Description
This course discusses random dynamics in terms of Markov processes and martingales using both analytic as well as probabilistic techniques. One particular emphasis is the asymptotic analysis. Applications are an important part of the course.

Pre-requisite
Elementary knowledge of probability and partial differential equations.

Texts

• Lectures on Random Evolution
  by Mark A. Pinsky, World Scientific Pub Co; (June 1991)
• Stochastic Processes
  Kiyosi Ito: Springer, c2004.
• Stochastic Differential Equations
  B. K. Ksendal: Springer-Verlag; 6th edition (2003)

Outline

• Markov chains and processes
• martingales
• limit theorems
• Brownian motion and stochastic integrals
• general random evolutions
• applications
  • kinetic theory of gases
  • transport on manifolds
  • random oscillators
  • random wave processes

Requirement
Students who sign up for the course should consult with me about the project/presentation that he/she would like to do.

List of Papers

• *Ambainis et al: 1-d quantum walks
• Bezuidenhout: A large deviation principle for small perturbations of random evolution equations
• *Chandresekar: Stochastic problems in physics and astronomy
• D. Duffie and P. Glynn, Estimation of continuous Markov processes sampled at random time intervals, Econometrica 72(2004), 1773-1808.
• Ellis and Pinsky: The first and second fluid approximations to the linearized Boltzman equation
  J. Math. Pure Appl. 54 (1975), 125-156.
• Evans: The perturbed test function method for viscosity solutions of nonlinear PDEs,
  Proc. Royal Soc. Edinburgh 111 (1989), 359-375.
• Hersh and Papanicolaou: Non-commuting random evolutions and an operator-valued Feymann-Kac formula
  Comm. Pure Appl. Math 25 (1972), 337-366.
• M. Kimura, Diffusion models in population genetics, Journal of Applied Probability 1 (1964), 177-232.
• Kurtz: A random Trotter product formula,
• *McKean: A simple model for the derivation of fluid mechanics from the Boltzmann equation
  Bull. Amer. Math. Soc. 75 (1969), 1-10.
• C. Neuhauser and S. Pacala, An explicit spatial version of the Lotka-Volterra model with interspecific competition.
• Papanicolaou and Kohler: Asymptotic theory of mixing stochastic ODEs
  Comm. Pure Appl. Math 27(1974): 641-688
• *Papanicolaou: Probabilistic problems and methods in singular perturbations
  Rocky Mountain Journal of Math. 6 (1976), 653-673.
• *Papanicolaou: Asymptotic analysis of stochastic equations
  Studies in Probability Theory, MAA Studies in Mathematics 18 (1978), 111-170.
• Papanicolaou, Stroock and Varadhan: Martingale approach to some limit theorems
  Statistical Mechanics, Dynamical Systems and Turbulence, Duke University Math Series, Vol 3
• Pinsky: On the Navier-Stokes approximation to the linearized Boltzman equation
  J. Math. Pure Appl 55 (1976), 217-231.