All readings will be listed here, organized by topic. Dates indicate the
class by which you should have read the indicated material. Some readings
may be announced here before they are announced in class, in case you want to
read ahead. Sometimes, particularly if I am traveling, you may find some
references posted at the library class reserves before they are linked here.

If a reading does not have a date, you don't have to worry about it yet.

Assigned Readings

- Kramer & Majda, "Fundamentals of Probability Theory" (PDF) (01/27/09)
- Kramer & Majda, "Fundamentals of Random Fields and Stochastic Processes" (PDF) (01/27/09)
- Minier & Peirano, "The PDF Approach to Turbulent Polydispersed Two-Phase Flows,"
Physics Reports352(2001), Sec. 2 (PDF) (01/27/09)- Kramer & Majda, "Wiener Process" (PDF) (01/27/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 2.1-2.5, 2.8 (PDF) (02/06/09)
- Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 1.3 (PDF) (02/13/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 3.4-3.6 (PDF) (02/24/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 4.1(PDF) (02/24/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 4.2-4.3 (PDF) (02/24/09)
- Gardiner, Handbook of Stochastic Methods, Sec. 4.4.4 (PDF) (02/24/09)

Optional Readings

- Simon,
Functional Integration & Quantum Physics, Sec. 5 (PDF)

- Discussion of Feynman path integral and other ways to define Brownian motion in a mathematically rigorous manner. Also some resuls on smoothness of stochastic processes.
- Oksendal,
Stochastic Differential Equations, Ch. 2 (PDF)

- Technical graduate-mathematics discussion of Wiener process
- Stroock, "Gaussian Measure on a Hilbert Space" (PDF)

- Notes from a graduate summer school on probability theory describing a direct definition of the Wiener process through a Gaussian probability measure on the function space of continuous functions.

- Kramer, "Brownian Motion"(PDF) (01/27/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 1.2 (PDF) (02/13/09)
- Gardiner, Handbook of Stochastic Methods, Secs. 3.8.1 & 3.8.2 (PDF) (02/13/09)
- Risken, The Fokker-Planck Equation, Sec. 5.10 (PDF) (04/03/09)

- Einstein, Introduction to the Theory of Brownian Motion (PDF)
- original article
- Risken, The Fokker-Planck Equation, Sec. 1.1 (PDF)
- Brownian motion approached through delta-correlated force (rather than SDE)
- Risken, The Fokker-Planck Equation, Sec. 3.1 (PDF)
- Langevin equation approached through delta-correlated force (rather than SDE)
- Chandrasekhar, "Stochastic Problems in Physics and Astronomy," Reviews of Modern Physics 15 (1943), pp. 1-89 (PDF)
- detailed calculations of solutions of Fokker-Planck equation for Brownian motion without force and with harmonic potential energy
- Kubo, "The Fluctuation-Dissipation Theorem," Reports on Progress in Physics 29 (1966), 255-284 (PDF)
- a technical review on fluctuation-dissipation theory
- Doi and Edwards, The Theory of Polymer Dynamics, Sec. 3.4 (PDF)
- a somewhat more conceptual review of fluctuation-dissipation theory
- Bocquet, "From a Stochastic to a Microscopic Approach to Brownian Motion," Acta Physica Polonica B 29 (1998), 1551-1564 (PDF)
- stochastic mode reduction for Brownian motion through terse and formal procedure
- Deutch & Oppenheim, "The Concept of Brownian Motion in Modern Statistical Mechanics," Faraday Discuss. Chem. Soc. 83 (1987), 1-20 (PDF)
- summary of hierarchy of models of Brownian motion and assumptions behind them
- Kramer and Majda, "Stochastic mode reduction for particle-based simulation methods for complex microfluid systems," SIAM J. Appl. Math. 64 (2), 2003: 401-422 (PDF)
- stochastic mode reduction for system of particles interacting with each other and fluid

- Higham, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations,"SIAM Review 43 (2001), 525-546 (PDF)
- Basic introduction to numerical simulation of stochastic differential equations
- Arnold, "Hasselmann's Program Revisited: The Analysis of Stochasticity in Deterministic Climate Models," Progress in Probability 49 (2001), 141-158 (PDF)
- Overview of method of averaging
- Majda, Timofeyev & Vanden-Eijnden, "Stochastic models for selected
slow variables in large deterministic system,"
*Nonlinearity***19**, 2006: 769--694 (PDF) - Illustration of most recent stochastic mode reduction procedure "seamless MTV" on a model problem
- Franzke, Majda & Vanden-Eijnden, "Low-Order Stochastic Mode Reduction for a Realistic Barotropic Model Climate," Journal of the Atmospheric Sciences 62, 2005: 1722-1745
- Application of stochastic mode reduction to realistic climate model system

- Risken, The Fokker-Planck Equation, Sec. 5.10 (PDF) (04/28/09)
- Risken, The Fokker-Planck Equation, Secs. 11.1-11.3 (PDF) (05/06/09)

- Risken, Fokker-Planck Equation Sec. 5.1 & 5.2 (PDF)
- Astumian & Hänggi, "Brownian motors," Physics Today (November 2002), 33-39 (PDF)
- Reimann & Hänggi, "Introduction to the physics of Brownian motors," Appl. Phys. A 75 (2002), 169-178 (PDF)
- Pavliotis, "A multiscale approach to Brownian motors," Physics Letters A 344 (2005), 331-345 (PDF)
- Hänggi & Marchesoni, "Artificial Brownian motors: Controlling transport on the nanoscale," Rev. Mod. Phys. 81 (2009), 387-442 (PDF)

- Tuckwell, Stochatic Processes in the Neurosciences, Secs. 4.1-4.3 (PDF) (05/06/09)