| 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 |
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| 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 |
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| 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 |
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| 4 |
Asymptotics Inside the Central Region
Gram-Charlier expansions for random walks, Berry-Esseen theorem, width of the "central region", "fat" power-law tails |
Problem set 1 due |
| 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 |
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| 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 |
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| 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 |
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| 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 |
Problem set 2 due |
| 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 |
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| 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 |
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| 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 |
Exam 1 due |
| 12 |
Black-Scholes and Beyond
Riskless hedging and pricing on a binomial tree, Black-Scholes equation in the continuum limit, risk neutral valuation |
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| 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 |
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| 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 |
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| 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) |
Problem set 3 due |
| 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 |
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| 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) |
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| 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 |
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| 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 |
Exam 2 due |
| 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 |
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| 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 |
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| II. Anomalous Diffusion |
| 22 |
Levy Flights
Steps with infinite variance, Levy stability, Levy distributions, generalized central limit theorems (Guest lecture by Chris Rycroft) |
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| 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 |
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| 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 |
Problem set 4 due |
| 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 |
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| 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 |
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