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The exhaustive list of topics in Random Walks and Diffusion in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics
  • Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation
  • Modified Kramers-MoyallCumulant Expansion for Identical Steps
  • Flagellar Bacteria
  • General Formulation in Higher Dimensions, Moments of First Passage Time, Eventual Hitting Probability, Electrostatic Analogy for Diffusion, First Passage to a Sphere
  • Run and Tumble Motion, Chemotaxis
  • Corrections to the CLT for Power-law Tails
  • Polymer Models: Persistence and Self-avoidance
  • I.C. First Passage and Exploration
  • Application to Flagellar Bacteria
  • Width of the Central Region when Third and Fourth Moments Exist
  • Nonlinear Diffusion
  • Arcsine Distribution
  • Conformal Invariance
  • General Formulation in One Dimension
  • Return Probability on a Lattice
  • Leaper Example: Polymer Surface Adsorption Sites and Cross-sections of a Random Walk
  • Financial Time Series
  • Weakly Non-identical Steps
  • Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation
  • Levy Flights
  • Fractional Diffusion Equations
  • Multi-dimensional CLT for Sums of IID Random Vectors
  • Surface Growth, Kardar-Parisi-Zhang Equation
  • Stochastic Differentials, Wiener Process
  • Power-law "Fat Tails"
  • Non-separable Continuous-time Random Walks
  • Nonlinear Drift
  • Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution
  • Hughes' Leaper and Creeper Models
  • Parabolic Cylinder Functions and Dawson's Integral
  • First Passage to a Circle, Wedge/Corner,
  • Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya's Theorem
  • From Random Walks to Diffusion
  • Minimum First Passage Time of a Set of N Random Walkers
  • Continuous-Time Random Walks
  • Central Limit Theorem and the Diffusion Equation
  • Normal vs. Anomalous Diffusion
  • Laplace Transform
  • Berry-Esseen Theorem
  • Fat Tails and Riesz Fractional Derivatives
  • Interacting Random Walkers, Concentration-dependent Drift
  • Leapers and Creepers
  • I.B. Nonlinear Diffusion
  • Cole-Hopf Transformation, General Solution of Burgers Equation
  • "Phase Diagram" for Anomalous Diffusion: Large Steps Versus Long Waiting Times
  • Smirnov Density
  • CLT for CTRW
  • Applications of Conformal Mapping
  • Asymptotics with Fat Tails
  • Method of Steepest Descent (Saddle-Point Method) for Asymptotic Approximation of Integrals
  • Power-law Tails, Diverging Moments and Singular Characteristic Functions
  • Normal Diffusion
  • Discrete Versus Continuous Stochastic Processes
  • First Passage in the Continuum Limit
  • Additive Versus Multiplicative Processes
  • Asymptotic Shape of the Distribution
  • Montroll-Weiss Formulation of CTRW
  • Parabola. Continuous Laplacian Growth, Polubarinova-Galin Equation, Saffman-Taylor Fingers, Finite-time Singularities
  • Renewal Theory
  • Mechanisms for Anomalous Diffusion. Non-identical Steps
  • Hitting Probabilities in Two Dimensions
  • First Passage in Arbitrary Geometries
  • Continuum Derivation Involving the Diffusion Equation
  • Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects
  • DNA Gel Electrophoresis
  • Globally Valid Asymptotics
  • Application to Random Walks
  • Diffusion-limited Aggregation
  • Moments, Cumulants, and Scaling
  • Probability Flux
  • 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
  • Hughes' General Formulation of CTRW with Motion between "turning points"
  • Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line
  • Mechanisms for Anomalous Diffusion
  • Additivity of Tail Amplitudes
  • Continuous-time Random Walks
  • Corrections to the Diffusion Equation Approximating Discrete Random Walks with IID Steps
  • Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers' Equation