density

Minimum First Passage Time of a Set of N Random Walkers


2005 Lecture 16 (PDF)

Exam 2 (problem 2)


First Passage in Arbitrary Geometries

General Formulation in Higher Dimensions, Moments of First Passage Time, Eventual Hitting Probability, Electrostatic Analogy for Diffusion, First Passage to a Sphere


2005 Lecture 18 (PDF)




Conformal Invariance

Conformal Transformations (Analytic Functions of the Plane, Stereographic Projection from the Plane to a Sphere,…), Conformally Invariant Transport Processes (Simple Diffusion, Advection-diffusion in a Potential Flow,…), Conformal Invariance of the Hitting Probability

(PDF) (Courtesy of Yee Lok Wong. Used with permission.)

2003 Lecture 23 (PDF)

An Article



Hitting Probabilities in Two Dimensions

Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line


2003 Lecture 23 (PDF)



Applications of Conformal Mapping

First Passage to a Circle, Wedge/Corner, Parabola. Continuous Laplacian Growth, Polubarinova-Galin Equation, Saffman-Taylor Fingers, Finite-time Singularities


2003 Lecture 23 (PDF)

2003 Lecture 24 (PDF)


Diffusion-limited Aggregation

Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics

Overview of Mechanisms for Anomalous Diffusion

Non-identical Steps


2003 Lectures 25 (PDF)

2003 Lecture 14 (PDF)

2003 Lecture 15 (PDF)

A Review Article

II. Anomalous Diffusion
II.A. Breakdown of the CLT

Polymer Models: Persistence and Self-avoidance

Random Walk to Model Entropic Effects in Polymers, Restoring Force for Stretching; Persistent Random Walk to Model Bond-bending Energetic Effects, Green-Kubo Relation, Persistence Length, Telegrapher’s Equation; Self-avoiding Walk to Model Steric Effects, Fisher-Flory Estimate of the Scaling Exponent


2005 Lectures 19 (PDF)

2005 Lecture 20 (PDF)

2003 Lecture 9 (PDF)

2003 Lecture 10 (PDF)

2003 Lecture 11 (PDF)


Levy Flights

Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution


2005 Lecture 22 (PDF)

2003 Lecture 12 (PDF)

2003 Lecture 13 (PDF)


II.B. Continuous-Time Random Walks

Continuous-time Random Walks

Laplace Transform. Renewal Theory

Montroll-Weiss Formulation of CTRW

DNA Gel Electrophoresis

(PDF) (Courtesy of Michael Vahey. Used with permission.)

2005 Lecture 23 (PDF)

2003 Lecture 15 (PDF)

2003 Lecture 16 (PDF)

2003 Lecture 17 (PDF)


Fractional Diffusion Equations


Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation


2005 Lecture 24 (PDF)

2003 Lecture 18 (PDF)


Non-separable Continuous-time Random Walks

“Phase Diagram” for Anomalous Diffusion: Large Steps Versus Long Waiting Times

Application to Flagellar Bacteria. Hughes’ General Formulation of CTRW with Motion between “turning points”


2005 Lecture 25 (PDF)

2005 Lecture 26 (PDF)



Leapers and Creepers

Hughes’ Leaper and Creeper Models

Leaper Example: Polymer Surface Adsorption Sites and Cross-sections of a Random Walk

Creeper Examples: Levy Walks, Bacterial Motion, Turbulent Dispersion


2005 Lecture 26 (PDF)


Course Info

As Taught In
Fall 2006
Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes