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.

Fundamentals of Probability Theory

Assigned Readings

• Kramer & Majda, "Fundamentals of Probability Theory" (PDF) (01/23/07)
• Kramer & Majda, "Conditional Probability and Expectation" (PDF) (01/25/07)
• Kramer & Majda, "Fundamentals of Random Fields and Stochastic Processes" (PDF) (01/30/07)
• Minier & Peirano, "The PDF Approach to Turbulent Polydispersed Two-Phase Flows," Physics Reports 352 (2001), Sec. 2 (PDF) (01/30/07)
• Bertsekas & Tsitsiklis, Introduction to Probability Theory, Sec. 3.5 (PDF) (02/07/07)
• Grigoriu, Stochastic Calculus, Sec. 2.11 (PDF) (02/07/07)
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 1.3 (PDF) (02/07/07)
• Oksendal, Stochastic Differential Equations, Appendix A (PDF) (02/27/07)

Fundamentals of Stochastic Processes

Assigned Readings

• Kramer & Majda, "Wiener Process" (PDF) (02/16/07)
• Grigoriu, Stochastic Calculus, Sec. 2.17 (PDF) (02/16/07)
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 1.7 (PDF) (02/27/07)
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 2.4 (PDF) (02/27/07)

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)
  • A brief encyclopedia article on use of Brownian motion in nonlinear science models

Analytical Solution of Stochastic Differential Equations

Optional Readings

• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Secs. 4.3--4.4 (PDF)
  • Methods of solving some classes of nonlinear stochastic differential equations

Numerical Simulation of Stochastic Differential Equations

Assigned Readings

• Higham, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," SIAM Review 43 (3), 2001: 525-546 (PDF) (04/13/07)

Optional Readings

• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 9.8 (PDF)
  • Stability issues in numerical simulations of stochastic differential equations
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 11.1 (PDF)
  • Strong first order Runge-Kutta-like (derivative free) method

Kolmogorov and Fokker-Planck Equations

Assigned Readings

• Minier & Peirano, "The PDF Approach to Turbulent Polydispersed Two-Phase Flows," Physics Reports 352 (2001), Secs. 2&4 (PDF) (04/17/07)
• Kramer & Majda, "Diffusion Equation"(PDF) (04/20/07)
• Kramer & Majda, "Method of Characteristics" (PDF)(04/20/07)
• Kramer & Majda, "Random Method of Characteristics"(PDF) (04/20/07)
• Kramer, "Fokker-Planck Equation" (PDF) (04/24/07)
• Oksendal, Stochastic Differential Equations, Sections 8 B & C (PDF) (05/08/07)
• Oksendal, Stochastic Differential Equations, Sections 7C & D (PDF) (05/08/07)

Optional Readings

• Risken, The Fokker-Planck Equation, Section 6.6 (PDF)
  • Summary of methods for obtaining solutions or approximate solutions to Fokker-Planck Equation
• Oksendal, Stochastic Differential Equations, Sections 7A & B (PDF)
  • Rigorous explanation of Markov and strong Markov properties of solutions to stochastic differential equations
• Risken, The Fokker-Planck Equation, Section 8.1 (PDF)
  • Further methods for calculating exit time statistics

Stochastic Stability

Optional Readings

• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 6.3 (PDF)
  • Methods for analyzing stability of SDE's; useful background for Problem 2.3 in Homework 4

Stochastic Filtering and Parameter Estimation

Optional Readings

• Oksendal, Stochastic Differential Equations, Chapter 6 (PDF)
  • Derivation of Kalman-Bucy filter for linear stochastic differential equation models
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 6.6 (PDF)
  • Overview of nonlinear filtering
• Kloeden & Platen, Numerical Solution of Stochastic Differential Equations, Sec. 6.4 (PDF)
  • Parameter estimation by Maximum Likelihood method, relation to Girsanov formula
• Pavliotis & Stuart, "Parameter Estimation for Multiscale Diffusions,"Journal of Statistical Physics, 127 (4), 2007: 741-781 (PDF)
  • Parameter estimation in more complex multiscale SDE models

Monte Carlo Methods

Optional Readings

• Mascagni & Simonov, "Monte Carlo Methods for Calculating Some Physical Properties of Large Molecules" SIAM J. Sci. Comp.26(1), 2004: 339-357 (PDF)
  • Application and development of Monte Carlo methods for solving elliptic equations for electrostatic properties associated to biomolecules in fluid

Stochastic Mode Reduction

Optional Readings

• 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