| LEC # | TOPICS | NOTES |
|---|---|---|
| I. Normal Diffusion: 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 | Chris Rycroft (PDF) |
| 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 | Ernst van Nierop (PDF) |
| 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 | Jacy Bird (PDF) |
| 4 | Asymptotics Inside the Central Region Gram-Charlier Expansions for Random Walks, Berry-Esseen Theorem, Width of the "Central Region", "Fat" Power-law Tails | Erik Allen (PDF) |
| 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 | (PDF) |
| 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 | Mustafa Sabri Kilic (PDF) |
| 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 | (PDF) |
| 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 | Ernst van Nierop (PDF) |
| 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 | Jacy Bird (PDF) |
| I. Normal Diffusion: Some Finance | ||
| 10 | Applications in Finance Models for Financial Time Series, Additive and Multiplicative Noise, Derivative Securities, Bachelier's Fair-game Price | Erik Allen (PDF) |
| 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 | J. F. (PDF) Additional Notes (PDF) |
| 12 | Black-Scholes and Beyond Riskless Hedging and Pricing on a Binomial Tree, Black-Scholes Equation in the Continuum Limit, Risk Neutral Valuation | Sergiy Sidenko (PDF) Additional notes on "Gram-Charlier" corrections for residual risk in Bouchaud-Sornette theory, by Ken Gosier (PDF) 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 | Sergiy Sidenko (PDF) |
| I. Normal Diffusion: Some Physics | ||
| 14 | Applications in Statistical Mechanics Random Walk in an External Force Field, Einstein Relation, Boltzmann Equilibrium, Ornstein-Uhlenbeck Process, Ehrenfest Model | Kirill Titievsky (PDF) |
| 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) | J. F. (PDF) |
| I. Normal Diffusion: 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 | Mustafa Sabri Kilic (PDF) |
| 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) | Ken Kamrin (PDF) |
| 18 | First Passage in Higher Dimensions Return and First Passage on a Lattice, Polya's Theorem, Continuous First Passage in Complicated Geometries, Moments of the Time and the Location of First Passage, Electrostatic Analogy | Kirill Titievsky (PDF) |
| I. Normal Diffusion: 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 | Allison Ferguson (PDF) |
| 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 | Neville Sanjana (PDF) |
| 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 | Ken Kamrin (PDF) |
| II. Anomalous Diffusion | ||
| 22 | Levy Flights Steps with Infinite Variance, Levy Stability, Levy Distributions, Generalized Central Limit Theorems (Guest Lecture by Chris Rycroft) | Neville Sanjana (PDF) |
| 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 | Chris Rycroft (PDF) |
| 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 | Yuxing Ben (PDF) |
| 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 | Geraint Jones (PDF) |
| 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 | Geraint Jones (PDF) |