LEC # | TOPICS | NOTES |
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I. Normal Diffusion: Fundamental Theory |
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1 |
## IntroductionHistory; 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 ScalingMarkov 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 TheoremMulti-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 RegionGram-Charlier Expansions for Random Walks, Berry-Esseen Theorem, Width of the "Central Region", "Fat" Power-law Tails |
Erik Allen (PDF) |

5 |
## Asymptotics with Fat TailsSingular 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 RegionAdditivity 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 WalkExample 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 LimitApplication 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 ExpansionRecursive 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 |
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10 |
## Applications in FinanceModels for Financial Time Series, Additive and Multiplicative Noise, Derivative Securities, Bachelier's Fair-game Price |
Erik Allen (PDF) |

11 |
## Pricing and Hedging Derivative SecuritiesStatic 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 BeyondRiskless 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 ProcessesDiscrete 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 |
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14 |
## Applications in Statistical MechanicsRandom Walk in an External Force Field, Einstein Relation, Boltzmann Equilibrium, Ornstein-Uhlenbeck Process, Ehrenfest Model |
Kirill Titievsky (PDF) |

15 |
## Brownian Motion in Energy LandscapesKramers 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 |
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16 |
## First Passage in the Continuum LimitGeneral 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 LatticeReturn 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 DimensionsReturn 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 |
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19 |
## Polymer Models: Persistence and Self-AvoidanceRandom Walk Models of Polymers, Radius of Gyration, Persistent Random Walk, Self-avoiding Walk, Flory's Scaling Theory |
Allison Ferguson (PDF) |

20 |
## (Physical) Brownian Motion IBallistic 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 IILangevin 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 |
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22 |
## Levy FlightsSteps with Infinite Variance, Levy Stability, Levy Distributions, Generalized Central Limit Theorems (Guest Lecture by Chris Rycroft) |
Neville Sanjana (PDF) |

23 |
## Continuous-Time Random WalksRandom 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 EquationsContinuum 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 TimesCTRW 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 CreepersHughes' 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) |