Tag - Probability theory

An LMS online lecture course in Markov chains.

We will start the course by presenting various results using the semimartingale approach for Markov chains. These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent. We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.

   1.  Irrational rotations on torus;
   2.  Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
   3.  Furstenberg's ×2,×3 theorem;
   4.  Results and problems on digit expansions of integers;
   5.  Furstenberg's theorem on 2-dimensional torus (if time permits).

Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.