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Applied Mathematics & Statistics, and Scientific Computation


Fall semester 2010
Instructor: Mark Freidlin (


  • Wiener process (Brownian motion) and its properties.

  • Stochastic integral and Ito's formula.

  • Stochastic differential equations; existence and uniqueness, Markov property.

  • Langevin's equation and Kramers approximation.

  • Semigroup related to a diffusion process, Generator of the process.

  • Backward and forward Kolmogorov's equations. Invariant density. Gaussian diffusion processes.

  • Long-time behavior of the process; recurrent and transient processes. Convergence to the limiting distribution.

  • Markov times. Exit problems. Elliptic boundary value problems for expected values of various functionals of diffusion processes.

  • Diffusion processes with reflection on the boundary. Other boundary conditions preserving the Markov property.

  • Introductions to asymptotic methods for stochastic differential equations and related PDE's: expansion in a small parameter, averaging, homogenization, large deviations.


B.Oksendal, Stochastic Differential Equations, Springer, Fifth edition.


I plan also to consider some examples and applications.

I assume that the students in the class know a course of probability, calculus. I will try to minimize the use of measure theory and pay the main attention to various methods of calculation of characteristics of stochastic processes.