Courant Institute of Mathematical Sciences

New York University

**Lecture 1**. Random walks and Diffusions in a Random environment.
General Introduction and a survey of what is known.

**Lecture 2**. Large deviation aspects of walks and diffusions in a
random environment. The annealed and the quenched versions.

**Lecture 3**. Connection with Homogenization of Hamilton-Jacobi-Bellman
equations with a quadratic Hamiltonian through Hopf-Cole
transformation. More general equations with a convex Hamiltonian.

**Lecture 4**. Details of Homogenization in the time independent case.
Lower bounds and the construction of super-solutions in the elliptic
case for proving upper bounds.

**Lecture 5**. Extension to the time dependent case and the construction
of parabolic super-solutions. Open problems.