Content: Stochastics IV: "Stochastic Homogenisation"
First we will recall some basic results about Markov processes and ergodic theory that will be needed in the rest of the course. We will then study the equations on a microscale level with random (periodic) coefficients and characterize the solutions as the scale of the microstructure tends to zero. In the second part of the course we will consider averaging and homogenization of stochastic differential equations. We will first derive the limiting equations and then prove the corresponding convergence theorems. Along the way we will comment on some applications of the stochastic homogenization theory. Moreover, we will mention the numerical methods in stochastic homogenization. In particular, approximations of effective coefficients.
Topics:
- Basics about Markov processes and ergodic theory;
- Homogenization of partial differential equations with random coefficients;
- Averaging for Markov chains;
- Avergaing for SDEs: derivation of the limiting equation and convergence theorem;
- Homogenization for SDEs: derivation of the limiting equation and convergence theorem;
- Applications;
- Numerical methods
Detailed Information can be found on the Homepage of 19242101 Stochastics IV.