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.
Recommended Texts
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 FokkerPlanck Equation. 2nd ed. New York, NY: SpringerVerlag, 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: SpringerVerlag, 2001. ISBN: 0387951547.
LEC #  TOPICS  2006 Lecture NOTES  2006 READINGS 

1 
Overview History (Pearson, Rayleigh, Einstein, Bachelier) Normal vs. Anomalous Diffusion Mechanisms for Anomalous Diffusion 
2005 Lecture 1 (PDF) Hughes 

I. Normal Diffusion  
I.A. Linear Diffusion  
2 
Moments, Cumulants, and ScalingMarkov Chain for the Position (in d Dimensions), Exact Solution by Fourier Transform, Moment and Cumulant Tensors, Additivity of Cumulants, “Squareroot Scaling” of Normal Diffusion 
2005 Lecture 2 (PDF) Hughes 

3 
The Central Limit Theorem and the Diffusion EquationMultidimensional CLT for Ssums of IID Random Vectors Continuum Derivation Involving the Diffusion Equation 
2005 Lecture 1 (PDF) 2005 Lecture 3 (PDF) 

4 
Asymptotic Shape of the DistributionBerryEsseen Theorem Asymptotic Analysis Leading to Edgeworth Expansions, Governing Convergence to the CLT (in one Dimension), and more Generally GramCharlier Expansions for Random Walks Width of the Central Region when Third and Fourth Moments Exist 
2005 Lecture 3 (PDF) 2005 Lecture 4 (PDF) Hughes Feller 

5 
Globally Valid AsymptoticsMethod of Steepest Descent (SaddlePoint 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) Hughes 

6 
Powerlaw “Fat Tails”Powerlaw Tails, Diverging Moments and Singular Characteristic Functions Additivity of Tail Amplitudes 
2005 Lecture 5 (PDF) 2005 Lecture 6 (PDF) Bouchaud and Potters 

7 
Asymptotics with Fat TailsCorrections to the CLT for Powerlaw 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) 
8 
From Random Walks to DiffusionExamples of Random Walks Modeled by Diffusion Equations
Run and Tumble Motion, Chemotaxis
Additive Versus Multiplicative Processes 
(PDF) (Courtesy of Daniel Rudoy. Used with permission.) 
2005 Lecture 10 (PDF) Bouchaud and Potters 
9 
Discrete Versus Continuous Stochastic ProcessesCorrections 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) Risken 
10 
Weakly Nonidentical StepsChapmanKolmogorov Equation, KramersMoyall Expansion, FokkerPlanck Equation. Probability Flux Modified KramersMoyall Cumulant Expansion for Identical Steps 
2005 Lecture 8 (PDF) 2005 Lecture 9 (PDF) 2005 Lecture 13 (PDF) Risken 

I.B. Nonlinear Diffusion  
11 
Nonlinear DriftInteracting Random Walkers, Concentrationdependent Drift Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers’ Equation Surface Growth, KardarParisiZhang Equation 
(PDF) (Courtesy of Lou Odette. Used with permission.)  
12 
Nonlinear DiffusionColeHopf Transformation, General Solution of Burgers Equation Concentrationdependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects 
Problem set 3 solutions 

I.C. First Passage and Exploration  
13 
Return Probability on a LatticeProbability 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 
14 
The Arcsine DistributionReflection 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 
15 
First Passage in the Continuum LimitGeneral 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) 

16 
First Passage in Arbitrary GeometriesGeneral 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) Redner Risken 

17 
Conformal InvarianceConformal Transformations (Analytic Functions of the Plane, Stereographic Projection from the Plane to a Sphere,…), Conformally Invariant Transport Processes (Simple Diffusion, Advectiondiffusion 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 Redner 
18 
Hitting Probabilities in Two DimensionsPotential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line 
2003 Lecture 23 (PDF) Redner 

19 
Applications of Conformal MappingFirst Passage to a Circle, Wedge/Corner, Parabola. Continuous Laplacian Growth, PolubarinovaGalin Equation, SaffmanTaylor Fingers, Finitetime Singularities 
2003 Lecture 23 (PDF) 2003 Lecture 24 (PDF) 

20 
Diffusionlimited AggregationHarmonic Measure, HastingsLevitov Algorithm, Comparison of Discrete and Continuous Dynamics Overview of Mechanisms for Anomalous Diffusion Nonidentical Steps 
2003 Lectures 25 (PDF) 2003 Lecture 14 (PDF) 2003 Lecture 15 (PDF) 

II. Anomalous Diffusion  
II.A. Breakdown of the CLT  
21 
Polymer Models: Persistence and SelfavoidanceRandom Walk to Model Entropic Effects in Polymers, Restoring Force for Stretching; Persistent Random Walk to Model Bondbending Energetic Effects, GreenKubo Relation, Persistence Length, Telegrapher’s Equation; Selfavoiding Walk to Model Steric Effects, FisherFlory 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) 

22 
Levy FlightsSuperdiffusion 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) Hughes 

II.B. ContinuousTime Random Walks  
23 
Continuoustime Random WalksLaplace Transform. Renewal Theory MontrollWeiss 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) 
24 
Fractional Diffusion EquationsCLT for CTRW Infinite Man Waiting Time, MittagLeffler Decay of Fourier Modes, Timedelayed Flux, Fractional Diffusion Equation 
2005 Lecture 24 (PDF) 2003 Lecture 18 (PDF) 

25 
Nonseparable Continuoustime 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) Hughes 

26 
Leapers and CreepersHughes’ Leaper and Creeper Models Leaper Example: Polymer Surface Adsorption Sites and Crosssections of a Random Walk Creeper Examples: Levy Walks, Bacterial Motion, Turbulent Dispersion 
2005 Lecture 26 (PDF) Hughes 