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.

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- Kramer & Majda, "Fundamentals of Probability Theory" (PDF) (01/29/13)
- Kramer & Majda, "Fundamentals of Random Fields and Stochastic Processes" (PDF) (01/29/13)
- Minier & Peirano, "The PDF Approach to Turbulent
Polydispersed Two-Phase Flows,"
*Physics Reports***352**(2001), Sec. 2 (PDF) (01/29/13) - Kramer & Majda, "Wiener Process" (PDF) (01/29/13)
- Gardiner, Handbook of Stochastic Methods, Secs. 2.1-2.5, 2.8 (PDF) (01/29/13)
- Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 1.3 (PDF) (02/22/13)
- Gardiner, Handbook of Stochastic Methods, Secs. 3.4-3.6 (PDF)
- Gardiner, Handbook of Stochastic Methods, Secs. 4.1 (PDF) (03/05/13)
- Gardiner, Handbook of Stochastic Methods, Secs. 4.2-4.3 (PDF) (03/08/13)
- Gardiner, Handbook of Stochastic Methods, Sec. 4.4.4 (PDF) (03/08/13)

- 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 results 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.

- Higham, "An Algorithmic Introduction to Numerical
Simulation of Stochastic Differential Equations,"
*SIAM Review***43**(3), 2001: 525-546 (PDF) - Gentle, concise introduction to the issues involved in numerically simulating stochastic differential equations, along with some basic algorithms
- Talay, "Simulation of Stochastic Differential Equations," in Probabilistic Methods in Applied Physics, Kree and Wedig (eds), Lecture Notes in Physics 451 (PDF)
- More advanced but practical discussion of numerical methods for approximately solving stochastic differential equations

- Kramer, "Brownian Motion"(PDF) (01/29/13)
- Gardiner, Handbook of Stochastic Methods, Secs. 1.2 (PDF) (01/29/13)
- Gardiner, Handbook of Stochastic Methods, Secs. 3.8.1 & 3.8.2 (PDF) (02/15/13)

- Nelson, Dynamical Theories of Brownian Motion Secs. 1-4 (PDF)
- Overview of history of understanding of Brownian motion
- 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
- Majda and Kramer, "Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena," Physics Reports 314 (1999), 237-574 (PDF)
- Section 2 discusses transport phenomena, including ballistic to diffusive transitions and anomalous diffusion, in the context of turbulent transport
- Didier, Fricks, et al, "Statistical challenges in microrheology," Journal of Time Series Analysis 33 (2012), 724-743 (PDF)
- Discusses the statistical characterization of transport of particles in complex fluids
- Barkai, Garini, & Metzler, "Strange kinetics of single molecules in living cells," Physics Today (August 2012), 29-35 (PDF)
- Discusses observation and modeling of anomalous diffusion of subcellular molecules
- James, Plank, and Edwards, "Assessing Levy walks as models of animal foraging," Journal of the Royal Society Interface 8, 2011: 1233-1247 (PDF)
- Anomalous diffusion models for animal movement
- 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
- Liu, "Strong Convergence of Principle of Averaging for Multiscale Stochastic Dynamical Systems," Commun. Math. Sci. 8 (2010), 999-1020. (PDF)
- Extension of averaging techniques to slow-fast systems of SDE's when the noise term on the slow variables depends on the fast variables
- Abdulle, E, Engquist & Vanden-Eijnden , "The heterogeneous multiscale method," Acta Numerica 21 (2012), 1-87. (PDF)
- E, Liu, and Vanden-Eijnden, "Analysis of Multiscale Methods for Stochastic Differential Equations," Communications on Pure and Applied Mathematics LVIII (2005), 1544-1585. (PDF)
- Multiscale computation for fast-slow systems of SDE's with two kinds of asymptotic structure
- Xing, Majda, and Grabowski, "New Efficient Sparse Space-Time Algorithms for Superparameterization on Mesoscales," Monthly Weather Review 137 (2009), 4307-4324 (PDF)
- Mitran, "Time-parallel kinetic-molecular interaction algorithm for CPU/GPU Computers," Procedia Computer Science 1 (2010), 745-752 (PDF)
- Young and Mitran, "A numerical model of cellular blebbing: A volume-conserving, fluid-structure interaction model of the entire cell," Journal of Biomechanics 43 (2012), 210-220. (PDF)

- Risken, The Fokker-Planck Equation, Sec. 5.10 (PDF) (05/03/13)
- Risken, The Fokker-Planck Equation, Secs. 11.1-11.3 (PDF) (05/07/13)

- Risken, Fokker-Planck Equation Sec. 5.1 & 5.2 (PDF)
- McKenzie, Lewis, and Merrill, "First Passage Time Analysis of Animal Movement and Insights into the Functional Response," Bulletin of Mathematical Biology 71 (2008), 107-129. (PDF)
- Djurdjevac, Sarich, and Schuette, "Estimating the Eigenvalue Error of Markov State Models," Multiscale Modeling & Simulation 10 (2012), 61-81. (PDF)
- Berglund & Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, App. A5, A6 (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)